chaotic synchronization application to living

WORLD SCIENTIFIC

ONLINEAR SCIENC Series Editor: Leon O. Chua

HiET HPPLICflTIONS TO LIVING SYSTEMS

Erik Mosekilde, Yuri Maistrenko & Dmitry Postnov

World Scientific

CHAOTIC SVNCHRONIZRTIOI APPLICATIONS TO LIVING SYSTEHS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE

Editor: Leon O. Chua University of California, Berkeley

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NONLINEAR SCIENCE •* S^WSA vol.** Series Editor; Leon O. Chua

Erik Mosekilde The Technical University of Denmark

Yuri Maistrenko National Academy of Sciences, Ukraine

Dmitry Postnov Saratov state University, Russia

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CHAOTIC SYNCHRONIZATION Applications to Living Systems

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PREFACE

The cooperative behavior of coupled nonlinear oscillators is of interest in connection with a wide variety of different phenomena in physics, engineering, biology, and economics. Networks of coupled nonlinear oscillators have served as models of spatio-temporal pattern formation and simple forms of turbulence. Systems of coupled nonlinear oscillators may be used to explain how different sectors of the economy adjust their individual commodity cycles relative to one another through the exchange of goods and capital units or via aggregate signals in the form of varying interest rates or raw materials prices. Similarly, in the biological sciences it is important to understand how a group of cells or functional units, each displaying complicated nonlinear dynamic phenomena, can interact with each other to produce a coordinated response on a higher organizational level. It is well-known, for instance, that waves of synchronized behavior that propagate across the surface of the heart are essential for the muscle cells to act in unison and produce a regular contraction. Waves of synchronized behavior can also be observed to propagate across the insulin producing beta-cells of the pancreas.

In many cases the individual oscillators display chaotic dynamics. It has long been recognized, for instance, that the ability of the kidneys to compensate for variations in the arterial blood pressure partly rests with controls associated with the individual functional unit (the nephron). The main control is the so-called tubuloglomerular feedback that regulates the incoming blood flow in response to variations in the ionic composition of the fluid leaving the nephron. For rats with normal blood pressure, the individual nephron typically

vi Preface

exhibits regular limit cycle oscillations in the incoming blood flow. For such rats, both in-phase and antiphase synchronization can be observed between adjacent nephrons. For spontaneously hypertensive rats, where the pressure variations for the individual nephron are highly irregular, signs of chaotic phase synchronization are observed.

In the early 1980's, Fujisaka and Yamada showed how two identical chaotic oscillators under variation of the coupling strength can attain a state of complete synchronization in which the motion of the coupled system takes place on an invariant subspace of total phase space. This type of chaotic synchronization has subsequently been studied by a significant number of investigators, and a variety of applications for chaos suppression, for monitoring and control of dynamical systems, and for different communication purposes have been suggested.

Important questions that arise in this connection concern the stability of the synchronized state to noise or to a small parameter mismatch between the interacting oscillators. Other questions relate to the form of the basin of attraction for the synchronized chaotic state and to the bifurcations through which this state loses its stability. Recent studies of these problems have led to the discovery of a large number of new phenomena, including riddled basins of attraction, attractor bubbling, blowout bifurcations, and on-off intermittency.

In addition to various electronic systems, synchronization of interacting chaotic oscillators has been observed for laser systems, for coupled superconducting Josephson junctions, and for interacting electrochemical reactors. For systems of three or more coupled oscillators, one can observe the phenomenon of partial synchronization where some of the oscillators synchronize while others do not. This phenomenon is of interest in connection with the development of new types of communication systems where one mixes a message with a chaotic signal.

Primarily through the works of Rosenblum and Pikovsky it has become clear that even systems that are quite different in nature (or oscillators that have different parameter settings) can exhibit a form of chaotic synchronization where the phases of the interacting oscillators are locked to move in synchrony whereas the amplitudes can develop quite differently. This phenomenon, referred to as chaotic phase synchronization, is of particular importance for living systems where the interacting functional units cannot be assumed to be identical.

Kuramoto and Kaneko have initiated the study of clustering in large en-

Preface vii

sembles of interacting chaotic oscillators with a so-called global (i.e., all-to-all) coupling structure. This type of analyses is relevant for instance to economic sectors that interact via the above mentioned aggregate variations in interest rates and raw materials prices. However, biological systems also display many examples of globally coupled oscillators. The beta-cells in the pancreas, for instance, respond to variations in the blood glucose concentration, variations that at least partly are brought about by changes in the cells' aggregate release of insulin. Important questions that arise in this connection relate to the way in which the clusters are formed and break up as the coupling between the oscillators is varied.

The purpose of the book is to present and analyze some of the many interesting new phenomena that arise in connection with the interaction of two or more chaotic oscillators. Among the subjects that we treat are periodic orbit threshold theory, weak stability of chaotic states, and the formation of riddled basins of attraction. In this connection we discuss local and global riddling, the roles of the absorbing and mixed absorbing areas, attractor bubbling, on-off intermit-tency, and the influence of a small parameter mismatch or of an asymmetry in the coupling structure. We also consider partial synchronization, transitions to chaotic phase synchronization, the role of multistability, coherence resonance, and clustering in ensembles of many noise induced oscillators.

However, our aim is also to illustrate how all of these concepts can be applied to improve our understanding of systems of interacting biological oscillators. In-phase synchronization, for instance, where the nephrons of the kidney simultaneously perform the same regulatory adjustments of the incoming blood flow, is expected to produce fast and strong overall reactions to a change in the external conditions. In the absence of synchronization, on the other hand, the response of the system in the aggregate is likely to be slower and less pronounced. Hence, part of the regulation of the kidney may be associated with transitions between different states of synchronization among the functional units.

Besides synchronization of interacting nephrons, the book also discusses chaotic synchronization and riddled basins of attractions for coupled pancreatic cells, homoclinic transitions to chaotic phase synchronization in coupled microbiological reactors, and clustering in systems of noise excited nerve cells.

To a large extent the book is based on contributions that have been made over the last few years by the Chaos Group at the Technical University of Den-

viii Preface

mark, by the Department of Mathematics, the National Academy of Sciences of Ukraine in Kiev, and by the Department of Physics, Saratov State University. We would like to thank our collaborators and students Brian Lading, Alexander Balanov, Tanya Vadivasova, Natasha Janson, Alexey Pavlov, Jacob Laugesen, Alexey Taborov, Vladimir Astakhov, Morten Dines Andersen, Niclas Carlsson, Christian Haxholdt, Christian Kampmann, and Carsten Knudsen for the many contributions they have made to the present work. Arkady Pikovsky, Jiirgen Kurths, Michael Rosenblum, Vladimir Belykh, Igor Belykh, Sergey Kuznetsov, Vadim Anishchenko, Morten Colding-J0rgensen, Jeppe Sturis, John D. Ster-man, Laura Gardini, and Christian Mira are acknowledged for many helpful suggestions.

We would also like to thank Niels-Henrik Holstein-Rathlau and Kay-Pong Yip who have made their experimented data on coupled nephrons available to us. Most of all, however, we would like to thank Vladimir Maistrenko, Oleksandr Popovych, Sergiy Yanchuk, and Olga Sosnovtseva who have been our closest collaborators in the study of chaotic synchronization. Without the enthusiastic help from these friend and colleagues, the book would never have been possible.

The book has appeared at a time when research in chaotic synchronization is virtually exploding, and new concepts and ideas emerge from week to week. Hence, it is clear that we have not been able to cover all the relevant aspects of the field. We hope that the combination of mathematical theory, model formulation, computer simulations, and experimental results can inspire other researchers in this fascinating area. We have tried to make the book readable to students and young scientists without the highest expertise in chaos theory. On the other hand, the reader is assumed to have a good knowledge about the basic concepts and methods of nonlinear dynamics from previous studies.

The book is dedicated to Lis Mosekilde. In her short scientific career she became the internationally most respected Danish expert in the fields of bone remodelling and osteoporosis.

Lyngby, November 2001 Erik Mosekilde, Yuri Maistrenko and Dmitry Postnov

Contents

PREFACE v

1 COUPLED NONLINEAR OSCILLATORS 1 1.1 The Role of Synchronization 1 1.2 Synchronization Measures 7 1.3 Mode-Locking of Endogenous Economic Cycles 13

2 T R A N S V E R S E STABILITY OF COUPLED M A P S 33 2.1 Riddling, Bubbling, and On-Off Intermittency 33 2.2 Weak Stability of the Synchronized Chaotic State 37 2.3 Formation of Riddled Basins of Attraction 41 2.4 Destabilization of Low-Periodic Orbits 44 2.5 Different Riddling Scenarios 49 2.6 Intermingled Basins of Attraction 54 2.7 Partial Synchronization for Three Coupled Maps 56

3 UNFOLDING THE RIDDLING BIFURCATION 75 3.1 Locally and Globally Riddled Basins of Attraction 75 3.2 Conditions for Soft and Hard Riddling 80 3.3 Example of a Soft Riddling Bifurcation 88 3.4 Example of a Hard Riddling Bifurcation 93 3.5 Destabilization Scenario for a — a,\ 95 3.6 Coupled Intermittency-III Maps 104 3.7 The Contact Bifurcation 109 3.8 Conclusions 116

4 TIME-CONTINUOUS SYSTEMS 123 4.1 Two Coupled Rossler Oscillators 123 4.2 Transverse Destabilization of Low-Periodic Orbits 125

ix

x Contents

4.3 Riddled Basins 130 4.4 Bifurcation Scenarios for Asynchronous Cycles 134 4.5 The Role of a Small Parameter Mismatch 140 4.6 Influence of Asymmetries in the Coupled System 145 4.7 Transverse Stability of the Equilibrium Point 147 4.8 Partial Synchronization of Coupled Oscillators 154 4.9 Clustering in a System of Four Coupled Oscillators 162 4.10 Arrays of Coupled Rossler Oscillators 166

5 COUPLED PANCREATIC CELLS 177 5.1 The Insulin Producing Beta-Cells 177 5.2 The Bursting Cell Model 181 5.3 Bifurcation Diagrams for the Cell Model 185

5.4 Coupled Chaotically Spiking Cells 192 5.5 Locally Riddled Basins of Attraction 196 5.6 Globally Riddled Basins of Attraction 200 5.7 Effects of Cell Inhomogeneities 203

6 CHAOTIC P H A S E SYNCHRONIZATION 211 6.1 Signatures of Phase Synchronization 211

6.2 Bifurcational Analysis 217 6.3 Role of Multistability 222 6.4 Mapping Approach to Multistability 227 6.5 Suppression of the Natural Dynamics 233 6.6 Chaotic Hierarchy in High Dimensions 239 6.7 A Route to High-Order Chaos 249

7 POPULATION D Y N A M I C SYSTEMS 259 7.1 A System of Cascaded Microbiological Reactors 259

7.2 The Microbiological Oscillator 262

7.3 Nonautonomous Single-Pool System 265 7.4 Cascaded Two-Pool System 270 7.5 Homoclinic Synchronization Mechanism 274

7.6 One-Dimensional Array of Population Pools 280

7.7 Conclusions 284

Contents xi

8 CLUSTERING OF GLOBALLY COUPLED M A P S 291 8.1 Ensembles of Coupled Chaotic Oscillators 291 8.2 The Transcritical Riddling Bifurcation 296 8.3 Global Dynamics after a Transcritical Riddling 302 8.4 Riddling and Blowout Scenarios 307 8.5 Influence of a Parameter Mismatch 313 8.6 Stability of tf-Cluster States 318 8.7 Desynchronization of the Coherent Chaotic State 321 8.8 Formation of Nearly Symmetric Clusters 326 8.9 Transverse Stability of Chaotic Clusters 329 8.10 Strongly Asymmetric Two-Cluster Dynamics 334

9 INTERACTING N E P H R O N S 349 9.1 Kidney Pressure and Flow Regulation 349 9.2 Single-Nephron Model 354 9.3 Bifurcation Structure of the Single-Nephron Model 359 9.4 Coupled Nephrons 365 9.5 Experimental Results 370 9.6 Phase Multistability 375 9.7 Transition to Synchronous Chaotic Behavior 382

10 COHERENCE R E S O N A N C E OSCILLATORS 395 10.1 But What about the Noise? 395 10.2 Coherence Resonance 400 10.3 Mutual Synchronization 404 10.4 Forced Synchronization 408 10.5 Clustering of Noise-Induced Oscillations 412

INDEX 425

Chapter 1

COUPLED NONLINEAR OSCILLATORS

1.1 The Role of Synchronization

Synchronization occurs when oscillatory (or repetitive) systems via some kind of interaction adjust their behaviors relative to one another so as to attain a state where they work in unison. An essential aspect of many of the games we play as children is to teach us to coordinate our motions. We skip and learn to jump in synchrony with the swinging rope. We run along the beach and learn to avoid the waves that role ashore, and we take dancing lessons to learn to move in step with the music.

One of the main problems in the swimming class is to learn to breath in synchrony with the strokes. Not necessarily one-to-one, as there are circumstances where it is advantageous to take two (or more) strokes per inhalation. However, the phase relations must be correct if not to drown. In much the same way, a horse has different forms of motion (such as walk, trot, and gallup), and each of these gaits corresponds to a particular rhythm in the movement of its legs [1,2]. At the trotting course, the jockey tries to keep the horse in trot to the highest possible speed. In its free motion, however, a horse is likely to choose the mode that is most comfortable to it (and, perhaps, least energy demanding). As the speed increases the horse will make transitions from walk to trot and from trot to gallup.

Synchronization is a universal phenomenon in nonlinear systems [3]. Well-known examples are the synchronization of two (pendulum) clocks hanging on a

l

2 Chaotic Synchronization: Applications to Living Systems

wall, and the synchronization of the moon's rotation with its orbital motion so that the moon always turns the same side towards the earth. A radio receiver functions by synchronizing its internal oscillator with the period of the radio wave so that the difference, i.e. the transmitted signal, can be detected and converted into sound. A microwave emitting diode is placed in a cavity of a specific form and size to make it synchronize with a particular resonance frequency of the cavity. In a previous book [4] we presented results on synchronization of coupled thermostatically controlled radiators and coupled household refrigerators. Synchronization can also be observed between coupled laser systems and coupled biochemical reactors, and it is clear that one can find thousands of other examples in engineering and physics. At the assembly line one has to ensure an effective synchronization of the various processes for the production to proceed in an efficient manner, and engineers and scientists over and over again exploit the technique of modulating (or chopping) a test signal in order to benefit from the increased sensitivity of phase detection.

The history of synchronization dates back at least to Huygens' observations some 300 years ago [5], and both the history and the basic theory are recapitulated in a significant number of books and articles [6, 7]. For regular (e.g., limit cycle) oscillators, synchronization implies that the periodicities of the interacting systems precisely coincide and that differences in phase remain constant. In the presence of noise (or for chaotic systems) one can weaken the requirements such that the periodicities only have to coincide on average, and the phase differences are allowed to move within certain bounds. One may also accept occasional phase slips, provided that they do not occur too often [8].

One-to-one synchronization is only a simple manifestation of a much more general phenomenon, also known as entrainment, mode locking, or frequency locking. In nonlinear systems, a periodic motion is usually accompanied by a series of harmonics at frequencies of p times the fundamental frequency, where p is an integer. When two nonlinear oscillators interact, mode locking may occur whenever a harmonic frequency of one mode is close to a harmonic of the other. As a result, nonlinear oscillators tend to lock to one another so that one subsystem completes precisely p cycles each time the other subsystem completes q cycles, with p and q as integers [9, 10]. An early experience with this type of phenomenon is the way one excites a swing by forcing it at twice its characteristic frequency, i.e., we move the body through two cycles of a bending and stretching mode for each cycle of the swing. A similar phenomenon is

Coupled Nonlinear Oscillators 3

utilized (in optics, electronics, etc.) in a wide range of so-called parametric devices.

Contrary to the conventional assumption of homeostasis, many physiological systems are unstable and operate in a pulsatile or oscillatory mode [10, 11]. This s the case, for instance, for the production of luteinizing hormone and insulin hat are typically released in two-hour intervals [4]. In several cases it has been >bserved that the cellular response to a pulsatile hormonal signal is stronger han the response to a constant signal of the same magnitude, suggesting that he oscillatory dynamics plays a role in the control of the system [12]. Hormonal elease processes may also become synchronized, and it has been reported, or instance, that the so-called hot flashes that complicate the lives of many vomen during menopause are related to the synchronized release of 5-7 different lormones [13, 14].

The beating of the heart, the respiratory cycle, the circadian rhythm, and the avarian cycle are all examples of more or less regular self-sustained oscillations. The ventilatory signal is clearly visible in spectral analyses of the beat-to-beat variability of the heart signal, and in particular circumstances the two oscillators may lock together so that, for instance, the heart beats three or four times for each respiratory cycle [15]. The jet lag that we experience after a flight to a different time zone is related to the synchronization of our internal (circadian) rhythm to the local day-and-night cycles, and it is often said that women can synchronize their menstrual cycles via specific scents (pheromones) if they live close together.

Rhythmic and pulsatile signals are also encountered in intercellular communication [16]. Besides neurons and muscle cells that communicate by trains of electric pulses, examples include the generation of cyclic AMP pulses in slime mold cultures of Dictostelium discoideum [17] and the newly discovered synchronization of the metabolic processes in suspensions of yeast cells [18]. Synchronization of the activity of the muscle cells in the heart is necessary for the cells to act in unison and produce a regular contraction. Similarly, groups of nerve cells must synchronize to produce the characteristic rhythms of the brain or to act as pacemakers for the glands of the hormonal systems [19]. On the other hand, it is well-known that synchronization of the electrical activity of larger groups of cells in the brain plays as essential role in the development of epileptic seizures [20].

However, nonlinear oscillators may also display more complicated forms of


dynamics, and an interesting question that arises over and over again in the biological sciences concerns the collective behavior of a group of cells or functional units that each display strongly nonlinear phenomena [21].

The human kidney, for instance, contains of the order of one million functional units, the nephrons. In order to protect its function, the individual nephron disposes of a negative feedback regulation by which it can control the incoming blood flow. However, because of the delay associated with the flow of fluid through the nephron, this regulation tends to be unstable and produce self-sustained oscillations in the various pressures and concentrations [22]. If the arterial blood pressure is high enough, the pressure oscillations in the nephron may become irregular and chaotic [23]. Neighboring nephrons interact with one another through signals that propagate along the afferent arterioles (incoming blood vessels) and, as experiments show, this interaction can lead to a synchronization of the regular pressure oscillations for adjacent nephrons [24].

Fig 1.1. Pressure variations in two neighboring nephrons for a hypertensive rat. Note that there is a certain degree of synchronization between the irregular (chaotic) signals. This synchronization is found to arise from interactions between the nephrons and not from common external influences.

'"*" 0 500 1000 Time (sec)

It is obviously of interest to examine to what extent similar synchronization phenomena are manifest in the irregular oscillations at higher blood pressures. Figure 1.1 shows an example of the chaotic pressure variations that one can observe (in the proximal tubule) for neighboring nephrons in a hypertensive rat. Although the two signals are strongly irregular, one is tempted to admit that there is a certain degree of synchronization: The most pronounced maxima and minima in the pressure variations occur almost simultaneously. Figure 1.2


shows a scanning electron microscope picture of the interaction structure for a couple of nephrons. Here, one can see how the common interlobular artery (IA) branches into separate afferent arterioles (af) for the two nephrons. The two ball-formed bundles are the capillary systems (the glomeruli) of the two nephrons. Here, blood constituents like water and salts are filtered into the tubular system of the nephrons and the remaining blood passes out through the efferent arterioles (e / ) . See Fig. 9.1 for a more detailed sketch of the structure of the nephron.

We would like to understand how the interaction between the nephrons influence the overall functioning of the kidney. Will there be circumstances, for instance, where the coupling produces a global synchronization of all the nephrons or will we see the formation of clusters of nephrons in different synchronization states? Will transitions between different states of synchronization play a role in the regulation of the kidney or will such transitions be related to the development of particular diseases?

Similarly, each of the insulin producing /3-cells of the pancreas exhibits a complicated pattern of oscillations and bursts in its membrane potential [25]. Presumably through their relation to the exchange of calcium between the cell and its surroundings, these bursts control the release of insulin. The /3-cells are arranged in a spiral structure along capillaries and small veins. Via insulin receptors in the cell membrane, each cell can thus react to the release of insulin from cells that are upstream to it. At the same time, the /3-cells are coupled

Fig 1.2. Scanning electron microscope picture of the arteriolar system for a couple of adjacent nephrons. The nephrons are assumed to interact with one another via muscular contractions that propagate along the afferent arterioles (af).


via gap junctions through which ions and small molecules can pass from cell to cell. Again it is of interest to understand how the collective behavior of a group of cells is related to the dynamics of individual cells. Experiments indicate that there will be waves of synchronization moving across larger groups of cells in an islet of Langerhans [26].

In the economic realm, each individual production sector with its characteristic capital life time and inventory coverage parameters tends to exhibit an oscillatory response to changes in the external conditions [27]. Overreac-tion, time delays, and reinforcing positive feedback mechanisms may cause the behavior to become destabilized and lead to complicated nonlinear dynamic phenomena. The sectors interact via the exchange of goods and services and via the competition for labor and other resources. A basic problem for the establishment of a dynamic macroeconomic theory is therefore to describe how the various interactions lead to a more or less complete entrainment of the sectors [28].

The problems associated with chaotic synchronization have also attracted a considerable interest in the fields of electronics and radio engineering. Here, the attention centers around the possibilities of developing new types of communication systems that exploit the particular properties of deterministic chaos [29, 30, 31]. Important questions that arise in this connection pertain to the sensitivity of the synchronized state to noise or to a parameter mismatch between the interacting oscillators. Other questions relate to the behavior of the coupled system, once the synchronization breaks down, and to the initial conditions for which entrainment can be attained. It is a problem of considerable interest, whether or not one can mask a message by mixing it with a chaotic signal [32].

In order to discuss some of the problems that arise in connection with chaotic synchronization we shall apply a variety of different simple mathematical models. We start in Chapters 2 and 3 by considering a system of two (or three) coupled logistic maps. This leads us to a discussion of the conditions for synchronization in systems of coupled Rossler oscillators (Chapter 4) and in a system of two (nearly) identical /3-cells (Chapter 5). In this connection we show that a /3-cell has regions of chaotic dynamics between the different states of periodic bursting. Towards the end of the book the analysis will lead us to consider clustering in systems of many coupled chaotic oscillators (Chapter 8) and to examine interacting coherence resonance (i.e., stochastically excited) os-


dilators (Chapter 10). On the way we shall discuss the characteristics of chaotic phase synchronization (Chapter 6) and use the obtained results to examine experimental data for the tubular pressure variations in neighboring nephrons (Chapter 9). Let us start, however, by discussing some of the characteristic signatures of synchronization in regular and chaotic systems. Thereafter, we shall use a model of two interacting capital producing sectors of the economy [28] to recall some of the basic concepts of the classical synchronization theory [33] and to illustrate the role of synchronization in macroeconomic systems.

1.2 Synchronization Measures

Let us consider some of the phenomena that one can observe in connection with chaotic phase synchronization [34, 35]. This is the type of synchronization that we expect to find between two coupled chaotic oscillators with different parameters such as, for instance, between neighboring nephrons in a hypertensive rat. The idea is to focus on the similarities between chaotic phase synchronization and the synchronization phenomena we know for regular oscillators. Among of the questions we would like to discuss are: What are the signatures of chaotic phase synchronization? Can we use similar diagnostic tools as we use for regular oscillators? What are the main bifurcation scenarios? First, however, we should perhaps recall some of the characteristics of the synchronization mechanism for regular oscillators [4, 9, 10].

From a mathematical point of view we understand the synchronization of two periodic oscillators as a transition from quasiperiodic motion to regular periodic behavior for the system as a whole. The quasiperiodic behavior is usually described as the motion on a torus. This motion is characterized by the presence of two incommensurate periods, asscociated with the motions of the individual oscillators. As coupling between the oscillators is introduced, both oscillators adjust their motions in response to the motion of the other, and when the coupling becomes strong enough a transition typically occurs where the two periods start to coincide. In the absence of coupling, the phase of each oscillator is a neutrally stable variable. There are no mechanisms that act to correct for a shift in phase. The amplitude, on the other hand, is controlled by a balance between instability and nonlinearity, and dissipation leads to a rapid decay of any pertubation of the amplitude. Hence, we conclude that mutual phase adjustments will be more significant than amplitude modulation [3]


At least for relatively small coupling strengths, the synchronization takes place via a saddle-node bifurcation [33]. On the surface of the torus a stable (node) and an unstable (saddle) cycle simultaneously emerge. Under variation of a control parameter (for instance, a parameter that controls the uncoupled period of one of the oscillators), the two cycles move away from one another along the torus surface to meet again and become annihilated on the opposite side. As a result, the synchronized state exists in a finite range of the control parameter.

The typical situations where synchronization occurs are mutually coupled oscillators and periodically forced oscillators. Glass and Mackey [10] have discussed, for instance, how different forms of synchronization can be observed for periodically forced chicken heart cells. Sturis et al. [36] have described how the release of insulin from the pancreas in normal subjects can be synchronized to an external variation in the supply of glucose, and Bindschadler and Sneyd [37] have described how oscillations in the intracellular concentrations of calcium in biological cells can be synchronized via the exchange of ions through the gap junctions. Under such conditions, the relevant control parameters are the frequency mismatch Aw for the uncoupled oscillators and the coupling strength (or forcing amplitude) K.

The phenomenon of synchronization can be described from different perspectives. If we look at the Fourier spectra of the oscillations, synchronization can be seen as a characteristic evolution of the amplitudes and frequencies of the spectral components. On the other hand, in terms of a phase space analysis, the synchronization mechanisms are the possible ways of transition from an ergodic (or nonresonant) two-dimensional torus (which, as mentioned above, is the image of quasiperiodic behavior) to a limit cycle, being the image of periodic oscillations.

Figure 1.3 shows the typical structure of the 1:1 synchronization regime [38, 39]. To be specific, let us talk about the case of forced synchronization. The representative ways on the (Aw, .^-parameter plane are denoted with arrows:

Route A: At weak coupling and a relatively small parameter mismatch the onset of resonance on the two-torus corresponds to the crossing of one of the saddle-node bifurcation lines SN. At this point, a pair of limit cycles (a stable and a saddle cycle) emerge on the torus surface. In terms of the Fourier spectra, this transition can be diagnosed from the approach and final merging


of the spectral peaks corresponding to the forcing signal and the self-sustained oscillations.

Route B: In the case of a considerable frequency mismatch, increasing the forcing amplitude leads to the gradual suppression of the self-sustained oscillations of the forced system. The two-torus decreases in size and collapses into a limit cycle. Hence, the synchronized state arises at the curve of torus bifurcation. In the Fourier spectra, the spectral component of the self-sustained oscillations decreases in amplitude and disappears when the bifurcation line T is reached.

B

Fig 1.3. Typical structure of 1:1 synchronization region. SN is a curve of saddle-Eode bifurcation for a pair of stable and saddle limit cycles and SSN is a curve of saddle-node bifurcation for a pair of saddle and unstable limit cycles. BT is the so-called Bogdanov-Takens point and C denotes another codimension-2 bifurcation point. T is the torus bifurcation line, and H denotes a line of homoclinic bifurcation.

Route C: In the resonant parameter area, increasing the forcing amplitude does not lead to a qualitative change of the stable limit cycle. However, at the curve of saddle-saddle-node bifurcation SSN the saddle limit cycle and an unstable limit cycle (from the inside of the torus) are annihilated. Thus, the invariant torus surface (which is defined by the unstable manifolds of the saddle cycle) no longer exists above this line. This transition cannot be diagnosed by means of Fourier spectra for the synchronous oscillations.

Route D: In some (usually narrow) parameter region a specific kind of transition can be detected in which a homoclinic bifurcation plays the key role.

"*/\V.;- f -i v ! :

Aco


Here, one observes a region of bistability, where the stable synchronous solution and the stable nonresonant torus coexist. Under variation of the control parameters, this bistability manifests itself in terms of a hysteretic behavior.

Most of the published work on chaotic phase synchronization refers to the case where the chaotic dynamics has appeared through a cascade of period-doubling bifurcations. This type of chaotic dynamics has a characteristic structure which manifests itself both in the rotation of the trajectory around some center point (Fig. 1.4(a)) and in the presence of a characteristic time scale which can be easily measured from the power spectrum (see Fig. 1.4(b)). These features of period-doubling chaos are important in relation to the problem of phase synchronization because they make it possible to introduce a simple measure of the instantaneous phase of the chaotic oscillations and to consider the mean return time to some Poincare secant as representative of the internal rhythm of the dynamics.

(b)

0 0.0 0.5 1.0 1.5 2.0 (0

Fig 1.4. The chaotic attractor of the Rossler model demonstrates the typical features of period doubling chaos. The phase trajectory rotates around some center in the phase space projection (a), and one can see the corresponding peak of the fundamental frequency in the power spectrum (b).

A relatively small number of model dynamical systems have served an important role in the investigation of period-doubling chaos. The same is true for the problem of chaotic phase synchronization. Here, we can mention the Rossler model [40, 41], the Chua circuit [42, 43], and the Anishchenko-Astakhov generator [35, 44]. All of these models are three-dimensional oscillators demonstrating the period-doubling route to chaos, with the high-periodic and chaotic


regimes located around a single unstable equilibrium point of saddle-focus type. During the period-doubling cascade, no additional complexity occurs, and these models thus describe a generic case.

By way of example let us base our discussion on the Rossler model [45]:

x = —u>y — z,

y = UJX + ay,

z = (3 + z(x-n), (1.1)

where a, (3, and \x are control parameters, and UJ defines the characteristic frequency of the oscillations. The chaotic dynamics of this model is well-studied. There are two possible types of chaotic attractor. With \i increasing, a cascade of period-doubling bifurcations leads to the emergence of chaos. This is called "spiral chaos". This type of chaos is illustrated in Fig. 1.4(a). If /x increases further, the more complicated chaotic motion referred to as "screw chaos" can be observed [44].

To study the synchronization mechanisms we rewrite the model (1.1) as a nonautonomous system:

x = —ujy — z + Ksinujft,

y = UJX + ay,

z = p + z{x-fi), (1.2)

where K is the amplitude of a external harmonic drive, and the forcing frequency ojj is fixed at 1.0.

As illustrated in Fig. 1.4 (b) there is usually a well developed peak in the power spectrum of period-doubling chaos at a frequency close to the frequency of the period-one limit cycle from which the chaotic dynamics has originated. Following the changes in the peak position, one can find an interval for the control parameter UJ where the peak frequency UJQ is in a rational relation with the frequency of the forcing signal ujf. This method works well both for numerical simulations and for full scale experiments. The first observation of frequency locking for chaotic oscillations was made with a similar approach [34, 35].

Figure 1.5 illustrates the locking of the fundamental frequency of the chaotic oscillations for the nonautonomous Rossler model. For UJ G [0.922; 0.929], a 1:1 locking region is observed. Chaotic oscillations inside and outside this re-


Fig 1.5. Frequency locking region for the chaotic oscillations in the nonautonomous Rossler model with a = 0.2, /? = 0.2, // = 4.0, and K = 0.02.

""'6.920 0.925 0.930

CO

gion can be classified as synchronous and asynchronous, respectively. This is confirmed by Fig. 1.6 where the Poincare sections for both cases are plotted.

It is interesting to note how Fig. 1.5 reproduces the variation that one observes in connection with the synchroniztion of a regular oscillator, forced by a periodic signal. In the interval of chaotic phase synchronization (0.922 < u> < 0.929), the characteristic frequency of the chaotic oscillations coincides precisely with the forcing frequency. This corresponds to the 1:1 step of the well-know devil's staircase for regular oscillations [4, 9, 10]. On both sides of the synchronization interval, the characteristic frequency of the chaotic oscillations shifts away from the forcing frequency. This region of asynchronous chaos replaces the quasiperiodic region for regular oscillators.

Inspection of Fig. 1.6(b) shows how the structure of the Poincare section for the asynchronous chaotic dynamics reproduces the characteristic structure of the phase space projection for the autonomous Rossler oscillator (see Fig. 1.4). This implies that our stroboscopic measurements (the Poincare section for a forced system) catch the Rossler oscillator more or less at random in all different positions along its trajectory. In the synchronized chaotic case (Fig. 1.6(a)), on the other hand, although there is a certain scatter in the position of the Rossler system in the stroboscopic map, the system is always found in a relatively narrow region of phase space. Whenever the forcing signal has completed a full period, the synchronized Rossler oscillator is back to nearly the same position in phase space.

In the following chapters (particularly in Chapter 6) we shall return to the problem of chaotic phase synchronization to discuss additional signatures of synchronization (e.g., the variation of the Lyapunov exponents) and to examine


10.0 | (a)

5.0

-5.0 •

-10.0

-10.0 -5.0 0.0 5.0 10.0 -10.0 -5.0 0.0 5.0 10.0

X X

Fig 1.6. Phase projections of the Poincare section in the nonautonomous Rossler model for synchronous (a) and asynchronous (b) chaos at fj, = 4.0, a = 0.2, /? = 0.2, and K = 0.02.

the bifurcations involved in the transition to chaotic phase synchronization. We shall also show that chaotic phase sychronization can develop along the same routes as we have illustrated for regular oscillations in Fig. 1.3. However, let us complete the present chapter with a discussion of the role of synchronization in economic systems. This is an area of research which appears so far to have attracted far too little attention.

1.3 Mode-Locking of Endogenous Economic Cycles

Macroeconomic models normally aggregate the individual firms of the economy into sectors with similar products, manufacturing processes and decision rules. Sometimes, only a single sector is considered (e.g., [46, 47]). This simplification is justified on pragmatic grounds by noting that it is impractical to portray separately all the firms in an industry or all the products on the market, and by arguing that the phenomena of interest are captured in sufficient detail by the aggregate formulation [48, 49]. Nevertheless, there are instances where aggregation is not justified. Economic sectors associated with the new information and communication technologies, for example, may show rapid growth while other, more traditional sectors show little increase or even decline. Here, disaggregation is required to provide a proper perspective of future developments. The oscillatory patterns that one can observe in many economic variables also display widely different periodicities. Commodity cycles in pork prices and slaughter rates typically exhibit a period of 3-4 years while similar cycles for

-10.0


chicken and cattle have periodicities of about 30 months and 15 years, respectively [50]. These cycles are related to the feeding periods for the various animals. In other sectors, such as the auto industry or the tanker market, one can find periodicities that relate to the lifetime of the capital [4, 51].

Obviously, the various sectors interact with one another. However, the economic modeling literature is weak in providing guidelines for appropriate aggregation of dynamic systems, particularly when there are significant interactions between the individual entities. Models of capital investment, for instance, typically represent the average lead time and lifetime of the plant and equipment used by each firm. In reality there are many types of plant and equipment acquired from many vendors operating with a wide range of lead times. In response to changes in its external conditions, each firm will generate cyclic behaviors whose frequency, damping, and other properties are determined by the parameters characterizing the particular mix of lead times and lifetimes the firm faces. Because the individual firms are coupled to one another via the input-output structure of the economy, each acts as a source of perturbations on the others.

How do the different lifetimes and lead times of plant and equipment affect the frequency, phase, amplitude, and coherence of economic cycles? How valid is aggregation of individual firms into single sectors for the purpose of studying macroeconomic fluctuations? Here, the issue of coherence or synchronization becomes important. The economy as a whole experiences aggregate business fluctuations of various frequencies from the short-term business cycle to the long-term Kondratieff cycle [4, 27]. Yet why should the oscillations of the individual firms move in phase so as to produce an aggregate cycle? Given the distribution of parameters among individual firms, why do we observe only a few distinct cycles rather than cycles at all frequencies - cycles which might cancel out at the aggregate level?

A common approach to the question of synchronization in economics is to assume that fluctuations in aggregates such as gross domestic product or unemployment arise from external shocks, for example sudden changes in resource supply conditions or variations in fiscal or monetary policy [52]. Forrester [53] suggested instead that synchronization could arise from the endogenous interaction of multiple nonlinear oscillators, i.e., that the cycles generated by individual firms become reinforced and entrained with one another. Forrester also proposed that such entrainment could account for the uniqueness of the eco-


nomic cycles. Oscillatory tendencies of similar periodicity in different parts of the economy would be drawn together to form a subset of distinct modes, such as business cycles, construction cycles, and long waves, and each of these modes would be separated from the next by a wide enough margin to avoid synchronization. Until recently, however, these suggestions have not been subjected to rigorous analysis.

Roughly speaking, synchronization occurs because the nonlinear structure of the interacting parts of a system creates forces that " nudge" the parts of the system into phase with one another. As described by Huygens [5], two mechanical clocks, hanging on the same wall, are sometimes observed to synchronize their pendulum movements. Each clock has an escapement mechanisms, a highly nonlinear mechanical devise, that transfers power from the weights to the rod of the pendulum. When a pendulum is close to the position where the escapement releases, a small disturbance, such as the faint click from the release of the adjacent clock's escapement, may be enough to trigger the release. Hence, the weak coupling of the clocks, through vibrations in the wall, can bring individual oscillations into phase, provided that the two uncoupled frequences are not too different (see [3] for a more complete discussion of Huygens results).

We have previously described how mode-locking and other nonlinear dynamic phenomena arise in a simple model of the economic long wave [4, 27, 54]. As described by Sterman [55], the model explains the long wave as a self-sustained oscillation arising from instabilities in the ordering and production of capital. An increase in the demand for capital leads to further increases through the investment accelerator or "capital self-ordering", because the aggregate capital-producing sector depends on its own output to build up its stock of productive capital. Once a capital expansion gets under way, self-reinforcing processes sustain it beyond its long-term equilibrium, until production catches up with orders. At this point, however, the economy has acquired considerable excess capital, forcing capital production to remain below the level needed for replacement until the excess has been fully depreciated, and room for a new expansion has been created.

The concern of the present discussion is the model's aggregation of capital into a single type. The real economy consists of many sectors employing different kinds of capital in different amounts. Parameters, such as the average productive life of capital and the relative amounts of different capital components employed, may vary from sector to sector. In isolation, the buildings


industry may show a temporal variation significantly different from that of, for instance, the machinery industry.

An early study by Kampmann [56] took a first step in this direction by disaggregating the simple long-wave model into a system of several capital producing sectors with different characteristics. Kampmann showed that the multi-sector system could produce a range of different behaviors, at times quite different from the original one-sector model. The present analysis provides a more formal approach, using a two-sector model. One sector can be construed as producing buildings and infrastructure with very long lifetimes, while the other could represent the production of machines, transportation equipment, etc., with much shorter lifetimes. In isolation, each sector produces a self-sustained oscillation with a period and amplitude determined by the sector's parameter values. However, when the two sectors are coupled together through their dependence on each other's output, they tend to synchronize with a rational ratio between the two periods of oscillation.

The extended long wave model [28] describes the flows of capital plant and equipment in two capital producing sectors. Each sector uses capital from itself and from the other sector as the only factors of production. Each sector receives orders for capital, from itself, from the other sector, and from the consumer goods sector. Production is made to order (no inventories are kept), and orders reside in a backlog until capital is produced and delivered.

Each sector i —1,2 maintains a stock Kij of each capital type j = 1,2. The capital stock is increased by deliveries of new capital and reduced by physical depreciation. The stock of capital type j depreciates exponentially with and average lifetime of Tj. The difference in lifetime between the two sectors A T will be used as a bifurcation parameter to explore the robustness of the aggregated model.

Output is distributed "fairly" between customers, i.e., the delivery of capital type j to sector i is the share of total output Xj from sector j , distributed according to how much sector i has on order with sector j , relative to sector j ' s total order backlog Bj. Hence,

Kit = x& - ^ (1.3) Bj Tj

and


S--Sij = 0{j -Xj-j*-, (1.4)

where a dot denotes time derivative. Sij represents the orders that sector i has placed with sector j but not yet recieved, and Oij represents the rate of sector i's new orders for capital from sector j . Each sector receives orders from itself On, from the other capital sector Oji, and from the consumer goods sector, y,. It accumulates these orders in a backlog S;, which it then depleted by the sector's deliveries of capital X{. Hence,

Bi = {on + Oji - y{) - Xi, j ^ i. (1.5)

Production capacity in each sector is determined by a constant-returns-to scale Cobb-Douglas function of the individual stocks of the two capital types, with a factor share a £ [0,1] of the other sector's capital type and a share 1 — a of the sector's own capital type, i.e.,

* = K?K}rKfj, j + i. (1.6)

where the capital-output ratio K, is a constant. The parameter a determines the degree of coupling between the two sectors. In the simulation studies a is varied between 0, indicating no interdependence between the sectors, and 1, indicating the strongest possible coupling where each sector is completely dependent on capital from the other sector. A characteristic aspect of the Cobb-Douglas function is that it allows substitution between the two production factors, i.e., the same production capacity q can be achieved with different combinations of Ku and Kij. In this perspective, a is referred to as the elasticity of substitution.

The output Xi from sector i depends on the sector's production capacity Q, compared to the sector's desired output x*. If desired output is much lower than capacity, production is cut back, ultimately to zero if no output is desired. Conversely, if desired output exceeds capacity, output can be increased beyond capacity, up to a certain limit. In our model, the sector's output is formulated as

xi = f(^jci (1.7)


where the capacity-utilization function /(•) has the form

/(r)=7(l-(l^y), 7>1. (1.8)

With this formulation /(0) = 0, / ( l ) = 1, and lim^oo / ( r ) = 7. Thus the parameter 7 determines the maximum production possible. In the present analysis we take 7 = 1.1. Note that / ( r ) > r, r G [0,1], implying that firms are reluctant to cut back their output when capacity exceeds demand. Instead, they deplete their backlogs and reduce their delivery delays.

Sector i's desired orders o*j for new capital of type j consist of three components. First, all other things being equal, firms will order to replace depreciation of their existing capital stock, K^/TJ. Second, if their current capital stock is below (above) its desired level &*• firms will order more (less) capital in order to correct the discrepancy over time. Third, firms consider the current supply line Sij of capital and compare it to its desired level s*-. If the supply line is below (above) that desired, firms order more (less) in order to increase (decrease) the supply line over time. In total, our expression for the desired ordering rate becomes

dl. = L+KiZ^L + i i (1.9)

where the parameters rf and rf are the characteristic adjustment times for the capital stock and the supply line, respectively. This decision rule is supported by extensive empirical [57] and experimental [58, 59] work. The rule is based on a so-called anchoring-and-adjustment approach that is believed to capture the bounded rationality of real decision makers.

Actual orders are constrained to be non-zero (cancellation of orders is not considered) and the fractional rate of expansion of the capital stock is also assumed to be limited by bottlenecks related to labor, market development, and other factors not represented in the production function. These constraints are accounted for through the expression


where orders for new capital are expressed as a factor g(-) times the rate of capital depreciation. For the function g(-) we have assumed the form

9(u) = r-TT r ^ r (1-11)

where the parameters have the following values /3 = 6, /ii = 27/7, \ii — 8/7, v\ = 2/3, and 1/2 = 3. These parameters are specified so that #(1) = 1, g'(l) = 1, and g"(l) = 0. Furthermore limu^00g(u) = f3 and l im^-oo g(u) = 0. Note that g(u) has a neutral interval around the equilibrium point (u = 1) where actual orders equal desired orders.

The desired capital stock fc*- is proportional to the desired production rate x* with a constant capital-output ratio. Thus, it is implicitly assumed that the relative prices of the two types of capital are constant, so there is no variation in desired factor proportions. Hence,

tyj ~ Kijxi (.I'l^,)

where Ky is the capital-output ratio of capital type j in sector i. In calculating the desired supply line s*-, firms are assumed to account for

the delivery delay for each type of capital. The target supply line is taken to be the level at which the deliveries of capital, given the current delivery delay, would equal the current depreciation of the capital stock. The current delivery delay of capital from a sector is the sector's backlog divided by its output. Thus,

4 = ^ , (1-13)

Finally, orders from the consumer goods sector j/j are assumed to be exogenous, constant, and equal for both sectors. The last assumption is not without consequence, since the relative size of the demands for the two types of capital can change the dynamics of the model considerably [56].

The capital-output ratios and average capital lifetimes are formulated in such a way that the aggregate equilibrium values of these parameters for the model economy as a whole remain constant and equal to the values in Sterman's original model. Specifically, the average capital lifetimes in the two sectors are


AT , AT

TI = T + — a n d T2 = T - — (1.14)

and the capital-output ratios are

T' T-

KU = (l-a)K-, Kij^an-1, i ^ j , and nt = K}raKfj. (1.15)

Fig 1.7. Simulation of the one-sector model. The steady-state behavior is a limit cycle with a period of approximately 47 years. The plot shows production capacity, production, and desired production of capital equipment, respectively. All variables are shown on the same scale. Maxima are reached in the order: Desired production, actual production, capacity.

The average lifetime of capital r is taken to be 20 years and the average capital-output ratio K = 3 years.

The above formulation assures that capacity equals desired output when both capital stocks equal their desired levels and that the equilibrium aggregate lifetime of capital and equilibrium aggregate capital-output ratio equal the corresponding original parameters in the one-sector model. Furthermore, parameters in the decision rules were scaled to the average lifetime of capital produced by that sector. Thus, when there is no coupling between the sectors (a = 0), one sector is simply a time-scaled version of the other. Hence, the parameters are

r f = r*!*, rf = ^Zl a n d j . = ,jZi (Ll6) T T T

where (as in the original model) TK - 1.5 y, r 5 = 1.5 y, and S = 1.5 y. Figure 1.7 shows a simulation of the limit cycle of the one-sector model

(a = 0, AT = 0). Even with the modifications we have introduced, the behavior

100 150 Time (Years)


of our model is virtually indistinguishable from that of the original model [55]. With the above parameters, the equilibrium point is unstable and the system quickly settles into a limit cycle with a period of approximately 47 years. Each new cycle begins with a period of rapid growth, where desired output exceeds capacity. The capital sector is thereby induced to order more capital, which, by further swelling order books, fuels the upturn in a self-reinforcing process. Eventually, capacity catches up with demand, but at this point it far exceeds the equilibrium level. The self-ordering process is now reversed, as falling orders from the capital sector lead to falling demand, which collapses to the point where only the exogenous goods sector places new orders. A long period of depression follows, during which the excess capital is gradually depleted, until capacity reaches demand. At this point, the capital sector finally raises enough orders to offset its own discards, increasing orders above capacity and initiating the next cycle.

To explore the robustness of the single-sector model to differences in the parameters governing the individual sectors, we now simulate the model when some parameters differ between the two sectors. In spite of its simplicity, the model contains a considerable number of parameters which may differ from sector to sector. In the present analysis, we vary the difference A T in capital lifetimes for different values of the coupling parameter a. As described above, we have scaled all other parameters with the capital-lifetime parameters in such a way that, when a = 0, each sector is simply a time-scaled version of the original one-sector model.

Fig 1.8. Synchronization (1:1 mode-locking) in the coupled two-sector model. The figure shows the capacity in each of the two sectors as a function of time in the steady state. The difference in capital lifetimes AT is 6 years. The coupling parameter a is 0.25. The machinery sector leads the oscillations.



In the simulations that follow, sector 1 is always the sector with the longer lifetime of its capital output, corresponding to such industries as housing and infrastructure, while sector 2 has the shorter lifetime parameter, corresponding to the machine and equipment sector. Introducing a coupling between the sectors will not only link the behaviors together, but also change the stability properties of the individual sectors. A high value of the coupling parameter a implies that the strength of the capital self-ordering loop in any sector is small. In the extreme case a = 1, each sector will not order any capital from itself. If the delivery delay for capital from the other sector is taken as exogenous and constant, the behavior of an individual sector changes to a highly damped oscillation. Indeed, a linear stability analysis around the steady-state equilibrium of an individual sector shows that the equilibrium becomes stable for sufficiently high values of a. As will become evident below, this stability at high values of the coupling parameter has significant effects on the mode-locking behavior of the coupled system.

Fig 1.9. 2:2 mode-locking resulting from a period-doubling bifurcation. As the difference AT is increased to 9 years, the 1:1 mode is replaced by an alternating pattern of smaller and larger swings, so that the total period is doubled. As in the previous figure, a = 0.25.

0 100 200 300 400 500 Time (Years)

As long as the parameters of the two sectors are close enough, we expect synchronization (or 1:1 frequency locking) to occur, i.e., we expect that the different cycles generated by the individual sectors will adjust to one another and exhibit a single aggregate economic long wave with the same period for both sectors. The stronger the coupling a, the stronger the forces of synchronization are expected to be. As an example of such synchronization, Fig. 1.8 shows the outcome of a simulation performed with a difference in capital lifetimes between the two sectors of A T = 6 years and a coupling parameter a = 0.25. The two sectors, although not quite in phase, have identical periods of oscillations. The


larger excursions in production capacity are found for sector 2 (the "machinery" sector), which is also the sector that leads in phase.

The lifetime difference of 6 years corresponds to a lifetime for machinery capital of 17 years and a lifetime of buildings and infrastructure of 23 years. If, with the same coupling parameter, the difference in capital lifetimes is increased to A r = 9 years, we observe a doubling of the period. The two sectors now alternate between high and low maxima for their production capacities. This type of behavior is referred to as a 2:2 mode. It has developed out of the synchronous 1:1 mode through a period-doubling bifurcation [60]. The 2:2 solution is illustrated in Fig. 1.9.

As the difference in lifetimes is further increased, the model passes through a Feigenbaum cascade of period-doubling bifurcations (4:4:, 8:8, etc.) and becomes chaotic at approximately A T = 10.4. years. Figure 1.10 shows the chaotic solution generated when A r = 10.7 years. Calculation of the largest Lyapunov exponent confirms that the solution in Fig. 1.10 is chaotic. We conclude that deterministic chaos can arise in a macroeconomic model that in its aggregated form supports self-sustained oscillations, if the various sectors (because of differences in parameter values) fail to synchronize in a regular motion.

Fig 1.10. Synchronized chaotic behavior. As the difference in capital lifetimes is increased further, the behavior becomes chaotic. For Ar = 10.7 years, the model shows irregular behavior, and initial conditions close to each other quickly diverge. Nonetheless, the two sectors remain locked with a ratio of unity between their average periods.

A more detailed illustration of the route to chaos is provided by the bifurcation diagram in Fig. 1.11. Here we have plotted the maximum production capacity attained in sector 1 over each cycle as a function of the lifetime difference Ar . The difference in capital lifetimes spans the interval 6 y < AT < 30 y. When A r = 30 y, the lifetime of the short-lived capital is just five years while the lifetime of the long lived capital is 35 years. The coupling parameter is kept


24 Chaotic Synchronization; Applications to Living Systems

Fig 1.11. Bifurcation diagram for increasing lifetime difference Ar and constant a. The figure shows the local maxima attained for the capacity of the long-lived capital producer in the steady state for varying values of the lifetime difference. From left to right the main regions of periodic behavior correspond to the 1:1, 1:2, 1:3, and 1:4 synchronization regions.

constant equal to a = 0.2. Inspection of the figure shows that the 1:1 frequency-locking, in which the production capacity of sector 1 reaches the same maximum in each long-wave upswing, is maintained up to A r « 6.4 years, where the first period-doubling bifurcation occurs. (Identification of the various periodic modes cannot be made from the bifurcation diagram alone, but involves the time and phase plots as well.) In the interval 6.4 y < AT < 7.9 y , the longwave upswings alternate between a high and a low maximum. Hereafter follows an interval up to approximately A r = 8.1 years with 4:4 locking, an interval of 8:8 locking, etc. Within the interval approximately 8.2 y < AT < 11.8 y small windows of periodic motion are visible between regions of chaos. In the region around 12.4 y < AT < 13.0 y chaos gives way to the 2:3 mode-locked solution and the associated period-doubling cascade 4:6, 8:12, etc. Another region of chaotic behavior follows until about A r « 15.2 years, where the system locks into 1:2 motion. Similarly the regions of 1:3 and 1:4 entrainment are clearly visible as A r continues to increase. Note that the 1:4 region bifurcates into 2:8 at around A r ft* 27.6 years, but then returns to 1:4 motion at A r as 28.3 years, rather than cascading through further doublings to chaos.

The phase diagram in Fig. 1.12 gives an overview of the dominant modes for different combinations of the lifetime difference A r and the coupling parameter or. The zones of mode-locked (i.e., periodic) solutions in this diagram are the well-known ArnoPd tongues [33, 38, 39]. Besides the 1:1 tongue, the figure shows a series of l:n tongues, i.e., regions in parameter space where the buildings industry completes precisely one long-wave oscillation each time the machinery

0.0 -tromtmim n iii,|« prmm, m») | 6 8 10 12 14 16 18 20 22 24 26 28 30

Lifetime difference (years)


industry completes n oscillations. Between these tongues, regions with other commensurate wave periods may be observed. An example is the 2:3 tongue found in the area around a = 0.15 and A r = 12 years. Similar to the 2:2 period-doubled solution on the right-hand side of the 1:1 tongue, there is a 2:4 period-doubled solution along part of the right-hand edge of the 1:2 tongue.

1.0

0.9

0.8

0.7

0.6

SacMr I Ua*tnt4 Seao. 2 Mori-livid capital produce) it ruble capital producer) • ruble la ieiaUioa hr alpha, ta kraWoa Ac aloha

abova Hue too.

IIH|llllimi|lllllllll|lllllllll|lllllllll|lllllllll|lllllllll|Tllllllll|lllllllll|lllllllll 12 14 16 18 20 22 24 26 28 30

Lifetime difference (Years)

Fig 1.12. Parameter phase diagram. The figure summarizes the steady-state behavior of the two-sector model for different combinations of the coupling parameter a and the lifetime difference Ar. A region labeled p : q indicates the area in parameter space where the model shows periodic mode-locked behavior of p cycles for sector 1 and q cycles for sector 2. The dashed curves across the diagram indicate the value of a above which each sector in isolation becomes stable. Above these lines synchronous 1:1 behavior prevails.

The phase diagram in Fig. 1.12 also reveals that the synchronous 1:1 solution extends to the full range of the lifetime differences A r for sufficiently high values of the coupling parameter a. When a is large enough, the individual sectors become stable, if the delivery delay and demand from the other sector are taken as exogenous. For reference, two curves have been drawn in Fig. 1.12, defining the regions in which one or both of the individual sectors are stable. As a increases, the overall behavior is increasingly derived from the coupling between the sectors and less and less from the autonomous self-ordering


mechanism in each individual sector. Thus, for high values of a, there is less competition between the two individual, autonomous oscillations and stronger synchronization. For large differences in capital lifetimes and low values of the coupling parameter a, the short-lived capital sector (sector 2) completes several cycles for each oscillation of the long-lived sector (sector 1). However, as a is increased, the short-term cycle is reduced in amplitude and, for sufficiently high values of a, it disappears altogether, resulting in a synchronous 1:1 solution. The locally stabilizing effect of high values of a creates an interesting distortion of the Arnol'd tongues in Fig. 1.12. For instance, the figure reveals that both the 1:1 region and the 2:2 region stretch above and around the other regions for high values of a.

By employing only a single capital-producing sector, the original long-wave model [55] represents a simplification of the structure of capital and production. In reality, "capital" is composed of diverse components with different characteristics. We have focused on the difference in the average lifetime of capital and it is clear from our analysis that a disaggregated system with diverse capital lifetimes exhibits a much wider variety of fluctuations. For moderate differences in parameters between the sectors, the coupling between sectors has the effect of merging distinct individual cycles into a more uniform aggregate cycle. The period of the cycle remains in the 50-year range, although the amplitude may vary greatly form one cycle to the next. The behavior of the two-sector model thus retains the essential features of the simple model and is robust to the aggregation of all firms into a single sector.

Entrainment in the disaggregated model arises only via the coupling introduced by the input-output structure of capital production. Other sources of coupling were ignored. The most obvious links are created by the price system. If, for instance, one type of capital is in short supply, one would expect the relative price of that factor to rise. To the extent that sectors can substitute one type of capital for another, one would expect demand for the relatively cheaper capital components to rise. In this way, the price system will cause local imbalances between order and capacity across the sectors to equalize, thus helping to bring the individual sectors into phase. (We have performed a few preliminary simulations of a version of the model that includes a price system and these simulations show an increased tendency for synchronization). The degree of substitution between capital types in the production function may well be an important factor: One would expect high elasticities of substitution to yield


stronger synchronization. A next step in the study of coupled economic oscillators could therefore involve introducing relative prices and differing degrees of substitution.

Another, more immediate extension of the above discussion would involve looking at more than two sectors. On the one hand, a wider variety of capital producers would introduce more variability in the behavior and, hence, less uniformity. On the other hand, as the system is disaggregated further, the strength of the individual self-ordering loops is reduced to near zero, and overall dynamics will more and more be determined by the interaction between sectors.

It would also be interesting to consider the influence of other (more global) macroeconomic linkages, such as the Keynesian consumption multiplier. Our preliminary results demonstrate the importance of studying non-linear entrapment in the economy. The intricacies of such phenomena suggest that there is a vast unexplored domain of research in the area of economic cycles. We suppose that nonlinear interactions could play as large a role in shaping economic cycles as do the external random shocks on which much of mainstream business cycle theory relies. At the same time, our discussion points to the similarities in nature between the problems we meet in macroeconomic systems and in the biologically oriented problems discribed in other chapters of this book.

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[4] E. Mosekilde, Topics in Nonlinear Dynamics (World Scientific, Singapore, 1996).


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Chapter 2 TRANSVERSE STABILITY OF COUPLED

MAPS

2.1 Riddling, Bubbling, and On-Off Intermittency

In the early 1980's, Fujisaka and Yamada [1] showed how two identical chaotic systems under variation of the coupling strength can attain a state of full synchronization where the motion of the coupled system takes place on an invariant subspace of total phase space. In spite of the fact that the systems are chaotic, their interaction allow them to move precisely in step. For two coupled identical, one-dimensional maps, for instance, the synchronous motion is one-dimensional and occurs along the main diagonal in the phase plane. The transverse Lya-punov exponent Ax provides a measure of the stability of the chaotic attractor perpendicular to this direction. As long as Ax is negative, a trajectory moving in the neighborhood of the synchronized chaotic state will, on average, be attracted towards this state.

Chaotic synchronization has subsequently been studied by a large number of investigators [2, 3, 4], and a variety of applications for chaos suppression, for monitoring and control of dynamical systems, and for different communication purposes have been suggested [5, 6, 7]. Important questions that arise in this connection concern the form of the basin of attraction for the synchronized chaotic state and the bifurcations through which this basin, or the attractor itself, undergoes qualitative changes as a parameter is varied. Under what conditions will the interacting chaotic oscillators be able to synchronize if they are

33


started out of synchrony? What is the response of the synchronized state to noise or to a small parameter mismatch, and what happens when the synchronization breaks down? Recent studies of these and related problems have led to the discovery of a variety of new phenomena, including riddled basins of attraction [8, 9, 10, 11, 12] and on-off intermittency [13] that can be observed on either side of the so-called blowout bifurcation [10, 11], where the transverse Lyapunov exponent Aj. changes sign.

On-off intermittency is an extreme form of intermittent bursting [14, 15] that can occur in the presence of a small positive value of Ax. In this case the chaotic set on the invariant subspace is no longer transversely attracting on average. However, immediately above the blowout bifurcation, where Aj_ is still small, a trajectory may spend a relatively long time in the neighborhood of the invariant subspace. From time to time, the repulsive character of the chaotic set manifests itself, and the trajectory exhibits a burst in which it moves far away from the invariant subspace, to be reinjected again into the proximity of this subspace. Besides a reinjection mechanism, the occurrence of on-off intermittency hinges on the fact that the positive value of the transverse Lyapunov exponent applies on average over long periods of time. For shorter time intervals, the net contribution to Aj. may be negative, and the trajectory is attracted to the chaotic set [13]. If there is no reinjection mechanism, a trajectory started near the invariant subspace may exhibit a superpersistent chaotic transient in which the behavior initially resembles the chaotic motion before the blowout bifurcation. Eventually, however, almost all trajectories will move away from this region and approach some other attractor (or go to infinity).

When on-off intermittency occurs, the blowout bifurcation leads to a dynamics that is confined by nonlinear mechanisms to a region of phase space (the trapping zone or absorbing area) situated inside the original basin of attraction [16, 17, 18, 19], and this basin remains essentially unaffected by the blowout bifurcation. Variation of a parameter that causes the attractor to grow (or the basin to shrink) may then produce a crisis in which the attractor abruptly disappears as it makes contact with the basin boundary [20, 21, 22].

Riddled basins of attraction denote a characteristic form of fractal domain of attraction that can be observed on the other side of the blowout bifurcation, where the transverse Lyapunov exponent is negative [23, 24, 25, 26]. Even though the chaotic set is now attractive on average, particular orbits (usually of

Transverse Stability of Coupled Maps 35

low periodicity) embedded in the chaotic attractor may be transversely unstable. In this case we talk about weak attraction or attraction in the Milnor sense [27]. The chaotic set in the invariant subspace then attracts a set of points of positive Lebesgue measure in phase space. However, arbitrarily close to any such point one may find a (small but) positive Lebesgue measure set of points that are repelled by the chaotic attractor.

The emergence of riddled basins of attraction occurs through a so-called riddling bifurcation (sometimes referred to as a bubbling transition [23]) in which the first orbit embedded in the synchronized chaotic attractor loses its transverse stability. This phenomenon was described in more detail by Lai et al. [28] and by Venkataramani et al. [29, 30]. They suggested that the riddling bifurcation takes place as two repelling orbits located symmetrically on either side of the invariant subspace approach the chaotic attractor and collide with a saddle cycle embedded in this attractor (in a so-called subcritical pitchfork bifurcation). Alternatively, a point cycle embedded in the synchronized state can lose its transverse stability through a period-doubling bifurcation [26]. In both cases, i.e., for the pitchfork as well as for the period-doubling bifurcation, the transition may be subcritical or supercritical depending on the sign of a certain quantifier (the so-called Lyapunov value) [31]. After a supercritical bifurcation, the transversely destabilized orbit will be surrounded by saddle cycles with their unstable manifolds stretching along the invariant subspace of the synchronized state. This leads to attractor bubbling and to the phenomenon of local riddling [23, 25]. In this case, the trajectories cannot escape a region around the synchronized state until one or two additional local [32] or global [33] bifurcations have occurred.

The riddling bifurcation causes tongues of finite width to open up along the transversely unstable directions from each point on the desynchronized periodic orbit [28]. In these tongues trajectories move away from the chaotic attractor. Similar tongues open up from each preimage of the points on the repelling cycle, and since these preimages are dense in the invariant subspace an infinite number of tongues emerge, creating the characteristic riddled structure in which the basin of attraction locally becomes a fat fractal.

Maistrenko et al. [34, 35] have studied chaotic synchronization and the formation of riddled basins of attraction for a system of two coupled piecewise linear maps. They related the various phenomena to the different types of instability for the chaotic set in the invariant subspace. Also investigating cou-


pled piecewise linear maps, Pikovsky and Grassberger [36] discovered that even when the coupled system exhibits a stable synchronized behavior as indicated by a negative value of the transverse Lyapunov exponent, the basin of attraction may have full measure and yet be densely filled with unstable periodic orbits. Hence, the synchronized attractor is surrounded by a strange invariant set which is dense in a region around the attractor, and the synchronized state is not asymptotically stable. Pikovsky and Grassberger [36] also observed a bifurcation in which a synchronized one-dimensional attractor explodes into a two-dimensional attractor that contains the strange invariant set. Gardini et al. [37] investigated a one-parameter family of twisted, logistic maps. Particularly interesting in the present context is their description of the contact bifurcations in which the boundary of the absorbing area for a chaotic set touches the basin boundary. As we shall see in Chapter 3, global bifurcations of this type are often involved in the transition from local to global riddling [33].

The purpose of the present chapter is to provide a first, phenomenological introduction to some of the interesting phenomena that are involved in the loss of synchronization and the emergence of riddled basins of attraction. With this purpose we shall study a simple two-dimensional map F = Fafi : R2 —> R2 given by the recurrence formula

xn+l = Ja\xn) + £\Vn ~ xn) M . \

J/n+1 = fa(yn)+e{Xn-yn)-

Here, fa is the one-dimensional logistic map

fa:x>-Kix(l-x), a; €[0,1] , o £ [ 0 , 4 ] , (2.2)

and e € R is the coupling parameter. We first determine the regions of weak stability in the (a, e) plane for the synchronized chaotic attractor and calculate the transverse Lyapunov exponent Ax as a function of e for values of the nonlin-earity parameter a where the individual map displays a homoclinic bifurcation. These are the parameter values at which fa{x) is known to produce a chaotic attractor with an absolutely continuous invariant measure.

The formation of riddled basins of attraction is discussed in Sec. 2.3, and in Sec. 2.4 we determine the bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the chaotic attractor. For one-band, two-band, and four-band chaotic dynamics we hereafter follow the changes that take place in the attractor and its basin of attraction under passage of the riddling and


blowout bifurcations [26]. It is shown that the loss of weak stability does not necessarily affect the basin of attraction. Instead, the chaotic attractor may spontaneously break the symmetry and spread into the two-dimensional phase space. It is also shown that the emergence of transversely unstable trajectories, while being a necessary condition for basin riddling, is not sufficient for global riddling to occur [33].

In Sec. 2.6 we illustrate the phenomenon of intermingled basins of attraction [8, 9] for a situation where the system has two coexisting four-band attractors, each displaying a riddled basin structure. Finally, in Sec. 2.7, we consider the phenomenon of partial synchronization in a system of three coupled logistic maps [38]. Here, we shall see how two of the chaotic oscillators can synchronize with one another, while the third oscillator remains unsynchronized. As described in Chapter 1, this problem is of particular interest in connection with the development of new types of communication systems.

2.2 Weak Stability of the Synchronized Chaotic State

The metric and topological properties of the logistic map have been studied extensively over the last two or three decades [39, 40]. It is known, for instance, that for any a G [0,4], fa has no more than a single attractor, which may be a cycle of points or of chaotic intervals. The parameter set C = {a G [0,4] : fa has an attracting cycle} is open and everywhere dense. At the same time, the parameter set K, = {a G [0,4] : fa has an absolutely continuous invariant measure} is nowhere dense (i.e., it has a Cantor-like structure), and the measure n{IC) > 0 [41]. Figure 2.1 shows a part of the bifurcation diagram for the logistic map with the well-known period-doubling cascade of attracting cycles 72". Increasing the parameter a beyond the accumulation point a* = 3.569... for this cascade, a reverse cascade of homoclinic bifurcations of the cycles ^2" takes place at the parameter values a„. At ao, for instance, the unstable fixed point XQ = 1 — 1/a undergoes its first homoclinic bifurcation as the critical point xc = 1/2 is mapped into XQ in three interations of fa.

The bifurcation points an can easily be determined numerically (oo can also be calculated explicitly). The first four are given by

a0 = 3.67857351042832...,

ai = 3.59257218410697...,


i.O

X

0.5

00

-====5Sl

-c:^^8l - = = =

a' « 2 , a, an 3.54 3.59 3.64 3.69

Fig 2.1. Bifurcation diagram for the individual map fa:x^ ax(l - x) for 3.54 < o < 3.89. At each of the homoclinic bifurcation points on, the map has an absolutely continuous invariant measure.

a2 - 3.57480493875920...,

a 3 = 3.57098594034161....

In Fig. 2.1, a = ao is the parameter point at which the two bands of the chaotic attractor merge into a single band. At this point, /„ has an attracting interval To = [/0(oo/4); (OQ/4)] consisting of two subintervals TQ = [/o(oo/4); XQ] U [XQ\ (ao/4)] that are mapped one into the other under the action of /„. Moreover, fa has an absolutely continuous invariant measure in IV Hence the map is chaotic [39, 40]. Similarly, for a — an, fa has an attracting cycle F n of 2n intervals consisting of 2"+1 subintervals that are pairwisely mapped one into another under the action of /„". Moreover, having an absolutely continuous invariant measure, /„ is chaotic in Fan.

The main diagonal {x = y} is a one-dimensional invariant manifold of the two-dimensional map F. This implies that a point on the diagonal will be mapped into another point on this line, or, in other words, F({x = y}) C {x — y}. The existence of such a one-dimensional invariant manifold is clearly a consequence of the restrictions imposed by the symmetric coupling of two identical one-dimensional maps. Any small mismatch between the maps leads, in general, to the disappearance of the one-dimensional manifold, with the result that the dynamics becomes two dimensional.

By subtracting the two one-dimensional maps in Eq. (2.1), one finds a


transverse line {x + y = 1 — 2e/a} that also maps into the main diagonal under the action of F [42]. This line is the preimage of the main diagonal, and part of the line (which maps into the attractive interval on the main diagonal) will belong to the basin of attraction for stable solutions on the diagonal. By adding the two one-dimensional maps in Eq. (2.1), one can show that the preimage of the transverse line is a circle centered in (a;, y) = (1/2,1/2) and with a radius {a2-2a + 4eY^/aV2.

For any point on the main diagonal, F0j£ has an eigendirection ui = (1,1) along the diagonal, and an eigendirection u-i — (1, —1) perpendicular to it. The corresponding eigenvalues are

i/i = f'a{x) = a(l - 2x) (2.3)

and z,2 = fa(x) - 2e = a( l - 2s) - 2e. (2.4)

Along the diagonal the coupling vanishes, and the dynamics coincides with that of the one-dimensional map fa{x). Let I C M. be a chaotic attractor for fa{x), then A — {x = y G / } will be a one-dimensional invariant chaotic set for the coupled map system. A attracts points from its one-dimensional neighborhood along the diagonal. Does it also attract points from its two-dimensional neighborhood Us(A)? In other words, is A an attractor in the plane? The answer to this question clearly depends on the values of the transverse eigenvalues v<i at all the different points of A.

As a first approach to addressing the above questions we may evaluate the transverse Lyapunov exponent

1 N

A x = U m - £ > ! £ ( * „ ) - 2 e | , (2.5) n = l

where {xn = fa(x)}^Li is a typical itinerary on A. If the set A has an absolutely continuous invariant measure, e.g., for the above mentioned parameter values a„, the value of Ax will be the same for almost all trajectories on A. If Ax is negative, we expect that A is attracting on average in a two-dimensional neighborhood. As previously mentioned, this type of stability is referred to as weak stability or stability in the Milnor sense [27], and the transition in which Ax changes sign is referred to as the blowout bifurcation [12].

Indeed, as shown by Alexander et al. [8, 9], in the case that A is a finite union of intervals and the invariant measure of fa on A is absolutely continuous, the


condition Ax < 0 guarantees that A attracts a positive Lebesgue measure set of points from any two-dimensional neighborhood Us(A). Moreover, weak stability in the Milnor sense implies that the measure fj, of points that are attracted to A approaches the whole measure of U$(A) as the width of the neighborhood 5 ->• 0, i.e.,

,,(R( A\ n rrj AW

(2.6) l i m M(g(A)n[ / , (A) ) L

<s-+o (J.{US(A))

Here B(A) denotes the basin of attraction of A, i.e., the set of points (x,y) £ K2

for which the stationary state (or UJ limit) is contained in A. When Ax becomes positive, the synchronized chaotic state loses its average attractivity.

4.0

a

T>

I

1 1

-

w

R. 1 V V

-

/

-2.0 2.0

Fig 2.2. Regions of parameter plane where the transverse Lyapunov exponent Ax < 0.

Figure 2.2 shows the regions of parameter space in which Ax < 0, so that the synchronized attractor is (at least) weakly stable. The figure was obtained by performing 1000 scans of X±(e) for different values of a with a similar resolution along the e axis [26]. The stability regions clearly reflect the complexity of the bifurcation structure. In particular, we notice the irregular variation with a in the chaotic regime. For values of a below the accumulation point a* = 3.569..., the individual map displays an attracting cycle, and the synchronized behavior is also periodic. In this case there is no distinction between weak and asymptotic (or strong) stability, and our results are similar to those obtained by Schult et at. [43].

In each of the periodic windows in the region a > a*, the distinction between weak and strong stability likewise disappears. Moreover, for an ./V-periodic


synchronous state 7^ = {x\, £2, •••, %N}, the criterion for transverse stability

N

J ] | / 'K) - 2e\ < 1, (2.7) n = l

is satisfied if and only if the coupling parameter e belongs to the union of N (possibly overlapping) intervals, each including one point from 7jy. Some of these intervals correspond to positive values for e, and some of them correspond to e < 0. The stability intervals are clearly seen in Fig. 2.2. We notice, for instance, the stability regions for the period-6 solution around a — 3.63, for the period-5 solution around a = 3.74, and for the period-3 solution in the interval around a = 3.84. For each of these windows we also observe the signature of the period-doubling cascade in which they end. In a similar manner, the two-band chaotic attractor existing for a = a\, may give rise to a two e-intervals with Ax < 0, and the four-band chaotic attractor existing for a = 0,2 may have four (partly overlapping) intervals for e with weak stability.

2.3 Formation of Riddled Basins of Attraction

The condition A^ < 0 guarantees that almost all trajectories on A are transversely attracting. However, there can still be an infinite set of trajectories (having positive Lebesgue measure) in the neighborhood of A that are repelled from it. As we shall see, this situation is quite generic for our system of coupled logistic maps. To obtain an attractor in the usual (topological) sense, we must ascertain that all trajectories on A are transversely attracting [44].

Consider, for example, the fixed point P(xo,xo). Inserting XQ = 1 — 1/a into Eq. (2.4) and requiring that the magnitude of vi be less than 1, we find a coupling interval

a — 1 a — 3

in which the fixed point is transversely stable. If a > 3 and e falls outside this interval, both v\ and v^ will be numerically larger than 1, and P becomes a repelling node.

Under the action of Fafi, a trajectory starting close to such a repelling node moves away from it along an integral curve. Under general conditions (yi 7 v^ and v\ ^ 1/2), these integral curves can be obtained in a neighborhood of P by


the smooth transformation

£ = XQ + 6?j2 + higher order terms

of the curves | | = q ^ M / M " ^ (2.8)

where f = a; + y — 2XQ and r) = y — x are new coordinates defined along and perpendicular to the diagonal, respectively. C is an arbitrary constant that depends on the initial conditions, and b = af{v\ — j/f). Expression (2.8) gives the integral curves of the linearized map DF around the fixed point. It follows from this expression that the integral curves are tangential to the »?-axis for I 21 < \v\\ and tangential to the £-axis for |i^| < \v\\. These two situations are illustrated in Fig. 2.3. Formulae for the invariant curves of the nonlinear system (2.1) for a cycle Pn of any period n in the diagonal will be derived in Chapter 3.

Fig 2.3. The two generic forms for the repelling tongues that develop from each point on a transversely unstable periodic cycle. Note that these tongues do not have sharp boundaries. They are made up by bundles of trajectories that happen to fall in the neighborhood of P.

Let us now suppose that the fixed point P{XQ,XQ) belongs to the chaotic attractor A on the main diagonal, and that A has an (ergodic) absolutely continuous invariant measure on it. The fixed point P will then have infinitely many preimages, and the set of these preimages will be dense in A. Hence, we expect that the set U$ of preimages of the tongues TlM a n d 11^ will be


dense in some neighborhood of A (except in some half-neighborhoods of the end points for the intervals that span the attractor).

Fig 2.4. The riddling bifurcation produces a dense set of repelling tongues.

This situation is schematically illustrated in Fig. 2.4. The region Iff (shown in gray) is the locally repelling set of the attractor A. Each point from U#\A leaves the neighborhood U$(A) in a finite number of iterations. It is important to note that this repelling property has a local character. Our analysis says nothing about the fate of the trajectories once they have left the neighborhood Us. This depends on the global dynamics of F, and two different scenarios may occur:

Scenario 1 Having left the locally repelling region U$(A), the trajectories wander around

in phase space. However, they are restricted by nonlinear mechanisms to moving within an absorbing area [18, 19, 33, 37] that lies strictly inside the basin of attraction. The trajectories can never escape the absorbing area. Hence, they cannot diverge or approach an attractor outside the absorbing area. If there is no other attractor, apart from A, inside the absorbing area, then, sooner or later, most of the trajectories will return to Us(A). Some of them may again be mapped into U%(A) and, hence, again leave the neighborhood of A. This


type of dynamics gives rise to the temporal bursting characteristic of on-off intermittency. As long as the transverse Lyapunov exponent is positive, the bursts will never stop, and in essence we have an attractor in two-dimensional phase space with a more or less pronounced maximum of its invariant density in the neighborhood of the original one-dimensional attractor. As Aj_ becomes negative, the bursts tend to stop. However, this transition may not be very sharp, and, particularly in the presence of noise, one may still see some bursting even for negative values of Ax (attractor bubbling). (If another attractor exists inside the absorbing area, one may observe a global riddling of the basin of attraction for the weakly stable synchronized chaotic state with initial conditions that lead to the other attractor. We shall return to this situation in more detail in Chapter 3. Examples of the types of basins of attraction that one can observe in this case are shown in Figs. 2.13 and 2.19. The absorbing area itself will be defined in more detail in the following sections.)

Scenario 2 If the nonlinear mechanisms are too weak to restrict the motion to an absorb

ing area inside the basin of attraction for the synchronized state, the alternative is that most points leaving U%{A) diverge or go to another attractor. This can be an attracting point cycle, an invariant torus, or a chaotic attractor. On the other hand, provided that Ax < 0 the measure of the complementary set f/| C U$\U]j of points that are attracted to A and never leave Us normally approaches the measure of Us for 5 —> 0 [8, 9]. Under these conditions we expect the basin of attraction to attain a globally riddled structure with holes that belong to the basin of another attractor. Mathematically expressed, this implies that, for any 8 > 0, (i) the complementary set Us(A)\B(A) will be everywhere dense in Us, and (ii) 0 < fi[B(A) C\ Us(A)] < /i[Us(A)]. (In contrast to this, the case of asymptotic stability of A is characterized by the existence of a 5 > 0 such that B{A) C Us — U\. This requires that all trajectories on A are transversely stable.)

2.4 Destabilization of Low-Periodic Orbits

In order to determine the boundaries in parameter space for the regions of absolute stability for the synchronized chaotic attractor A, we can start by considering the transverse stability of the various point cycles embedded in this attractor. We have already seen that the conditions for transverse stability for


the fixed point are — (a — l ) /2 < e < — (a — 3)/2. The slanting line denoted "fixed point" in the middle of the stability diagram of Fig. 2.5 represents the lower edge of this zone. The upper end falls outside the range of coupling strengths considered in the figure. Hence, the fixed point is transversely stable to the right of the fixed point line. Along this line, destabilization of the fixed point happens in a subcritical pitchfork bifurcation. This implies that two repelling fixpoints (period-1 cycles) existing symmetrically on either side of the synchronous fixed point before the bifurcation, meet and are annihilated while at the same time destabilizing the synchronous fixed point in the transverse direction.

3.75

3.55

Fig 2.5. Bifurcation curves for the transverse destabilization of various low-periodic cycles embedded in the synchronized chaotic attractor. Regions in which the synchronized chaotic state is asymptotically stable are hatched. If more than one cycle of a given period occur, the cycle arising at the lowest value of o was considered.

The stability intervals for a point cycle Pn of period n are formally given by Eq. (2.7). The period-2 cycle P2 arises at a = 3, and thereafter alternates between the points

a + l ± V ( o + l ) ( a - 3 ) Zl ,2 = 2/1,2

2a (2.9)

Evaluating the conditions for transverse stability for this cycle for a > 1 + \JZ


gives two intervals for e of which the interval to the left of e = —1/2 is given

by

- ^ [ 1 + \ / ( a + l)(a - 3) + 1] < e < ~[1 + y/(a + l )(a - 3) - 1]. (2.10)

The borderlines of this interval are represented by the two curves denoted "period 2" in Fig. 2.5. The period-2 cycle is transversely stable between these curves. To the right the destabilization happens through a supercritical period-doubling bifurcation, and to the left through a subcritical pitchfork bifurcation. The subcritical pitchfork bifurcation is qualitatively similar to the bifurcation we have just described for the synchronous fixed point. After the supercritical transverse period-doubling bifurcation, the points of the now repelling, symmetric period-2 orbit are each surrounded by a pair of points of an asymmetric period-4 saddle cycle with its unstable manifolds stretching along the synchronization manifold.

Similarly, the curves denoted "period 4", "period 6", and "period 8" in Fig. 2.5 bound the regions of transverse stability for these cycles [26]. The minimum of the curve for the period-6 cycle falls at a = 3.626, where this cycle first arises. With the values of a considered in the figure, the period-3 cycle has not yet appeared in the individual map. Hence synchronized behavior with this periodicity cannot occur, and the figure delineates the regions of transverse stability for the most important cycles with a period less than or equal to n = 8. There are reasons to believe that transverse destabilization of cycles of higher periodicity generally play a less important role [47], and this is strongly supported by our numerical calculations. From the information in Fig. 2.5 we can therefore determine the regions of asymptotic stability for the synchronized chaotic attractor in each of the intervals ai < a < ao and ao < a.

The first of these regions is bounded on both sides by the transverse destabilization of the period-2 cycle which occurs before destabilization of the period-8 and period-4 cycles. For higher values of a, destabilization of the period-6 cycle limits the region of asymptotic stability for the synchronized chaotic attractor, and there is also a region in parameter space where destabilization of the period-4 cycle is the first to take place. Below the homoclinic bifurcation point ao, destabilization of the fixed point is not significant because the two-band chaotic attractor existing in this range does not contain the fixed point. For a > ao, however, the synchronized chaotic state is no longer asymptotically stable to the left of the fixed point curve.


2

1

0

-1 -2 -1 0 1 2

Fig 2.6. Variation of the transverse Lyapunov exponent Ax with the coupling parameter e for o = a0. The synchronized chaotic attractor is absolutely stable for (approximately) —1.31 < e < -1.24.

Figure 2.6 shows a scan of the transverse Lyapunov exponent X± as a function of the coupling parameter for a = ao- For this value of a, the individual map exhibits a one-band chaotic attractor consisting of two subintervals at the moment when they merge. The points e = —1.544 and e = —0.478, where Aj_ changes sign, are the blowout bifurcation points. In the interval between these points, the synchronized chaotic state is at least weakly stable. Outside this interval, A is a so-called chaotic saddle [23, 45]. Also indicated in Fig. 2.6 is the interval from e = —1.31 to e = —1.24 in which the chaotic attractor is asymptotically stable.

Figures 2.7(a)-(e) portray the basins of attraction for the synchronized chaotic state Ao at various values of the coupling parameter e. In each of these figures initial conditions leading to the chaotic attractor are plotted as gray points, and initial conditions leading to another attractor (or infinity) are left blank.

Figure 2.7(a) shows the basin of attraction for e = —1.4, i.e., a little to the left of the region of asymptotic stability. In this region the synchronized period-6 cycle is transversely unstable, and the figure reveals the characteristic appearance of a globally riddled basin with a dense set of tongues with points that are repelled from the attractor emanating from the period-6 cycle and its preimages. As previously noted, the basin of attraction includes a section of a transverse line that maps into the attracting interval of the main diagonal, as well as sections of the circle that is a preimage of the transverse line. From each

48 Chaotic Synchronization. Applications to Living Systems

125 a = a 0 . E = - l 4 (a)

-0 051 -0 05 125

125

y

a = ao,&=-l 1 (c)

^M

-0 051 _ -0 05 125

125

y

nos

a = = a 0 , s = - l 2 (b)

tS

-0 05

125

-0 05t

a = ao,E=-l 0

-0 05

125

y

0 05

a = a(,,s=-0 5

• ^ 1 ~

1 v . ^ H K

""'- 1 r -.*t*i

(e)

I T S

yF% \ I:. *

125

"(d)

-0.05

Fig 2.7. Basins of attraction for the one-band chaotic attractor AQ with different values of the coupling constant for a = a0 :(a) global riddling for e = —1.4, (b) local riddling for s = - 1 . 2 , (c) global riddling for e — —1.1, (d) global riddling for s = —1.0, and (e) global riddling close to the blowout bifurcation for e = -0 .5 . The basin of attraction is plotted gray in (a)-(d) and black in (e).


of these structures we have a similar dense set of repelling tongues, contributing altogether to the complexity of the basin.

Figure 2.7(b) shows the basin of attraction for a coupling parameter immediately to the right of the region of asymptotic stability (e = —1.2). Here the in-phase period-2 cycle is transversely unstable. However, while the basin has a fractal boundary, there are no tongues in it belonging to the basin of another attractor. This is characteristic of a locally riddled basin of attraction where trajectories repelled from the synchronized chaotic state never reach the basin boundary and sooner or later return to the neighborhood of the attractor. In this case the synchronized state may be referred to as a Milnor attractor, i.e., it attracts all points from its neighborhood except a set of zero measure.

As e is further increased, a transition occurs in which trajectories repelled from the main diagonal start to make contact with the basin boundary. As illustrated in Fig. 2.7(c), we then recover the globally riddled structure. Here e = —1.1, and the synchronized state is a weak attractor in the Milnor sense, i.e., there is a finite measure of points in its neighborhood that are repelled from it.

For e — —1.0 (Fig. 2.7(d)), a new structure in the basin of attraction becomes manifest. Here we observe two lines parallel to the main axes and crossing the diagonal at a period-2 point. Each of these lines are invariant with respect to the second iterate of Fafi. A similar set of curves are found to cross the diagonal at the other point of the in-phase period-2 orbit. Finally, in Fig. 2.7(e), e = —0.5, and we are close to the blowout bifurcation at e = —0.478. Here the measure of the points that are attracted to the synchronized chaotic state becomes very small, and the majority of initial conditions lead to diverging orbits.

2.5 Different Riddling Scenarios

The above scenario for a — CLQ was characterized by a direct transition from asymptotic stability to global riddling at the left bifurcation point (e = —1.31). where the period-6 orbit becomes transversely unstable, and by a transition involving an interval with local riddling at the right bifurcation point (e = — 1.24), where the period-2 cycle undergoes a supercritical transverse period-doubling bifurcation. Let us now consider what happens at other values of the nonlinearity parameter [26].

Figure 2.8 shows a scan of the transverse Lyapunov exponent Ax for a = a\


2

1

0

-1 - 2 - 1 0 1 2

8

Fig 2.8. Variation of the transverse Lyapunov exponent Ax with the coupling parameter e for a = ai. Here we have two intervals of weak stability. The synchronized attractor is absolutely stable for (approximately) —1.46 < e < —1.16.

where the individual map fa exhibits a two-band chaotic attractor. Inspection of the figure shows that we now have two regions of weak stability for the synchronized chaotic state, one for positive values and one for negative values of e. In the region of negative coupling constants, blowout bifurcations occur at e = -1.472 and e = —1.0385. The region of absolute stability extends from e = -1.464 to e = —1.156. At both ends of this region, the in-phase period-2 cycle becomes transversely unstable.

Figures 2.9(a)-(c) show typical examples of the basins of attraction observed in this region. At the same time, they illustrate an interesting change in the chaotic dynamics. In Fig. 2.9(a) (e = —1.3) we have an absolutely stable two-band attractor A\ on the main diagonal. There are holes in the basin of attraction. However, these holes do not emanate from points embedded in the attractor. Hence the basin is not riddled, but has a fractal boundary.

Figure 2.9(b) was obtained for e = —1.1, i.e., immediately after the loss of asymptotic stability in the riddling bifurcation at e = —1.156. This provides a new example of a locally riddled basin. Trajectories that are repelled from the main diagonal do not reach the basin boundary, but sooner or later most of them return to the neighborhood of the synchronized state. As e is increased a little more, we observe a spontaneous breaking of the symmetry as the chaotic attractor spreads into two-dimensional phase space. The basin of attraction, on the other hand, changes only slightly. This is shown in Fig. 2.9(c) for e = —1.03.


1251 a = a, ,e =-1 3 (a) 125f

-OOSL -0 05

y

a =a , ,e=- l . l (b)

-0 051 _ -0 05

1 25ra~=a,,s=-l,03 (c)

-0 051 -0 05 125

Fig 2,9. Basins of attraction for different values of the coupling parameter for a = oi : (a) fractal basin boundary for absolutely stable attractor, (b) locally riddled basin for weakly stable attractor, and (c) two- dimensional attractor restricted to the absorbing area.

Hence, in this case there is no interval with global riddling, but we observe a transition from local riddling to on-off intermittency when Ax becomes positive. The two-dimensional attractor A[ is bounded in phase space to an absorbing area defined by the iterates of the critical curves for Fafi [37, 46]. In this way the absorbing area plays a similar role for a two-dimensional non-invertible map as does the iterates of the critical point for the one-dimensional logistic map. The absorbing area lies fully within the basin of attraction, and as long as there is no contact between the two boundaries, the two-dimensional attractor continues to exist. With further increase of e (approximately at e = —0.95) a crisis takes place m which the borderline of the absorbing area touches the basin boundary,


and the two-dimensional attractor suddenly disappears.

a = ai, s=0.33

-0.15 _ 1.1

Fig 2.10. Basin of attraction for o = Oi and e = 0.33.

Figure 2.10 shows an example of the basins of attraction that one can observe in the other parameter window of weak stability for a = a%. Here e = 0.33, and the density of points in phase space from which the trajectory is attracted by the synchronized chaotic state is quite low. Only globally riddled basins of attraction are observed in this parameter window.

For a = 02, a scan of the transverse Lyapunov exponent shows three regions of weak stability. This is illustrated in Fig. 2.11. The largest region extends from the blowout bifurcation at e = —1.462 to the blowout bifurcation at e == —1.0134. Within this region we find a region of asymptotic stability delineated by the riddling bifurcations at e = -1.461 and e = —1.015. In both of these bifurcations, the in-phase period-8 cycle becomes transversely unstable. (Transverse destabilization of the period-2 cycle is not significant since the points of this cycle do not belong to the chaotic set.)

Figure 2.12(a) shows the locally riddled basin of attraction with fractal boundaries that one can observe for e = —1.14. In Fig. 2.12(b) the coupling parameter is increased to e = —1.01. This is immediately after the blowout bifurcation. The basin of attraction remains practically unaffected by this change. However, the dynamics of the coupled map system spontaneously breaks the


J

a = ai

-1.462L f-1.013... V

0 1

8

Fig 2.11. Variation of the transverse Lyapunov exponent X± with the coupling parameter e for

125n7=" a = a2,s=-i.i4 (a)

-oos -0.05 125 12j

Fig 2.12. Basins of aitiaction for a = 02: (a) locally riddled basin with fractal basin boundary, and (b) two coexisting eight-band chaotic attractors.

symmetry, and the synchronized four-band attractor is replaced by two mutually symmetric two-dimensional eight-band attractors. The eight-band chaotic attractors have developed via a sequence of bifurcations from two mutually symmetric period-4 stable cycles produced in a saddle-node bifurcation away from the diagonal.

Finally, Pig. 2.13 shows the basin of attraction for the synchronized four-band attractor in the second parameter window of weak stability (a = 02, e = —0.2). For these parameter values the synchronized chaotic state coexists with a stable


a = a2,s =-0.2

-0.05 1.25

Fig 2.13. The basin of attraction for the synchronous chaotic state is riddled with holes that belong to the basin of a coexisting periodic cycle. Note that in this case the repelling tongues have sharp and well-defined edges.

period-2 point cycle shown by crosses and the basin of attraction for the chaotic state is riddled with holes that belong to the basin of attraction of the point cycle. Note that in this case the repelling tongues have sharp and well-defined edges, defined by the stable manifolds of asynchronous saddle cycles situated along these edges and produced in the stabilization of the asynchronous period-2 cycle. As we shall see in Chapter 3, such asymmetric point cycles play a fundamental role in the global dynamics of our two-dimensional system (2.1).

2.6 Intermingled Basins of Attraction

Consider now the map Fa<e for e = — 1:

F-1AJ*U(i-*)-(*-v)J- ( }

e = — 1 is the value of the coupling parameter for which we observed the particular basin structure depicted in Fig. 2.7(d). It is easy to show that for any a the two mutually perpendicular straight lines x = X2 and y = x% both remain invariant under two iterations of F_i, i.e., Ftx{{x = X2}) C {x = X2}

1.25

y

0 05


and F*i({y = £2}) C {y = ^2}. Here #2, as given by Eq. (2.9), is the larger of the two amplitudes 0:1,2 for the period-2 cycle 72.

Let g : x \-~¥ g(x), x G l denote the one-dimensional map that Flt induces along these lines, i.e.,

(g(x),x2) = Fl1(x,x2) (2.12)

Fig 2.14. Chaotic attractors off the main diagonal: (a) sketch of the one-dimensional map g(x), (b) its bifurcation diagram, (c) the corresponding longitudinal Lyapunov exponent, and (d) the transverse Lyapunov exponent [26].

Figure 2.14(a) portrays the function g(x) for a = 3.6, and Fig. 2.14(b) shows the bifurcation diagram obtained by varying a over the interval from 3.4 to 3.74. Figures 2.14(c) and (d) provide scans of the corresponding longitudinal and transverse Lyapunov exponents Ay and Ax, respectively. g(x) has a finite interval over which it is unimodal, and Fig. 2.14(b) demonstrates that it produces a transition to chaos in accordance with the usual Feigenbaum scenario.


Magnifications of Figs. 2.14(c) and (d) show that Ay is positive for a = 3.6, and that Ax is negative. Hence, for this value of a, the numerical calculations indicate that g{x) displays a two-band chaotic attractor.

By virtue of its symmetry with respect to the main diagonal, the map F-\ exhibits two one-dimensional four-band invariant chaotic sets A4 and A4. Figure 2.15 displays the basins of attraction for each of these sets. For a = 3.6 and e = — 1, the synchronized state on the main diagonal is a chaotic saddle, and almost all trajectories that remain finite approach either A4 or A4. However, for both of these sets the basin of attraction is riddled with holes that belong to the basin of the other set. Hence, the basins of attraction exhibit an intermingled structure [9] with one half measure set of points that are attracted to A4, and one half measure set that is attracted to A4. Figure 2.15(b) shows a magnification of the intermingled basin structure in a region around (x2, X2).

2.7 Partial Synchronization for Three Coupled Maps

The purpose of the present section is to illustrate the phenomenon of partial synchronization by considering a system of three nonlinearly coupled logistic maps [38]. For a specific form of asymmetric coupling, we determine the region in parameter space in which two of the maps can attain a state of chaotic synchronization while the third map exhibits independent chaotic behavior. We also study the bifurcations through which this state of partial synchronization arises and breaks down, and we follow the associated transformations of the basins of attraction.

Problems of this nature arise, for instance, in connection with the development of new principles for secure communication based on the so-called master-slave configuration [5, 48, 49, 50]

(2.13)

As before, f(x) is here a one-dimensional, chaotic map. When transmitting the signal zn from the master system we want to be able to read the message

Master: \ *n+i = f(xn) + e (f(zn)-{ Zn+1 = f(zn) + £ (f(xn) -

siave- { y»+i = f(y*)+e(f(z")-\ un+i = f(zn) + e (f(yn) -


(a)

i

(b) • • - " • • • ^ • • ' • l A . - ^ g ^ ' " " ' - . - ^

Fig 2.15. Intermingled basins of attraction for two coexisting four-band chaotic attractors: (a) overview, and (b) detail around one of the points of the period-2 cycle. Initial conditions from which the trajectory approaches one of the chaotic attractors are marked black, while initial conditions leading to the other attractor are left blank [26].


xn from the slave system. This will be possible if the parameters of the coupled system can be adjusted such that yn synchronizes with xn. On the other hand, to mask xn during the transmission process, synchronization between xn

and zn should be avoided. Whenever xn and yn synchronize, un and z„ will also synchronize, and by comparing the variables of the slave system, one can immediately detect whether synchronization has occurred or not. Hence, the fourth equation in (2.14) can be omitted, and we arrive at a system of three coupled maps:

x„+i = f{xn) + e(f(zn) - f(xn)) J/„+i = f(yn)+e(f(zn)-f(yn)) (2.14) Zn+l = f(z„) + S(f(xn) - f(zn))

with / = fa : R —> K. The essential aspects of the above problem may be formulated as follows: We would like to find a set of parameters {a, e} for which the chaotic variables x and y synchronize with one another while the variable z remains unsynchronized. Hence, the asymptotic state (x, y, z) should belong to the synchronization plane Uxy = {(x, y, z) \x = y } without becoming restricted to the main diagonal D = {(x,y,z)\x = y = z}. Since we want this situation to be realized for a massive set of initial conditions, the following criteria must be fulfilled [49, 50]:

1. The synchronization plane n ^ must be invariant under the action of F.

2. In order to avoid complete synchronization, the main diagonal must be transversely repelling in the plane Hxy (or it should attract only a zero Lebesgue measure set of points).

3. The two-dimensional chaotic attractor (or attractors) that under these conditions can exist in Uxy must be stable in the direction perpendicular to this plane.

Figure 2.16 illustrates the situation for a set of three coupled logistic maps with a = ao and e = 0.98. Here, we have three different two-dimensional chaotic attractors in the synchronization plane. In each of these attractors the variables x and y are synchronized while z is not. Actually, as we shall see below, the occurrence of partial synchronization in the system (2.14) is restricted to an interval around e = 1. Moreover, by contrast to the original two-dimensional coupled maps system (2.1), F is not symmetric with respect to permutation of the variables. The planes Uxy and Iixz = {(x,y,z)\ x — z) are invariant under


y "

Fig 2.16. Partial synchronization in a system of three coupled one-dimensional maps. The limiting states are three two-dimensional chaotic attractors situated in the synchronization plane {x — y, z}. The figure was obtained for the coopled map system (2.14) with fa(x) ~ oar(l - x), a = a0 = 3.678573... and e = 0.98.

F whereas the plane Uyz = {(x,y,z) \y = z} is not. The main diagonal D is also invariant.

Prom a stability point of view, the partial synchronization may be strong or weak, depending on whether the chaotic set embedded in the synchronization manifold 11^ is asymptotically stable (in the Lyapunov sense) or not. In the case of weak stability as defined by Milnor [27], the synchronized chaotic state is stable on average (its transverse Lyapunov exponent is negative), but particular orbits embedded in this state are unstable in the direction perpendicular to IX^.

The simplest form of asymptotic dynamics that can occur in the coupled map system (2.14) is the fully synchronized (or coherent) behavior in which all three oscillators display the same temporal variation, xn = yn = zn, so that the motion is restricted to the main diagonal in the three-dimensional phase space. Along this diagonal the dynamics is specified by the one-dimensional map f(x), and the Lyapunov exponent is

1 N

n=l (2.15)

where f(x) denotes the derivative of f(x) and xn, n = 1,2... represents a typical itinerary. Obviously, Ar> > 0 if the dynamics of f(x) is chaotic.


Let us first consider the stability of the fully synchronized motion. In a point (x, x, x) of D, the Jacobian matrix of the coupled map system (2.14) is [49, 50]

/ l - e 0 e \ DF = f{x) 0 l - e e . (2.16)

\ e 0 l - e )

Besides the direction ao = (1,1,1) along the diagonal, the Jacobian matrix DF has the eigenvectors axy — (1,1, —1) and axz = (0,1,0) which lie, respectively, in the plane Uxy and in the plane Uxz. The corresponding eigenvalues

— f'{x)(l — 2e) and vxz = f'(x)(l — e), and the associated Lyapunov exponents are [51]

AaJ!/ = Azj + l n | l - 2 e | (2.17)

for deviations away from the main diagonal in the plane IL^y, and

Axz = A£, + l n | l - e | (2.18)

for deviations away from D in the plane Hxz. If Xxy < 0, the fully synchronized state is at least weakly stable in the plane Il^j,. This implies that D attracts a positive Lebesgue measure set of points from its two-dimensional neighborhood in the synchronization plane. For the fully synchronized state to be asymptotically stable, all trajectories embedded in this state must be transversely stable. As we shall see below, regions of parameter space may exist in which Xxy < 0 while at the same time orbits embedded in the synchronized chaotic state are transversely unstable. As before, the transition in which A^ changes sign from negative to positive is referred to as the blowout bifurcation [8, 9, 10, 11].

In the partially synchronized state, the motion is restricted to the plane IIxy, and the three-dimensional map system (2.14) reduces to

The expression (2.18) for the Lyapunov exponent with respect to perturbations transverse to Il^y maintains its general form, but \o must be replaced by

^ = ^ ^ E l n i / ' w i ( 2 - 2 °) n=l

where xn, n = 1,2... now denotes the first coordinate of the points {(xn, zn)™=1\ of a typical trajectory on the partially synchronized chaotic set A embedded

F\ • X


in the synchronization plane TLxy. As illustrated in Pig. 2.16, several partially synchronized attractors may coexist for the same parameter values {a, e). Each attractor Am, rn= 1,2... will then be characterized by its own value of Xxz, A™ = \cz(An)- For the partially synchronized chaotic state to be at least weakly stable we must have A™ < 0. To proceed with the analysis let us now assume that the one-dimensional map fa(x) = ax(l — x), 3 < a < 4.

- 0 . 5 0 0.5 1 1.5 e

Fig 2.17. Overview of the main stability regioas in {a,e} parameter plane [38]. The hatched area is the region of asymptotic stability for the fully synchronized chaotic state. At the edge of this region, the fixed point becomes transversely unstable, and as we move further away from the line e = | , additional low periodic cycles become transversely unstable. The blowout bifurcation at the outer borderline of the black domain delineates the region of weak stability for the fully synchronized chaotic state. In the dark gray region around e = 1, partial synchronization may occur.

The two-dimensional phase diagram in Fig. 2.17 provides an overview of the main stability regions in {a, e} parameter plane for dynamics that is restricted to the synchronization plane Uxy [38]. The hatched area represents the region in which the fully synchronized motion on the main diagonal is asymptotically stable. In this area, all trajectories embedded in the fully synchronized state are transversely stable. At the right hand edge of the hatched region, the fixed point PQ — (XQ, XQ, XQ) , XQ = 1 — 1/a undergoes a supercritical pitchfork bifurcation and two mutually symmetric period-1 cycles (fixed points) are produced on either side of D. This occurs when the transverse eigenvalue J%(XQ)

= f'(xo)(l - 2e) becomes equal to 1, which happens at e = | (o — l)/(a — 2). For a = ao, the pitchfork bifurcation takes place at e = 0.7979. At the left


hand edge of the hatched region, the fixed point PQ undergoes a supercritical period-doubling bifurcation as the transverse eigenvalue vxy (XQ) becomes equal to —1, or for e = \{a — 3)/(a — 2). For a = a0, this occurs at e = 0.2021. Situated in the synchronization plane Hxy but off the main diagonal D these cycles represent various forms of partially synchronized behavior. To differentiate these cycles from the fully synchronized cycles, we shall refer to them as asynchronous.

As we move out of the light gray region, the period-2 cycle

a + l ± V ( q + l ) ( a - 3 ) * l , 2 = Ya

( 2 - 2 1 )

also turns transversely unstable. This occurs in period-doubling bifurcations at

e = ^ [ l ± ( ( a + l ) ( o - 3 ) - l ) s ] . (2.22)

For a = ao, the bifurcation points are e = 0.1610 and e = 0.8391. Both of these period-doubling bifurcations are supercritical and produce an asynchronous period-4 cycle off the main diagonal.

As we move farther away from the line e = \ where vxy = 0, the synchronized period-4 cycle also undergoes a supercritical transverse period-doubling to give birth to an asynchronous period-8 cycle. To the right hand side of the phase diagram this occurs for e = 0.8451 for a = ao. For slightly higher values of e (e = 0.851), the synchronized period-8 cycle likewise undergoes a supercritical transverse period-doubling giving rise to an asynchronous period-16 cycle.

Figure 2.18 illustrates the situation in the Uxy phase plane for a = ao and £ = 0.81, i.e., immediately after the transverse destabilization of the fixed point and before the transverse destabilization of the synchronous period-2 cycle has occurred [33]. Here, the synchronization manifold {x = z] represents the main diagonal D in K3. The repelling fixed point PQ on this diagonal is indicated by a full circle, and the two mutually symmetric period-1 saddle cycles produced in the initial pitchfork bifurcation are shown as small triangles. Pi represents one of these saddle cycles with its stable and unstable manifolds drawn as fine lines. The arrows on these manifolds indicate their forward direction. Also indicated in Fig. 2.18 are the points of the synchronous period-2 cycle (small squares). The curves denoted L\ and L2 are arcs of so-called critical curves [18, 19]. L\ and Z/2 are obtained from the first and the second iterates for the map F\x=y of the points [37]

L0 = {(x,z)eR2 : \DF\x=y\=0} (2.23)


Fig 2.18. Situation in phase space after the transverse destabilization of the fixed point Pa has occurred. The small squares represent the points of the synchronized period-2 cycle. Also shown in this figure is the mixed absorbing area A' as defined by the outer loops of the unstable manifolds to the asynchronous period-1 saddle cycles and by sections of the critical curves L\ and Li. The absorbing area A is bounded by sections of the critical curves [33]. a = a0 and e = 0.81.

at which the Jacobian determinant \DF\ of F\x=y vanishes. The critical curves define the absorbing area A, and, as previously noted, they play the same important role for restricting the dynamics of two-dimensional endomorphisms as the iterates of the critical point do for one-dimensional non-invertible maps.

Returning to Fig. 2.17, the fractal curve represents the moments of blowout bifurcation where Xxy becomes positive, and the fully synchronized state loses its average transverse stability. Finally, the dark gray domain delineates the region of partial synchronization. Since partial synchronization requires \xy > 0 and Axz < 0, this phenomenon can only occur to the right hand side of the diagram where e « 1. For a = ao, the blowout bifurcation at which \xy becomes positive and the fully synchronized chaotic state loses its weak stability occurs at E = 0.8563. At the rightmost edge of the dark gray region, the partially synchronized chaotic attractor disappears in a boundary crisis [52]. For a = ao, this occurs at s = 1.089.

Consider now the further development of the dynamics for a = ao- At e = 0.8486, the two asynchronous period-1 saddle cycles stabilize in an inverse,


™ **>*— _ \ "' -'«-v

1 , « ' , ' :

,' ', 4 ) ' 4-:

Fig 2.19. Basins of attraction for the synchronized chaotic state (white), for the coexisting asynchronous fixed points (light and dark gray), and for the asynchronous period-4 cycle (scattered black points). Also indicated on this figure (black circles) are the asynchronous period-2 saddle cycles whose stable manifolds define the basin boundaries for the asynchronous fixed points, o = oo and e = 0.8804.

subcritical period-doubling bifurcation in the direction of their unstable manifolds. This produces a pair of mutually symmetric period-1 stable nodes, each surrounded by an asynchronous period-2 saddle cycle. The stable manifolds of these saddle cycles delineate the basins of attraction for the stable period-1 orbits. Right after the stabilization, the immediate basins of attraction stretch as narrow tongues all the way down to the repelling fixed point PQ. With further increase of the coupling parameter, the points of the period-2 saddle cycles move farther apart, and the basins of attraction for the asynchronous period-1 nodes become wider.

This situation is illustrated in Fig. 2.19 for e = 0.8804. Here, the basin of attraction for the fully synchronized chaotic state is shown in white. The mutually symmetric stable period-1 orbits are indicated by crosses. The basins of attraction for these two solutions are shown in light and dark gray, respectively. In addition to the immediate basins issuing from the repelling fixed point (light gray cross), these basins consist of all the preimages of the immediate basins. On the boundaries of the immediate basins, the mutually symmetric

z


period-2 saddle cycles are indicated as black circles. Also shown in Fig. 2.19 are the transversely unstable synchronous period-2 cycle (light gray squares) surrounded by the points of the asynchronous period-4 cycle (crossed circles), and the transversely unstable synchronous period-4 cycle (dark gray circles).

0.92 s 1-089 0.92 S 1.089

Fig 2.20. Bifurcation diagram for solutions of family A\ that develop from one of the asynchronous period-1 cycles produced by stabilization of the saddle cycles arising in the transverse pitchfork bifurcation of the fixed point (a). Corresponding variation of the Lyapunov exponents (b). a = a0-

Each of the stable asynchronous solutions corresponds to a particular limiting state for the coupled map system. Under variation of the coupling parameter these states may develop into more complicated attractors. Let us consider the following different families of attractors in the synchronization plane Ylxy:

A\: attractors arising from the asynchronous period-1 cycles produced in the initial pitchfork bifurcation of the fixed point PQ = {x$, XQ, XQ)

AI'. attractors arising from the asynchronous period-4 cycle born in the transverse period-doubling bifurcation of the synchronous period-2 cycle

A3: two-dimensional chaotic attractors born after the blowout bifurcation of the synchronous chaotic attractor on the diagonal.

Let us start by considering the family A\. This family begins with the stabilization of the asynchronous period-1 cycles at e = 0.8486. The bifurcation diagram in Fig. 2.20(a) illustrates the development that takes place within this family as the coupling parameter is increased from 0.92 to 1.089, and Fig. 2.20(b) shows the corresponding variations of the Lyapunov exponents Ai,


Fig 2.21. Phase portrait of the two mutually symmetric four-piece chaotic attractors of family Ai that exist for a = oo and e = 0.965. Also shown is the two-dimensional attractor produced after the blowout bifurcation of the fully synchronized chaotic set. Each of these three attractors represents a state of partial synchronization [38].

and A2, in the synchronization plane 11^ and Xxz in a direction transverse to this plane. We immediately observe that A^ is negative and numerically fairly large in the interval around e = 1. On the other hand, both Ai and A2 are positive in most of the interval above e = 0.96. Hence, we have a two-dimensional chaotic attractor in the synchronization manifold, and this attractor is at least weakly stable in the direction transverse to Uxy.

Inspection of the bifurcation diagram in Fig. 2.20(a) shows that the asynchronous period-1 cycle first undergoes a Hopf bifurcation of a map (also referred to as a Neimark bifurcation) at e = 0.9305. This produces a quasiperiodic behavior on a non-resonant torus. Further developments result in the appearance of a resonant torus followed by torus destruction and a transition to chaos.

Figure 2.21 displays a phase portrait of the two mutually symmetric four-piece chaotic attractors that exist for e = 0.965 [38]. The dark and light gray regions represent parts of their basins of attraction, and the white region is part of the basin of attraction for the two-dimensional chaotic attractor of family A3 that has developed after a blowout bifurcation of the fully synchronized chaotic attractor. As e is further increased, the four pieces of the partially synchronized chaotic states merge with one another, and for e = 1 the three partially synchronized chaotic attractors merge into a single large attractor in the synchronization plane. This explains the dramatic change in the size of the attractor that can be seen in Fig. 2.20(a) for this value of the coupling


Fig 2.22. Large partially synchronized chaotic attractor that exist for a = O,Q and e = 1.07. For slightly larger values of the coupling parameter, this attractor disappears in a boundary crisis.

parameter. Figure 2.22 shows the partially synchronized chaotic attractor that exists for e — 1.07, i.e., after the merging of the three two-dimensional attractors displayed in Fig. 2.21 and before the boundary crisis.

Let us finally consider the process through which the large two dimensional attractor around the main diagonal (shown in black in Figs. 2.16 and 2.21) is born. This process is illustrated in Figs. 2.23(a)-(c) where the coupling parameter attains the values e — 0.9250,0.9255, and 0.9300, respectively. As before, the dark and light gray areas represent the basins of attraction for the mutually symmetric attractors of family A\. After the blowout bifurcation of the fully synchronized state at e = 0.8563, the four-band chaotic attractor of family A% controls the white basin of attraction. In Figs. 2.23(a)-(c), this attractor is colored black. In Fig. 2.23(a) the points of the repelling period-2 cycle on the main diagonal are indicated by squares. As the coupling parameter is increased we observe how the four-band chaotic attractor of Fig. 2.23(a) expands and merges in a process that first generates a set of so-called rare points [53J. It is also worth noticing how the final attractor in Fig. 2.23(c)


Fig 2.23. Formation of the large two-dimensional, partially synchronized attractor around the diagonal through merging of the branches of the four-band chaotic attractor: e = 0.9250 (a), 0.9255 (b), and 0.9300 (c). a = a0.

remains bounded by a set of critical curves.

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Chapter 3

UNFOLDING THE RIDDLING BIFURCATION

3.1 Locally and Globally Riddled Basins of Attraction

In the previous chapter we made a first presentation of some of the phenomena that one can observe in connection with the synchronization of two (or more) identical chaotic oscillators. Under variation of the coupling parameter we showed how a state of complete (or partial) synchronization can be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state itself remains attracting on the average.

The purpose of the present chapter is to continue our analysis of coupled non-invertible maps to show that the transition to riddling can be soft or hard depending on whether the first orbit to lose its transverse stability undergoes a super- or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to a locally riddled basin. As an illustration of this analysis we present three different scenarios for the onset of riddling and for the subsequent transformations of

75


the basins of attraction. However, let us start the discussion by recalling some of the basic concepts and ideas that we introduced in Chapter 2.

Riddled basins of attraction may be observed in regions of parameter space where the synchronized chaotic state is attracting on average (the typical transverse Lyapunov exponents are negative) while at the same time particular orbits embedded in the chaotic set are transversely unstable (the corresponding eigenvalues are numerically larger than one) [1, 2, 3, 4, 5]. The basin of attraction for the synchronized chaotic state may then become a fat fractal, riddled with initial conditions from which the trajectories diverge towards infinity or approach other asymptotic states. The transition in which the first orbit on the chaotic set becomes transversely unstable is referred to as the riddling bifurcation. For a system of two symmetrically coupled one-dimensional maps, this bifurcation can be either a pitchfork bifurcation (eigenvalue +1) [6] or a period-doubling bifurcation (eigenvalue —1) [7, 8].

However, transverse destabilization of orbits embedded in the chaotic set is not sufficient for an observable riddling to arise. This will depend on the global dynamics of the system. Having left the locally repelling regions in the neighborhood of the chaotic set, the trajectories may wander around in phase space without ever approaching another attractor (or escaping to infinity). Sooner or later most of them will return to the neighborhood of the synchronization manifold. Some may again be mapped into repelling tongues, while others will be attracted by the chaotic set, and eventually almost all trajectories starting close to the invariant manifold will end up in the synchronized state. This produces the phenomenon that has been called local riddling [3, 4, 5, 7, 8, 9, 10]. In the presence of noise, a locally riddled basin of attraction will manifest itself in the form of attractor bubbling [3, 4, 5, 11, 12], where intervals of desynchronized bursting behavior occur.

Denoting the synchronized chaotic state by A, its 5-neighborhood by Us(A), and its basin of attraction by B(A), the basin is said to be locally riddled if there exists a 6 > 0 such that A attracts almost all trajectories originating from points of Us(A), i.e., fi {B{A) D Us (A)} = fx {Us (A)}, where fi {•} denotes Lebesgue measure. In contrast to the case of asymptotic stability, the transverse repulsive character of orbits embedded in the synchronized chaotic set implies that the neighborhood of any point of A will contain a positive measure set of points that leave Us(A) in a finite number of iterations. These points still belong to B(A), however, since the trajectories will sooner or later return to

Unfolding the Riddling Bifurcation 77

the neighborhood of A.

Alternatively, the global dynamics of the system may be such that it allows direct access for trajectories repelled from the neighborhood of the transversely destabilized orbit (as well as from the neighborhoods of its dense set of preim-ages) to go to some other attractor (or infinity). This is the case of global riddling. A then attracts a positive Lebesgue measure set of points from U$(A), but not the full measure, i.e., 0 < n{B(A)r\Us(A)} < n{Us{A)}. We have previously described how the distinction between these two types of riddling depends on the existence of an absorbing area [13, 14, 15] that controls the global dynamics of the system and restrains trajectories starting near the synchronized chaotic set from reaching other limiting states. A number of definitions relating to the concepts of locally or globally riddled basins of attraction, weak attrac-tors and absorbing areas have recently been proposed by Ashwin and Terry [16]. Qualitative theoretical results for the scaling behavior of chaotic systems near the riddling transition were first reported by Ott et al. [17].

The structure of the globally riddled basins of attraction is clearly distinguishable between the cases where the trajectories that are repelled from the synchronized chaotic state approach some asynchronous state inside the absorbing area for A or diverge towards infinity. In the former case, illustrated by Fig. 2.19, the repelling tongues generally have sharp and well-defined edges that follow the stable manifolds of saddle cycles produced in the stabilization of the attracting asynchronous state. In the latter case, where the trajectories diverge towards infinity, the repelling tongues consist of bundles of trajectories issuing from the neighborhood of the unstable synchronous cycles and from their dense set of preimages. This is illustrated in Fig. 2.4.

The riddling (or bubbling) transition itself may be characterized as being either soft or hard. This distinction was introduced by Venkataramani et al. [11, 12] to describe two different situations that can be observed immediately after the first orbit has lost its transverse stability. After a soft transition, trajectories starting in the neighborhood of the synchronization manifold will remain close to this manifold. After a hard transition, on the other hand, trajectories starting close to the synchronization manifold can immediately move far away in phase space, and some may approach other attractors.

In Chapter 2 it was suggested that the distinction between a soft and a hard riddling bifurcation is related to the super- or subcritical nature of the transverse bifurcation in which the first orbit embedded in the synchronized chaotic


set loses its stability. In this chapter, we shall establish general analytical conditions for the transverse bifurcation of a periodic orbit to be either super- or subcritical. This derivation is based on the construction of an asymptotical one-dimensional map acting along the transverse invariant manifolds of the orbit that first loses its stability. In a supercritical transverse destabilization of a periodic orbit, the unstable manifolds of the asynchronous saddle cycle(s) born in the bifurcation (together with elements of the boundary of the absorbing area) will form a so-called mixed absorbing area that stretches along the synchronized chaotic set and from which trajectories starting near the chaotic set cannot escape. As the asynchronous saddle cycle(s) under variation of a control parameter move(s) away from the synchronization manifold, the width of the mixed absorbing area will grow. This leads to a synchronization error that increases as yj\e — ec\, with |e — ec| < < 1 denoting the distance of the control parameter from the bifurcation point.

As opposed to the distinction between locally and globally riddled basins of attraction, the distinction between soft and hard riddling bifurcations only involves local conditions close to the synchronization manifold. A hard riddling bifurcation may lead to locally or globally riddled basins of attraction, depending on the conditions far from the synchronization manifold. As we shall show, however, immediately after a soft riddling bifurcation, the basin of attraction can be locally riddled only.

The purpose of the subsequent sections is to illustrate these concepts in more detail by presenting three different scenarios for the onset of riddling and for the subsequent development of the basin of attraction for a system of two coupled logistic maps. We follow the location of the relevant asynchronous cycles and determine their stable and unstable invariant manifolds. We also determine both the absorbing and the mixed absorbing areas and discuss their significance for the observed dynamics. The first scenario illustrates how the transition from locally to globally riddled basins of attraction can occur via a contact bifurcation between the basin of attraction for the synchronized chaotic state and its absorbing area [13, 14, 18, 19]. The second scenario involves a direct transition to global riddling following a subcritical transverse bifurcation of a synchronized periodic orbit. In this case, the mixed absorbing area exists before the riddling bifurcation and disappears at the moment of bifurcation. Finally, in the third scenario, we follow a long and interesting sequence of bifurcations after the destabilization of the synchronous period-2 cycle in a


supercritical transverse period-doubling. In this case, the asynchronous period-4 saddle produced in the riddling bifurcation stabilizes in an inverse subcritical period-doubling bifurcation before the contact bifurcation between the basin of attraction and the absorbing area takes place. This gives rise to the emergence of a new attracting state inside the absorbing area for the synchronized chaotic set. In this connection, we also present a phase diagram delineating the regions in parameter space where the various solutions exist.

At the end of the chapter we consider a system of two coupled one-dimensional maps that each displays type-Ill intermittency [20]. This implies that the transition to chaos for the individual map takes place via a subcritical period-doubling bifurcation rather than via the usual Feigenbaum cascade of supercritical period-doubling bifurcations. We determine the regions in parameter space where the transverse Lyapunov exponent is negative so that the synchronized chaotic state is a least weakly stable. The bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the synchronized chaotic state are obtained, and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations. Our purpose is again to illustrate the role of the absorbing area in restraining the dynamics, once the transverse destabilization of the synchronized chaotic state has occurred. By following the variations in the boundaries of the absorbing area and the basin of attraction under variation of the coupling parameter we provide an alternative illustration of the contact bifurcation that causes the transition from local to global riddling. At the same time, the system of the coupled type-Ill intermittency maps exhibits a new example of the phenomenon of intermingled basins of attraction that we observed for a system of two coupled logistic maps in Chapter 2. Chaotic synchronization and riddled basins of attraction in a system of two coupled type-I intermittency maps were studied by Manscher et al. [21]. In this case, the transition to chaos in the individual map takes place via a tangent (or saddle-node) bifurcation.

Strictly speaking the weak attractor formed after a riddling bifurcation does not exist in the presence of noise or a small asymmetry (parameter mismatch). Hence, to observe experimentally the weakly attracting state the time that a trajectory on average spends near this state must be long compared with the temporal resolution of the experiment. To examine this problem, Venkatara-mani et al. [11, 12] constructed a particular two-dimensional map in which the stability transverse to a symmetric direction with chaotic dynamics could


be explicitly controlled. This approach also underlines the fact that, besides in coupled identical oscillators, riddled basins of attraction can also arise in systems with other types of symmetry, provided that this symmetry allows for a chaotic dynamics to take place in an invariant manifold of lower dimension than the full phase space. Venkataramani et al. derived scaling relations for the time that a trajectory on average spends in the neighborhood of the invariant manifold as a function of the asymmetry parameter and the amplitude of the applied noise.

As emphasised above, the critical curves and the so-called absorbing area play an important role in limiting the dynamics in phase space for two-dimensional endomorphisms (non-invertible maps). An absorbing area A for a two-dimensional map F is a closed subset of the plane, bounded by a finite number of arcs of critical curves, which is trapping in the sense that points within A can never escape, i.e., F(A) C A, and for which a neighborhood exists in which the points are mapped into A in a finite number of iterations. The absorbing area is invariant if F(A) = A.

The concept of an absorbing area has been worked out in considerable detail in a series of publications by Mira and coworkers [13, 14, 22, 23]. Here, one can also find a set of more precise definitions. (Note, however, that the French tradition with respect to the designations of the critical curves is a little different from ours which, we believe, coincide with common usage.)

In a detailed investigation of a one-parameter family of twisted logistic maps, Gardini et al. [24] have illustrated how the stationary dynamics is bounded by the critical curves, and how the global bifurcations that this dynamics undergoes can be explained in terms of collisions between the boundary of the absorbing area and the basin boundary. Finally, Celka [25] has used the concept of critical curves to estimate the synchronization regions for two unidirectionally coupled skew tent maps.

3.2 Conditions for Soft and Hard Riddling

Let us again consider the system

2-n+l = Ja\xn) T £\})n ~ xn) /o -i\

Vn+1 = fa{Vn) + £{x„ ~ Vn)

of two symmetrically coupled logistic maps fa{x) = ax(l — x) with 3 < a < 4 and - 2 < e < 2. At a = a0 = 3.678573..., the fixed point x0 = 1 - 1/a


undergoes its first homoclinic bifurcation, and two chaotic bands merge into a single band. Likewise, for a — a^ = 3.592572..., the period-2 cycle undergoes its first homoclinic bifurcation, and four chaotic bands merge into two. At each of these bifurcation points, fa(x) has an absolutely continuous invariant measure and the dynamics of fa(x) is chaotic.

In order to delineate the regions of parameter space where the synchronized chaotic state is asymptotically stable we have previously considered the transverse stability for each of the most important low-periodic point cycles 7„ (see Sec. 2.4). For a = ao we have found that the interval of asymptotic stability is bounded by a transverse period-doubling bifurcation of the period-6 cycle at e = —1.31 and by a transverse period-doubling of the period-2 cycle at e = —1.24. For a = oi, the interval of asymptotic stability for the synchronized chaotic state stretches from e = —1.46 to e = —1.16, with both the upper and the lower ends being associated with a destabilization of the period-2 cycle.

Let us now examine these bifurcations in more detail in order to establish the conditions for the transverse bifurcations to be either sub- or supercritical. The transverse destabilization of the fixed point is not associated with any riddling bifurcation in our model. However, to illustrate our method let us first examine the bifurcation in which the fixed point PQ = (XQ, XO),XQ = 1 — 1/a loses its transverse stability. The map F : M2 —> IR2 is non-invertible, and it is easy to see that the determinant \DF\ of the Jacobian vanishes along two branches of a hyperbola (the critical curves [13, 14])

L0={^v):y = + (f-^)} ^ that cross the diagonal {x = y} in the points xc\ = 1/2 and xC2 = 1/2 — e/a. It follows that F is a diffeomorphism in some neighborhood of PQ, provided that a ^ 2 and a ^ 2(1 — e). The same provisions imply that v\\ ^ 0 and v± 7 0. Moreover, the first non-resonant condition j/y ^ v±_ will be satisfied for non-vanishing values of the coupling parameter. Here, v\\ = f'(xo) = 2 — a and V]_ — f'(xo) — 2s = 2 — a — 2e are the longitudinal and transverse eigenvalues of F in PQ as given by Eqs. (2.3) and (2.4), respectively. There will then exist a neighborhood of the fixed point where F has two one-dimensional invariant manifolds W\\ and W± of Po- As illustrated in Fig. 3.1, the longitudinal manifold W\\ coincides with the synchronization manifold {x = y}, and


the transverse manifold Wj_ intersects it perpendicularly. The transverse manifold will be unique as long as the fixed point is a saddle, and the manifolds form a one-parameter family when PQ becomes an unstable node [26]. The first non-resonant condition (i/\\ ^ v±) guarantees C1-smoothness of W± in some neighborhood Us(Po) of PQ. In the interval of interest the fixed point is unstable in the direction along W\\. As discussed in Sec. 2.3, PQ loses its transverse stability when \v±_\ = 1, either in a pitchfork bifurcation for e = (1 — a)/2 or in a period-doubling bifurcation for e = (3 — a)/2.

Fig 3.1. Longitudinal and transverse invariant manifolds W\\ and W± for the map F in the

fixed point P0. US(PQ) denotes a small neighborhood around Po

ll! our analysis we approach the point of transverse bifurcation from the side

where the fixed point is transversely stable. To examine how our map acts

along the manifold W± we rewrite F in terms of the variables £ = (x + j/)/2

and ri = (y- x)/2,

where, as before, / = fa, and / ' denotes its derivative. In a neighborhood of PQ the transverse manifold may be expanded as W± = {(£, rj) : £ = <p{v)} with

ip(rj) = XQ + Br] + higher order terms. (3.4)


Linear contributions to ip(rj) are absent because W± is parallel to the r\-axis for r) = 0. To determine the coefficient B we substitute (3.4) into (3.3) and use the invariance of W±_ with respect to the action of F. This gives B = a/ (y\\ — v2^) from which B can be determined provided that the second non-resonant condition v\\ ^ v\ is satisfied. This condition guarantees C2-smoothness of W±.

In a neighborhood of PQ, the action of F along W± can now be represented by a one-dimensional map h : W± ->• W± with the asymptotic expression

h : i] ->• v\j) + Crf + higher order terms. (3.5)

Quadratic terms in this expansion vanish by virtue of the symmetry of the system. Inserting finally Eq. (3.5) into Eq. (3.3) and using our result for B we obtain C = —2a2/ (y\\ — u\).

It is easy to see that both the first and the second non-resonant conditions are satisfied for s j^ 0 and a > 3. Moreover, since un < 0 for a > 3, C is positive at both bifurcation points. Hence, the bifurcations of the one-dimensional map h in T] = 0 are sub- or supercritical depending on the sign of the product u±C. If V\C < 0, the bifurcation is supercritical and if v±C > 0, it is subcritical. In this way, we conclude that the transverse pitchfork bifurcation of the fixed point that occurs at e = (1 — a)/2 is subcritical and that the transverse period-doubling bifurcation at e = (3 — a)/2 is supercritical.

Let us now examine the bifurcations in which the period-2 cycle

a + l ± V / ( q + l ) ( a - 3 ) £1,2 = 2/1,2 = *—z (3-6)

la loses its transverse stability. It follows from the above analysis that F is a diffeomorphism in a neighborhood of both points of the symmetric period-2 cycle, provided that a ^ 1 + \/5 and o ^ l + y/A + (1 + 2e)2. These provisions also imply that i>\\ ^ 0 and v± ^ 0 where

"|| = / ' (s i ) / ' (*2) = 1 - (a + l )(a - 3) (3.7)

and

" i = (/ '(zi) - 2e)(f'(x2) - 2e) = (1 + 2ef - (a + l)(a - 3) (3.8)


are the longitudinal and transverse eigenvalues for the period-2 cycle. f'{x) denotes the derivative of f{x). Moreover, for e ^ 0 and e ^ —1, the first non-resonant condition v\\ ^ vL will be satisfied. For each of two period-2 points, a neighborhood will then exist in which the transverse invariant manifolds W±j are at least C2-smooth. Invariance in this case obviously applies with respect to the iterated map F2.

In the parameter interval of interest, the symmetric period-2 cycle is unstable in the direction of the longitudinal manifold W\\ = {x = y}. As discussed in Sec. 2.4, the cycle loses its transverse stability either in a pitchfork bifurcation (i/x = 1) for

1 £ = ~ 2

1 ± y/{a + l)(a-3) + l (3.9)

or in a period-doubling bifurcation (v± = — 1) for

1 £ = " 2

1 ± V(o + l ) ( o - 3 ) - l . (3.10)

To investigate how the map (3.1) acts along the transverse manifolds W±j we again rewrite F in the form (3.3). The term "transverse manifold" is meant to denote the manifold in which the asynchronous cycle(s) involved in the bifurcation is (are) situated. In the neighborhood of each period-2 cycle point P, we can expand the one-dimensional manifolds W±j = {(£, r]) : £ = <fi(r])} such that

ipi(rj) = Xi + BiT] + higher order terms. (3-11)

The coefficients Bi can be obtained by inserting Eq. (3.11) into Eq. (3.3) and using the invariance of W±i. This gives

Bt = a [%i+1 + i / y / foi - v\) (3.12)

where v\\ti = f'fa), v_L,i = f'{xi) - 2e, and xi+1 = f(xi). From Eq. (3.12), Bi can be calculated provided that the second non-resonant condition v\\ / v\ is satisfied. This condition will always be fulfilled since v« < 0 in the chaotic regime a > a*.

The one-dimensional mapping h, : Wj_j ->• W±j of F2 along the transverse manifolds of the period-2 cycle takes the asymptotic form


hi : 77 -» 1 77 + Ciif + higher order terms. (3.13)

As before, quadratic terms do not arise in this expansion because of the symmetry of the system. Inserting Eq. (3.13) into Eq. (3.3) and using our results (3.12) for B; we obtain [18]

Ci = —la {vl,i + u\\,i+i) ("-M+1 + "11,^-M)

-fi "x,i (3.14) "II ~ vl

The bifurcations of the symmetric one-dimensional map hi will again be super- or subcritical depending on the sign of the product u±Ci. (Obviously, Ci must have the same sign for the two transverse manifolds). If v±Ci < 0, the bifurcation is supercritical, and it is subcritical for v\Ci > 0. Direct calculation shows that Cj is positive in the relevant parameter interval. Hence, we conclude that the transverse pitchfork bifurcation of the symmetric period-2 cycle is subcritical and that the transverse period-doubling is supercritical. As an illustration to this discussion, Fig. 3.2 shows the region in parameter space in which the symmetric period-2 cycle is transversely stable. This region is delineated by the curves (3.9) and (3.10) along which the subcritical pitchfork and the supercritical period-doubling bifurcations take place.

Figure 3.3 shows the results of a numerical evaluation of the coefficients Ci for the upper point (x2, £2) of the period-2 cycle. The two curves denoted C[F and CPD refer to the transverse destabilization via a pitchfork and a period-doubling bifurcation, respectively. (It is a simple matter to obtain analytic expressions for these curves using the form (3.13). However, the expressions are somewhat complicated, and we omit them here). The curve for CPF passes through zero for a = 3 where the symmetric period-2 cycle first arises. The curve for CPD

diverges for a = 1 + y/Z = 3.236 where the longitudinal eigenvalue i/y vanishes. This is the point where the individual map fa{x) displays a superstable period-2 orbit. It is also the minimal value of a for which a transverse period-doubling bifurcation can occur (see Fig. 3.2).

The above approach can be extended to the case of an n-periodic symmetric cycle [27]. Then, the transverse manifolds W±ti asymptotically take the form (3.11) with

Bi = -^Ri{n). (3.15)


4.0

a

3.5

3.0 -2.0 -0.5 1.0

Fig 3.2. Region of transverse stability for the symmetric period-2 cycle. Destabilization occurs via a subcritical pitchfork bifurcation (lower curve) or via a supercritical period-doubling bifurcation (upper curve). For our discussion of chaotic synchronization, only the region a> a* ss 3.567 is of interest. Part of this stability regime can also be seen in Fig. 2.5.

40.0

C

0.0 3.0 1+V5 Q 4.0

Fig 3.3. Variation of the coefficients CfF and CfD associated, respectively, with the transverse pitchfork and the transverse period-doubling bifurcations of the symmetric period-2 cycle.


Here,

n n

11 = 1 1 ^ ) and "J-= I I tf'fc) - 2e) 2 = 1 8 = 1

are the longitudinal and transverse eigenvalues of the n-periodic cycle, respectively. The functions Ri(n) can be obtained from the following recurrence relations that express the results for an n-periodic cycle in terms of the results for cycles of lower periodicity [18]:

k

Ri(k + l) = vli+kRi(k) + J ] uli+j_, (3.16) J=I

with Ri(l) — 1 and k = 1,2,3...n — 1. Here, vy = / ' (x;) and I/_L,( = / ' fai) — 2e. The points x\ are supposed to be ordered in such a way that xi = fJ (a:;) with I = i + j , modulus n.

The one-dimensional mapping /i, : W±j -» W±j of F " along Wi,i takes the asymptotical form (3.13) with

n

Q = - a V - ^ ^ Q i ( f c ) (3.17)

where the functions Qi(k) can be obtained recurrently from

jfe-i

Qi(k + 1) = v\\,i+k-iQi{k) - 2a Y, rt,i+j-i (3.18)

with <9J(1) = 2Bj, (5,(2) = 21/p.Bj - 2a, and fc = 2, 3, ...n - 1.

Obviously, for a given cycle, all d must have the same sign at the moment of bifurcation \v±\ = 1- Depending on whether or not this sign agrees with the sign of u±, the transverse bifurcation will be sub- or supercritical. With this result in mind we show below that the riddling bifurcation will be soft if it arises from a supercritical bifurcation of some cycle. If it is associated with a subcritical bifurcation of a cycle, the riddling bifurcation will be hard. In the latter case the bifurcation may lead directly to a global riddling of the basin of attraction. The soft transition, on the other hand, only leads to a locally riddled basin.


3.3 Example of a Soft Riddling Bifurcation

Figure 3.4 illustrates the situation in the phase plane of our coupled map system for a = ao and s = —1.234. With this value of a the transverse period-doubling of the period-2 cycle occurs at e = —1.2373. The full line along the diagonal represents the synchronized chaotic state, and the two points on this line indicated by open circles are the points of the period-2 cycle that has just undergone a transverse period-doubling. In the vicinity of these points, the four points indicated by circles with crosses through are the points of the asynchronous period-4 cycle that has appeared in the bifurcation. The period-4 cycle is a saddle cycle with a stable manifold that connects it with the period-2 repeller. The unstable manifolds of the period-4 saddle stretch along the synchronization manifold. The arrows on the various stable and unstable manifolds denote their forward directions. It should be noted that since we are dealing with a non-invertible map, a (stable or unstable) manifold is allowed to intersect itself. Note also how the two point pairs of the period-4 cycle, in accordance with the above discussion, have moved different distances away from the period-2 cycle. Also shown in Fig. 3.4 is the fractal boundary of the basin of attraction for the synchronized chaotic state.

In order to demonstrate a number of important details, Fig. 3.5 displays a magnification of the upper part of the phase plane. Here e = —1.225. A denotes (part of) the synchronized chaotic set, and Pi and P4 are points on the period-2 repeller and the period-4 saddle cycle, respectively. Finally, on the boundary of the basin of attraction (and indicated by small triangles in Fig. 3.5) we find the points P$ of a period-8 repeller. Any neighborhood of this repeller contains a positive measure set of initial conditions from which the trajectories diverge towards infinity.

As bounded by the outmost loops of the unstable manifolds for the period-4 cycle (drawn as thin lines) and by segments of the critical curves L\ — L& (drawn as heavier lines), the mixed absorbing area A [13, 14, 28] completely surrounds the synchronized chaotic set. With the situation depicted in Fig. 3.5, the mixed absorbing area is a closed invariant set, i.e., F(A') = A', and trajectories starting inside the mixed absorbing area (or at its boundary) cannot leave it. A trajectory starting near the synchronization manifold {a; = y} will move along this manifold until it comes close enough to the point P2 of the unstable period-2 cycle. The trajectory will then move out towards P\. How-


1.25

y

-0.05 -0.05 x 1.25

Fig 3.4. Situation in phase plane immediately after the supercritical transverse period-doubling of the symmetric period-2 cycle. The figure shows the period-4 saddle cycle with its stable and unstable manifolds. Also indicated are the absorbing area as bounded by segments of critical curves L^ and the basin of attraction with its fractal boundary, e = —1.234.

ever, as it approaches this point it will be guided along the unstable manifold of Pi back towards the synchronization manifold. Moreover, most trajectories starting from a neighborhood outside A' will follow the unstable manifolds of the period-4 saddle, fold at the critical curves (L\ or L3), and cross into A'. We conclude that as long as a mixed absorbing area exists inside the basin of attraction (and no other attractor exists within the mixed absorbing area), the synchronized chaotic state will attract almost all points from its neighborhood. A is then a Milnor attractor, and its basin of attraction can be locally riddled only. In this way the formation in a supercritical transverse bifurcation of a mixed absorbing area that stretches along the synchronized chaotic state plays a major role for restraining the amplitude of the bursts away from the synchronized state.

Surrounding the mixed absorbing area in Fig. 3.5 we find the absorbing area A (also referred to as the trapping region [29, 30]). Bounded by a finite number of segments Lk = Fk(Lo) of images of the critical curves LQ as given by (3.2), the


1.0

y

0.66

0.66 X 1.0

Fig 3.5. Magnification of the upper part of Fig. 3.4. Here the coupling parameter is s = —1.225. A denotes (part of) the synchronized chaotic set, and Pi and P4 are points of the period-2 repeller and the period-4 saddle cycle, respectively. Points Pg denoted by a small triangle belong to a period-8 repeller that is situated at the boundary of the basin of attraction.

absorbing area [13, 14, 15] has the property that trajectories that enter this area cannot leave it again, i.e., F(A) Q A. Moreover, trajectories that start in the neighborhood of the absorbing area will enter it in a finite number of iterations. The presence of an absorbing area is a characteristic feature of non-invertible maps. For our system of two coupled logistic maps, the absorbing area exists in an interval of the coupling parameter that includes part of the interval of asymptotic stability for the synchronized chaotic state and can stretch beyond this interval. When, as illustrated in Fig. 3.5, the mixed absorbing area falls fully within the absorbing area (and no attracting state other than A exists in A), almost all trajectories starting in the absorbing area will enter the mixed absorbing area, to finally be attracted by A.

As we continue to increase the coupling parameter, the points P4 of the period-4 saddle cycle move further out along the transverse manifolds of the period-2 repeller, and the mixed absorbing area A' continues to grow until it fills out most of the absorbing area A- This is illustrated in Fig. 3.6, where e = —1.21. The critical curves Lfc will serve as an envelope to the unstable


0.97

y

0.86

0.67 x 0.86

Fig 3.6. For e = —1.21 the period-4 saddle cycle has moved further out along the transverse manifolds of the period-2 repeller, and the mixed absorbing area A' covers most of the absorbing area .4. At the same time the period-8 repeller has moved closer to the common boundary of A and A'.

manifolds, and as long as the period-4 cycle falls within the boundary of the absorbing area, its unstable manifolds will be restrained to this area. Compared with Fig. 3.5, the period-8 repeller (and hence the basin boundary) has moved closer to the boundary of the absorbing area. At e = —1.205 a crisis takes place as the boundary of the absorbing area comes in contact with the basin boundary, and the period-8 repeller crosses into the region delineated by the critical curves. We can see that this happens in points where the boundaries of A and A' coincide. This marks the transition from local to global riddling of the basin of attraction.

Figure 3.7 shows the situation immediately after this contact bifurcation has occurred. Here, e = —1.18. The points of the period-8 repeller now fall inside the region delineated by the images of the critical curves, the absorbing area has ceased to exist, and direct access has been opened for points starting near the transversely unstable period-2 cycle (and its dense set of preimages) to diverge to infinity. This is a typical example of global riddling. One can examine this transition in more detail by constructing the preimages of those tongues of the


1.0

y

0.55

0.55 X 1.0

Fig 3.7. Crossing of the period-8 repeller through the segments of critical curves and unstable manifolds that define .4, and A' marks the transition from locally to globally riddled basins of attraction. There is now direct access from the neighborhood of the transversely unstable period-2 cycle (and its dense set of preimages) to infinity.

basin of infinity that have penetrated into the region A fl A'. Immediately after the boundary crisis these preimages can be followed backwards along the unstable manifolds that define the boundary of A' towards the period-4 saddle cycle P4 and from here along the transverse manifolds back towards the period-2 repeller P<i on the main diagonal.

To complete the present section, let us note that while varying the coupling parameter from the point of transverse destabilization of the period-2 cycle to the final boundary crisis, we have not found evidence of the presence of attracting sets in A' other than the synchronized chaotic state. This is the reason why we have been able to say that the basin of attraction for A is locally riddled. However, we cannot exclude that this is a result only of our finite numerical accuracy. Our simulations have revealed how the stable and unstable manifolds of the period-4 saddle intersect in many points. This implies the existence of homoclinic trajectories and of the complicated dynamics associated with a Smale horseshoe. It is well-known that under specific conditions this can lead to the formation of so-called Newhouse regions in parameter space with


a dense set of systems having a large or even an infinite number of attracting cycles [31, 32]. The presence of such regions with unremovable homoclinic tangencies causes enormous problems, mathematically as well as numerically, and we shall not discuss their consequences here.

We consider the above scenario describes the generic transition from locally to globally riddled basins of attraction following a supercritical riddling bifurcation in coupled map systems when the asynchronous cycle born in the bifurcation does not stabilize before the contact bifurcation between the absorbing area and the basin of attraction has occurred. Note that when the contact bifurcation occurs, the boundaries of the absorbing and the mixed absorbing areas coincide. Hence, the bifurcation involves a minimal, invariant absorbing area.

In Sec. 3.5 we shall discuss a very different and much more complicated riddling scenario that arises in our system for a = a\. Here, the asynchronous saddle cycle born in the riddling bifurcation stabilizes in an inverse period-doubling bifurcation to produce an attracting state inside the absorbing area for the synchronized chaotic set. First, however, let us consider a generic scenario for the appearance of globally riddled basins of attraction following a hard riddling bifurcation that can also be observed for a = ao-

3.4 Example of a Hard Riddling Bifurcation

At the other end of the interval of asymptotic stability for a = ao, a transverse destabilization of the symmetric period-6 cycle takes place . This again occurs via a period-doubling bifurcation (I/_L = — 1). However, evaluation of the parameters Ci shows that the bifurcation is subcritical (d < 0). Hence, before the transition occurs, the points of a period-12 repeller are situated on either side of the symmetric period-6 cycle. This situation is illustrated in Fig. 3.8(a) where we also can see the fractal boundary of the basin of attraction for the synchronized chaotic state. Closer examination reveals that the points of the period-12 repeller fall at cusp points of the basin boundary, and any neighborhood of these points contains a positive measure set of initial conditions from which the trajectories diverge to infinity.

Figure 3.8(b) is a magnification of part of the structure in Fig. 3.8(a). Here, the coupling parameter e = —1.3. The figure shows (the upper part of) the synchronous chaotic state with three points of the symmetric period-6 cycle


-0.05 X 1.25 0.7 X 0.94

Fig 3.8. (a) Overview of the situation in phase plane before the transverse destabilization of the symmetric period-6 cycle. The figure shows the asymmetric period-12 repeller with its longitudinal manifolds, (b) Magnification of part of (a). Note how the absorbing area falls inside the mixed absorbing area, which now exists before the riddling bifurcation.

situated along the main diagonal. Also shown is the absorbing area as bounded by segments of the critical curves L\ — L\± (heavy lines). The dots indicate where these segments connect. Outside the absorbing area we find six points of the period-12 repeller (shown as circles with crosses through). As noted above, these points fall at cusp points of the basin boundary. In the longitudinal direction the unstable manifolds of the period-12 repeller again seem to wind around the synchronized chaotic state. However, since these manifolds now fall outside the absorbing area, we can not be sure that they will always remain bounded in the vicinity of the synchronous set. In the present situation this appears to be the case, since the unstable longitudinal manifolds all connect to the critical curves and, hence, define a mixed absorbing area (which now exists before the riddling bifurcation).

As we approach the bifurcation, the points of the period-12 repeller move through the critical curves Lk, and the absorbing area ceases to exist. Unstable manifolds now protrude out through the images of the critical curves, and trajectories starting outside of the longitudinal manifolds of the period-12 repeller will diverge. This situation is illustrated in Fig. 3.9 for e = —1.307. Finally, at the point of bifurcation where the period-12 repeller disappears, direct access is opened from the symmetric period-6 cycle (as well as from its dense set of preimages) to infinity. The emergence of global riddling in this way


0.94

y

0.7

0.7 x 0.94

Fig 3.9. For e — —1.307 the period-12 repeller has moved across the critical curves, and the absorbing area has ceased to exist.

occurs simultaneously with the destabilization of the period-6 cycle in a local bifurcation.

Based on the above results we conclude that the riddling bifurcation will be hard if it is associated with a subcritical destabilization of some orbit. The subcritical bifurcation does not produce asynchronous saddle cycles whose unstable manifolds can restrain the bursts of trajectories. It is possible, of course, that saddle cycle(s) of appropriate periodicity could exist outside the repelling cycle(s) involved in the transverse destabilization (an example is provided by the model considered by Venkataramani et al. [11, 12]). In this case the sub-critical riddling bifurcation might not directly produce a globally riddled basin of attraction.

3.5 Destabilization Scenario for a = oi

Our last scenario concerns the sequence of events that take place in connection with the destabilization of the synchronized chaotic state for a = a\ and e = — 1.1. This scenario starts with the transverse destabilization of the symmetric period-2 cycle in a supercritical period-doubling bifurcation for e = —1.1560. The blowout bifurcation at which the transverse Lyapunov exponent becomes


positive and the chaotic state loses its average attraction occurs at e = —1.0385. Hence, we shall follow the bifurcations that take place in the interval between e = -1.1560 and e = -1.0385.

For a — ai, the synchronous chaotic state consists of two separate bands. The absorbing and mixed absorbing areas each therefore also consists of two regions that are mapped one into the other under the action of F. Figure 3.10(a) shows the upper part of the phase plane after the transverse period-doubling has occurred. Here, e — —1.155. We note the two points of the asynchronous period-4 saddle cycle situated along the transverse manifolds of the symmetric period-2 repeller. The points of the period-4 saddle are indicated as circles with crosses through. Together with segments of the critical curves L\ and L3, the unstable manifolds of the period-4 saddle define a mixed absorbing area A'. Around this area we find a large absorbing area A as bounded by segments of the iterates L\, L2, £3 and L4 of LQ.

0.75 X 0.95 0.85 x 0.89

Fig 3.10. (a) Upper part of phase plane for a = a and e = —1.155. We observe a larger and a smaller absorbing area together with a mixed absorbing area delineated by the unstable manifolds of the asymmetric period-4 saddle, (b) Magnification of part of the phase plane for e = —1.15597. Note how the transverse manifolds of the period-2 repeller pass right between the critical curves.

Trajectories starting inside the mixed absorbing area cannot escape from it. Except for a measure zero set of trajectories starting from points on the external branches of the stable manifolds of the asynchronous period-4 cycle (and possible preimages of these manifolds), trajectories starting from a neighborhood of the mixed absorbing area will move along the unstable manifolds


of the period-4 saddle points towards the points where they meet the critical curve L3 (or L{). After a folding here they will be mapped into the mixed absorbing area, soon to be trapped in the smaller absorbing area denned by the critical curves L\, L3, L5, Lj, L$, L§, L\Q, and Lu- The transverse manifolds of the synchronous period-2 cycle pass right between the critical curves L5 and L7. Hence, in spite of the fact that the period-2 cycle has a dense set of preimages along the main diagonal, not a single trajectory will be able to leave the smaller absorbing area.

Comment: The smaller absorbing area represents the absorbing area that existed before the homoclinic bifurcation in which the four-band chaotic at-tractor for the individual map has merged into a two-band attractor. The smaller absorbing area is destroyed in the homoclinic bifurcation, and as soon as the nonlinearity parameter a exceeds a\ by as little as a part in 1011 one can start to observe how trajectories escape along the transverse manifold of the synchronous period-2 cycle.

Immediately after the transverse destabilization of the period-2 cycle (e.g., for £ = —1.15597 as shown in Fig. 3.10(b)) the period-4 saddle points will be situated very close to the main diagonal, and its unstable manifolds stretch along the synchronized chaotic state as a narrow band from which trajectories cannot escape. Hence, we again observe that a supercritical transverse bifurcation leads to a soft riddling transition with a locally riddled basin of attraction and with small and smoothly growing bursts of trajectories away from the synchronized state. As we move further away from the bifurcation point, however, a completely different sequence of events takes place. At e = —1.09571 the period-4 saddle cycle undergoes an inverse, subcritical period-doubling in the direction of its unstable manifold. This produces a stable asynchronous period-4 cycle surrounded on both sides by the points of a period-8 saddle cycle.

This situation is illustrated in Fig. 3.11, where a point P4 of the stable period-4 cycle is marked by a small square and the neighboring points Ps of the period-8 saddle by circles with small crosses. Situated along the main diagonal and embedded in the synchronized chaotic attractor A, the upper point of the period-2 repeller is indicated by a small triangle. Also shown in this figure are the stable and unstable manifolds of the period-8 saddle. The white region B(oo) in the top right corner belongs to the basin of infinity. With a gray shading, the area B(A) is the basin of attraction for the synchronized chaotic state. As in the previous figures, the boundary between these two basins is


Pig 3.11. The asymmetric period-4 cycle has undergone an inverse, subcritical period-doubling transition producing a stable period-4 cycle F4 and a period-8 saddle Pg. The stable manifolds of this saddle delineate the immediate basin for the period-4 node. The figure was obtained for e = -1.08.

fractal. The inverse period-doubling bifurcation at e = -1.09571 has produced a

new attracting state inside the former basin of attraction for the synchronized chaotic state. The immediate basin for the period-4 cycle (shown hatched in Fig. 3.11) is defined by the stable manifolds of the period-8 saddle. Emanating from the symmetric period-2 repeller, these manifolds delineate a basin that stretches as a set of tongues all the way down to the main diagonal. Immediately after the stabilization of the period-4 cycle these tongues will be very narrow and they will not intersect the smaller absorbing area. As a consequence, trajectories that start within this area will not be able to reach the stable period-4 cycle, and the basin of attraction for the synchronized chaotic state will remain locally riddled only.

Besides the immediate basin, the basin of attraction for the asynchronous period-4 cycle also consists of a set of secondary tongues. Most prominent among these are the tongues that emanate from the points where the critical curves L\ and L3 intersect the main diagonal. These secondary tongues are


also clearly visible (hatched) in Fig. 3.11. Like the primary tongues, they are delineated by sharp and well-defined edges.

If the value of the nonlinearity parameter a exceeds a\ by the smallest amount, the smaller absorbing area ceases to exist. The basin of attraction for the asynchronous period-4 cycle will then include tongues that emanate from the dense set of preimages of the period-2 repeller in the synchronized chaotic set A, and the basin of attraction for A will be globally riddled. In this case the transition to global riddling is accomplished via two local bifurcations: first the transverse supercritical period-doubling of the symmetric period-2 cycle and thereafter the stabilization of the asynchronous period-4 cycle in an inverse subcritical period-doubling. However, the new tongues tend to be extremely narrow, and in the numerical calculations they show up only as randomly scattered points within the area that has otherwise been assigned to B{A).

A similar transition to global riddling has recently been described by As-takhov et al. [33] for a pair of nonlinearly coupled logistic maps. In their main scenario it is the fixed point that first undergoes a transverse destabilization in the form of a supercritical period-doubling. After similar period-doubling bifurcations of the symmetric period-2, period-4, period-8, and period-16 cycles, the asynchronous period-2 saddle cycle produced in the first transverse bifurcation stabilizes in an inverse, subcritical pitchfork bifurcation, giving way to a globally riddled basin of attraction for the synchronous chaotic state.

It is worth noting, however, that the globally riddled basins of attraction created through these processes have a fairly unusual structure. Firstly, the main tongues are delineated by the stable manifolds of a saddle cycle (or a pair of saddle cycles). In contrast, in the commonly described form of global riddling [1, 2, 29, 30], the repelling tongues are defined only in terms of bundles of trajectories that follow an unstable manifold away from the synchronized chaotic state. In addition, there is a prominent set of secondary tongues that also have sharp and well defined edges. Finally, the remaining tongues form an extremely thin structure that shows up in the numerical calculations only as randomly scattered points from which the trajectories eventually reach the asymmetric point attractor.

As the coupling parameter is further increased, the phase portrait starts to become complicated. At e = —1.085, the asynchronous period-4 cycle is transformed from a stable node into a stable focus (i.e., the eigenvalues become


complex conjugated). At e = -1.0625 a subharmonic saddle-node bifurcation [34] takes place in which a stable period-12 cycle is born together with a period-12 saddle. This situation is illustrated in Fig. 3.12 where e = —1.0615.

Fig 3.12. A stable period-12 cycle has been born together with a period-12 saddle in a subharmonic saddle-node bifurcation. The stable manifold of the period-12 saddle defines disc-shaped immediate basins for the period-12 node. Here, e = —1.0615.

We now have two coexisting stable solutions in the repelling tongues emanating from the symmetric period-2 repeller. In Fig. 3.12, a point P4 of the period-4 focus is indicated by a square drawn with relatively heavy lines and the points Pn of the period-12 node by finer squares. The immediate basin of attraction for the period-12 cycle takes an unusual form namely as a set of discs surrounded by topological circles which are formed by the stable manifold of the period-12 saddle (points of which are shown as small circles). This peculiar structure, in which the same stable manifold as a closed curve approaches the saddle point from both sides, is made possible by virtue of the non-invertibility of the map F. This non-inverfcibility allows the preimages of the period-12 saddle cycle to serve as points of separation between the two directions of the manifolds. The complete basin of attraction for the period-12 node consists of these immediate disc-formed domains together with all their preimages in

Unfolding the Rtddhng Btfurcation 101

the main tongues as well as in their preimages. A similar basin structure was observed by Maistrenko et al. [35] for a two-dimensional, non-invertible map derived from a problem in radiophysics.

The next transformations of the phase portrait to occur involve a Hopf bifurcation of the period-4 focus at e = —1.0581 and a Hopf bifurcation of the stable period-12 solution at e = —1.0562. The second Hopf bifurcation produces a period-12 invariant circle (referred to as a torus) Tn corresponding initially (with high probability) to quasiperiodic motion. With further increase of the coupling parameter, the dynamics becomes non-invertible and the torus breaks down [36], producing chaos filled by windows of periodicity. In both cases we can have either a single or two coexisting and mutually symmetric attractors. For e € [—1.054, —1.053], for instance, the dynamics is reduced to two symmetrically located stable period-12 solutions and two saddle cycles of the same period.

1.03

y

0.94 0.6 X 0.78

Fig 3.13. For e = —1.0513, a single 12-piece chaotic attractor exists in the repelling tongues emanating from the symmetric period-2 repeller. As the coupling parameter is further increased, the 12-piece chaotic attractor merges into a 4-piece attractor finally to disappear together with its basin of attraction in a boundary crisis.

For other values of the coupling parameter, the dynamics produced through


the breakdown of Tn involves two coexisting chaotic attractors. For e = — 1.0517, for instance, two 5-12-piece chaotic attractors exist. At e = —1.05134, a single 12-piece chaotic attractor is born and for slightly higher values of the coupling parameter, the 12-piece attractor merges into a four-piece attractor. This last process gives rise to the appearance of so-called rare points [37], indicating that merging takes place across a fractal basin boundary. Finally, at e = —1.0458 the chaotic attractor and its basin of attraction disappear in a boundary crises, leaving only a chaotic saddle of period 4 in the region around the original asynchronous period-4 cycle. Figure 3.13 shows a phase portrait for £ = —1.0513. For this value of the control parameter, our coupled map system F displays a 12-piece chaotic attractor situated in the repelling tongues issued from the points of the symmetric period-2 cycles. As described above, this attractor has been produced via breakdown (loss of differentiability) of the 12-piece torus T\2- Also shown in the figure are the locations of the period-4 and period-12 cycles, both of which are now unstable focuses.

This completes our discussion of the complex scenario that unfolds after the supercritical transverse destabilization of the symmetric period-2 cycle. After the asynchronous chaotic attractor has disappeared in a boundary crisis, no other attracting set is observed inside the larger absorbing area, and the basin of attraction for the synchronized chaotic state remains locally riddled until, at e = —1.0385 a blowout bifurcation takes place. In this bifurcation the typical transverse Lyapunov exponent for trajectories on A becomes positive, and the synchronized chaotic state loses its average attraction. However, the larger absorbing area A still exists inside the basin of attraction, and the blowout bifurcation leads to on-off intermittency. The chaotic attractor spreads over the whole area of the absorbing set. However, trajectories starting near the synchronization manifold cannot diverge to infinity. For slightly higher values of the coupling parameter, the two-dimensional chaotic attractor undergoes an inverse boundary crisis by which it decomposes into two mutually symmetric chaotic attractors. For e = — 1, these attractors are both restricted to one-dimensional manifolds, and as discussed in Sec. 2.6 one can observe the phenomenon of intermingled basins of attraction. Each attractor here has a basin that is riddled with initial conditions that lead to the other attractor. The mutually symmetric chaotic attractors finally disappear in a boundary crisis at e = —0.935. When this occurs, the larger absorbing area in Fig. 3.10 has made contact with the basin boundary and, hence, has ceased to exist.


Fig 3.14. (a) Phase diagram illustrating the region (vertically hatched) in which attracting states exist inside the repelling tongues emanating from the symmetric period-2 cycle. The triangular region to the left is the region where the symmetric period-2 cycle is transversely stable (compare with Fig. 3.2). (b) Magnification illustrating the region of stability for the period-12 cycle (horizontally hatched). The curve marked CB represents the contact bifurcation.

The phase diagram in Fig. 3.14(a) gives an overview of the main bifurcations involved in the above scenarios and clarifies the difference between the scenarios observed for a — ao (Sec. 3.3) and for a = a\ (Sec. 3.5). In this diagram the hatched region to the left represents parameter values for which the symmetric period-2 cycle is transversely stable (compare with Fig. 3.2). At the right hand edge of this region, the period-2 cycle undergoes a supercritical transverse period-doubling producing a period-4 saddle cycle outside the main diagonal. The vertically hatched finger to the right of this bifurcation curve represents the regions of parameter space in which attracting states exist in the repelling tongues emanating from the points of the period-2 repeller. The left hand edge of this finger is the bifurcation curve in which the asymmetric period-4 saddle stabilizes in an inverse, subcritical period-doubling. The dashed curve running through the middle of the finger is the Hopf-bifurcation curve for the period-4 cycle, and the right hand edge of the finger represents the boundary crisis in which a 4-piece chaotic attractor disappears through collision with its basin boundary.

The curve marked CB represents the contact bifurcation between the absorbing area and the basin boundary for the synchronized chaotic state A. For a = ao, as the coupling parameter is increased, the contact bifurcation occurs


before the stabilization of the asynchronous period-4 cycle. This leads to the scenario that we have described in Sec. 3.5. Here, the transition from local to global riddling of the basin of attraction for A takes place via the contact bifurcation. For a = a\, on the other hand, the asymmetric period-4 cycle stabilizes before the contact bifurcation, and the scenario described in the present section takes place. If the nonlinearity parameter is precisely equal to a\, we do not observe a transition to global riddling. For a value of a slightly larger than a\, however, we observe a transition to the peculiar form of global riddling that manifests itself in the form of randomly scattered dots within the region otherwise assigned as the basin for A [33].

Figure 3.14(b) provides a little more detail to the bifurcation structure. Again, the vertically hatched region represents parameter values for which stable solutions exist inside the repelling tongues issued from the symmetric period-2 cycle. The left edge is the bifurcation curve in which the asynchronous period-4 cycle stabilizes, and the right edge represents the boundary crisis where the repelling tongues disappear. The narrower, horizontally hatched area is the region where the stable period-12 cycle exists. This cycle arises in a subhar-monic saddle-node bifurcation to be destabilized at higher values of the coupling parameter in a Hopf bifurcation leading to the above mentioned 12-piece torus T\2. The dashed curve running through the middle of the finger again represents the bifurcations in which the asynchronous period-4 cycle loses its stability via a Hopf bifurcation.

At the other end of the interval of asymptotic stability for a — a\, the transverse destabilization takes place in the form of a subcritical pitchfork bifurcation of the symmetric period-2 cycle at e = —1.46. This leads directly to a globally riddled basin of attraction. Let us finally note that for a = a\, there is an additional interval of weak stability for the synchronized chaotic state around e = 0.3. However, this interval does not contain regions of asymptotic stability for the chaotic set. Nor is there an absorbing area. Hence, the basin of attraction is globally riddled in the whole interval.

3.6 Coupled Intermittency-III Maps

The purpose of the following three sections is to discuss the same type of phenomena that we have examined for coupled logistic maps for a very different pair of maps in which the transition to chaos takes place via a subcritical


period-doubling rather than via the usual Feigenbaum cascade of supercritical period-doubling bifurcations. This also allows us to present some of the phenomena from a different point of view.

Let us consider the two-dimensional map F = F^£ : R2 —> R2 defined by

P. J x \ _ > J Ux)+£(y-x) \ / o 1 Q \ F-\yi^\U(y) + e(z-y)f (3'19)

where /M = / : R1 -> R1 is the one-dimensional map

/M : x -»• - ((1 + n + 7) x + x3) e~bx2 + jx (3.20)

and e 6 R1 is the coupling parameter, fi E R1 is used as the bifurcation parameter for the individual map, and in most of the analysis the remaining parameters are kept constant at b = 0.14 and 7 = 0.02, respectively.

The one-dimensional map /M has previously been investigated by Laugesen et al. [38] with the aim of explaining the anomalous statistics for the length of the laminar phases associated with the strongly nonuniform reinjection mechanism for 7 = 0. These authors have also examined part of the bifurcation structure for a system of two symmetrically coupled maps corresponding to F^. An antisymmetric version of Eq. (3.19) has been considered by Laugesen et al. [39] as an example of a map that exhibits type-II intermittency with a nearly homogeneous reinjection mechanism.

For 7 < 1, the map f^x) has a unique fixed point x* = 0. Here, the slope is / (0) — —(1 + A*), implying that the fixed point is stable for — 2 < n < 0. At fi = 0, a period-doubling bifurcation takes place, and since / (0) = 0 and / (0) = — 6 (1 — b (1 + 11 + 7)) this bifurcation is subcritical for 6(1 + 7) < 1. The function f^ix) also provides an efficient reinjection mechanism: When x becomes large, the factor e~bx becomes very small, and the next iterate may fall close to the unstable fixed point. The parameter 7 has been introduced to control the reinjection process. For b = 0.14 and 7 = 0.02, the iterate of the critical point f(xc) ( / (xc) = 0) falls outside the point XQ ( / (zo) = 0) where the map changes sign. Hence, the critical point is not mapped directly into the neighborhood of the unstable fixed point, and the reinjection is reasonably uniform. Figure 3.15 shows the map /M(a;) with a typical itinerary.

The bifurcation diagram in Fig. 3.16 illustrates how the stable fixed point x* = 0 for (j, = 0 undergoes a subcritical period-doubling. Hereafter, the map fn{x) displays type-Ill intermittency, interrupted by periodic windows. It


10

8

6

4

2

y o

-2

-4

-6

-8

-10 -10 -8 -6 -4 -2 0 2 4 6 8 10

x

Fig 3.15. Iteration of the map f^x) = - ((1 + n + 7) x + x3) e~bx2 + •yx for b = 0.14, 7 = 0.02, and n = 0.1. The iterate of the critical point xc falls outside the point xo where the map crosses the horizontal axis. This provides a reasonably homogeneous reinjection near the unstable fixed point x* = 0.

appears, however, that the prevalence of such windows is relatively low, and we shall assume that the individual map generates chaotic dynamics for the parameter values considered in the following analyses. The global dynamics of fn{x) is such that it allows for the existence of a dense set of unstable periodic orbits as required for deterministic chaos to occur. In the considered parameter range, the number of cycles of period 2, 4, and 6 are 3, 12, and 16, respectively.

Along the main diagonal {x = y}, the coupling vanishes, and the dynamics coincides with that of the one-dimensional map f^(x). If / C M1 denotes a chaotic attractor for /M(z), then A = {x = y G / } will be a one-dimensional invariant chaotic set for the coupled map system (3.19). This implies that the main diagonal is an invariant manifold for F^. For any point on this manifold, F^ has an eigenvector along the diagonal and an eigenvector perpendicular to it. In accordance with our results in Chapter 2, the corresponding eigenvalues are

/

':/

' ' ' ' LI S /

' \

V W"

\

•

/"Xo

-

"II = flip) (3.21)


X 0

Fig 3.16. One-parameter bifurcation diagram for f^(x) with b = 0.14 and 7 = 0.02. For H = 0, the stable fixed point at x* = 0 undergoes a subcritical period-doubling and the system hereafter exhibits type-Ill intermittency. Considered from the right, the observable windows have periodicities 16, 9, 20, 11, 24, 13, etc.

and

vx = / ; (*) - 2e, (3.22)

respectively. As previously noted, Alexander et al. [29, 30] have shown that if (i) A is a

finite union of intervals, and (ii) the invariant measure of fp{x) on A is absolutely continuous, then a negative value of the transverse Lyapunov exponent Ax guarantees that A attracts a positive Lebesgue measure set of points from any two-dimensional neighborhood Ug(A). This implies that the synchronized chaotic state is at least weakly attracting. For our system of two symmetrically coupled intermittency-III maps, the condition (ii) of an absolutely continuous invariant measure for /M(a?) is not necessarily fulfilled. Nonetheless, we shall use the transverse Lyapunov exponent as given by

1 N I < (3.23)

as indicator of the average transverse stability for the synchronized state. As before, [x„ = f™{x)}™ denotes a typical itinerary of A. Figure 3.17 shows the region of /ie-parameter plane in which Ax < 0 (hatched with downwards sloping lines). For /z = 0, the interval of weak stability extends from e = — 1 to e — 0. As /x increases, the interval gradually shifts towards more negative


2.0

1.5

1.0

0.5 £ 0.0

-0.5

-1.0

-1.5

-2.0 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

Fig 3.17. Region in parameter space where the synchronized chaotic state is weakly stable (downwards sloping hatch). Also shown are the regions in which the fixed point and the symmetric period-2 cycle (upwards sloping hatch) are transversely stable. Since there is no overlap between these regions, the synchronized chaotic state is never asymptotically stable.

values of the coupling parameter while at the same time decreasing slightly in width. Note the characteristic fluctuations of the borderlines for the region of negative transverse Lyapunov exponent. Additional stability regions arise whenever /M(a;) displays a periodic window.

For the synchronized chaotic state to be asymptotically (or strongly) stable all orbits embedded in this state must be transversely stable. For the fixed point, for instance, the transverse eigenvalue is u± — f (0) — 2e, and the condition of transverse stability for \i > 0 gives

- l - f < e < - f - (3-24)

The interval in which the fixed point is transversely stable is also shown in Fig. 3.17. To a large extent this interval coincides with the range of weak stability for the chaotic set A.

For comparison with the region of weak stability, Fig. 3.17 also shows the region in which the symmetric period-2 orbit is transversely stable (hatched with upwards sloping lines). This region falls entirely in the realm where e > 1. Since there is no overlap with the region in which Aj_ < 0, the synchronized chaotic state cannot be asymptotically stable. Neither can we observe the actual riddling bifurcation.


Fig 3.18. Regions of transverse stability for the symmetric period-4 cycles (a), and the symmetric period-6 cycles (b). Destabilization occurs either through a pitchfork or through a period-doubling bifurcation, and both of these bifurcations can be either sub- or supercritical.

To complete the investigation of the transverse stability of low-periodic orbits embedded in the synchronized chaotic state, Figs. 3.18(a) and (b) show the bifurcation curves in which the symmetric period-4, respectively the period-6 orbits lose their transverse stability. Again, the regions of transverse stability are hatched. We note that there are regions of parameter space where all of these cycles are transversely stable while at the same time Ax < 0.

3.7 The Contact Bifurcation

As previously noted, the emergence of an observed riddling after the transverse destabilization of the first low periodic orbit depends on the global dynamics of the system. Here, the existence of an absorbing area inside the basin of attraction for the synchronized chaotic state plays an important role.

Figure 3.19 shows the construction of the boundary dA of the absorbing area for the synchronized chaotic state of FMi£ with /x = 0.07 and e = —0.585. The critical curve LQ = {(x,y) : [DF^ = 0}, at which the Jacobian of F^ vanishes, does not directly play a role. Its first iterate L\ — F^E(Lo) produces the smooth curve that outlines the form of the absorbing area along the upper right periphery. L\ is also involved in defining dA near the lower left corner. The second iterate L2 demarcates part of the boundary in the upper left corner. It is clear that the absorbing area can have a very complicated boundary, and that


15

10

5

y o

-5

-10

-15 -15 -10 -5 0 5 10 15

x

Fig 3.19. Construction of the boundary of the absorbing area through iterates of the critical curves, /i = 0.07 and s = -0.585. Trajectories started within the absorbing area are restrained by the nonlinearities of the map F^e to remain in this area.

a large number of images of the critical curves may be involved in determining the precise border. In the present case, the construction of the absorbing area may be simplified considerably, however, by noting that dA is symmetric both with respect to the line {x — y} and with respect to {x = —y}. This symmetry allows us to determine the most essential aspects of dA from L\ and L%.

As the coupling parameter e varies, the absorbing area A(A) may grow or decrease. At the same time, the basin of attraction B{A) will also change its size, and at a certain parameter value a global bifurcation (a crisis) may occur in which dA makes contact with the basin boundary. When this occurs, the absorbing area ceases to exist, and trajectories repelled from the synchronized chaotic set are no longer restrained by nonlinear mechanisms from approaching other attractors (or to go to infinity). As discussed for two coupled logistic maps in Sec. 3.3, this contact bifurcation typically marks the transition from locally to globally riddled basins of attraction. For \i = 0.07, the contact bifurcation occurs approximately at e = —0.586.

Figure 3.20 shows the variation of dA with e for /J, = 0.07. This variation is nonuniform, such that increasing the numerical value of e causes the corners farthest away from the main diagonal to move outwards while at the same time


20

15

10

5

y o

-5

-10

-15

-20 -20 -15 -10 -5 0 5 10 15 20

x

Fig 3.20. Variation of the boundaries of the absorbing area (curves 1, 2, and 3) and the basin of attraction (curves 4 and 5) with the coupling parameter for fi = 0.07. The contact bifurcation occurs for e ~ —0.595. This bifurcation marks the transition from local to global riddling of the basin of attraction.

points near the line {x — —y} move in a nonmonotonous manner. Curve 1 applies for e = —0.4, and curves 2 and 3 for e = —0.5 and —0.585, respectively. Also plotted in Fig. 3.20 is the boundary dB of the basin of attraction for e = —0.585 (curve 4) and e = —0.5 (curve 5). Since fp(x) is attracting for all x, the basin of attraction for the coupled map system stretches to infinity along the main diagonal. For e = —0.585, the system is very close to the contact bifurcation, and as the coupling constant is reduced to —0.595, the border of the absorbing area touches the basin boundary.

For b = 0.14, 7 = 0.02, and /x = 0.07, the longitudinal Lyapunov exponent is Ay = 0.2542. The interval of negative transverse Lyapunov exponent stretches from a blowout bifurcation at s = —1.1 to a blowout bifurcation at e = —0.2. In this interval the synchronized chaotic state is either a Milnor attractor (that attracts almost all points from its neighborhood) or it is a weak Milnor attractor that attracts a positive (but not the full) Lebesgue measure set of points from its neighborhood. As shown above, the global bifurcation at which the absorbing area makes contact with the basin boundary occurs approximately at £ = —0.595, and for —1.1 < e < —0.595 the coupled map system displays global riddling.


Fig 3.21. Global riddling of the basin of attraction for the synchronized chaotic state, fj, = 0.07 and e = —0.7. Points that are attracted to the synchronized state are indicated by black dots, and points that are repelled from this state are left blank.

Figure 3.21 shows a typical example of the basin of attraction for the synchronized chaotic state in this coupling interval. Here, e = —0.7. Points that are attracted by the synchronized state are indicated by black dots, and points that are repelled from this state are left blank. The outer borders of the basin of attraction have remained nearly unaffected by the contact bifurcation (compare, e.g., with the basin of attraction in Fig. 3.20). However, we note the characteristic holes that have developed in the basin, some of which emanate directly from the main diagonal. Arbitrarily close to any point that is attracted by the synchronized chaotic state one can now find a positive measure set of points that are repelled from it, and the basin of attraction is a fat fractal.

As e is reduced to —1.1 (i.e., relatively close to the blowout bifurcation), the basin of attraction for the synchronized chaotic state takes the form depicted in Fig. 3.22(a). The transversely unstable period-2 orbit iterates between x = —5.67 and x = 5.67. In the magnification of part of the basin of attraction displayed in Fig. 3.22(b) we have indicated the position of some of the preimages (s_2 and X-i) to the unstable period-2 orbit. One can clearly observe how the basin of attraction is riddled with tongues emanating from the period-2 cycle and its preimages. As discussed in Sec. 2.3, such repelling tongues can take two


Fig 3.22. Basin of attraction for the synchronized chaotic state for /i = 0.07 and e = —1.1 (a). Note how the repelling tongues emanate from the preimages of the transversely unstable period-2 cycle (b).

different forms, depending on the relative sizes of the transverse and longitudinal eigenvalues [7, 8]. In the present case, the transverse eigenvalue is numerically the larger, and the tongues are tangential to the invariant manifold.

For comparison with the basin of attraction, Fig. 3.23 shows the first three preimages of the main diagonal. Since the map F is noninvertible, in addition to itself the diagonal has other sets of preimages. Points that map onto the invariant manifold in one iteration of F^E fall on the curves drawn by a heavy line. Points that map onto {a; = y} in two iterations of F^£ are indicated by a weaker line, and points that map onto the diagonal in three iterations are indicated by a thin line. We note how the structure outlined by these curves forms the backbone of the basin of attraction. Each of these curves contains a dense set of preimages of the transversely unstable period-2 cycle, and from each of these points repelling tongues emanate to both sides, contributing all together to the complexity of the attracting domain.

One may wonder what happens to trajectories that are not attracted to the synchronized state. Some of these go to infinity. However, as described for our system of coupled logistic maps in Sec. 3.5, others can go to other attractors in phase space provided that such attractors exist. To illustrate this point, Fig. 3.24 shows the basins of attraction for two coexisting chaotic attractors. Here, /J, = 0.2 and e = -0 .9 . The heavy line along the main diagonal represents the synchronized chaotic state AQ. In the lower left corner the black


10

5

y o

-5

-10 -10 -5 0 5 10

x

Fig 3.23. First three preimages of the invariant set {x = y} drawn with decreasing line width. These preimages form the backbone of the basin of attraction in Fig. 3.22.

points above the diagonal show part of its basin of attraction, the full basin being symmetric both with respect to {x = y} and with respect to {x = —y}. The black points below the diagonal represent the basin of attraction for a coexisting antisymmetric, two-dimensional chaotic attractor A\. This basin is also symmetric with respect to {x — y} and {x = —J/}. The antisymmetric chaotic attractor is shown in the middle of the figure around the line {x = —y}-

If fi is increased to /i = 0.22, the two-dimensional antisymmetric attractors are replaced by a one-dimensional chaotic attractor on the line {x = —y}, and for a certain interval of the coupling constant, this attractor is weakly stable. In the simultaneous presence of two chaotic attractors both restricted to one-dimensional manifolds in phase plane, the attracting domains take the form illustrated in Figs. 3.24(a) and (b). Here, e = -0.89 and fx = 0.219494. The basin of attraction for each of the two co-existing chaotic states is riddled with initial conditions that belong to the basin of attraction of the other state. This is a new example of intermingled basins of attraction [7, 8, 29, 30]. At the same time, the basins are riddled with holes that belong to the basin of infinity.


Fig 3.24. Basins of attraction for two coexisting chaotic attractors: The synchronized chaotic state A0 on the main diagonal, and the two-dimensional, antisymmetric chaotic state Ai around {x = —J/}. Both basins are symmetric with respect to {x = y} as well as {x = —y}.

Fig 3.25. Intermingled basins of attraction in the presence of two one-dimensional chaotic attractors. Black points are attracted to chaotic sets along {x — y} (a) and along {x = —y} (b), respectively . Here, e = -0 .89 and fj, = 0.219494.


3.8 Conclusions

As the parameters of the coupled map system are changed we may pass out through the blowout bifurcation where the synchronized state loses its average transverse stability and becomes a chaotic saddle. Almost all trajectories may then be repelled from the synchronized state. However, if the parameters are such that an absorbing area still exists, the dynamics will be restrained by nonlinear mechanisms to occur within this area.

To investigate the phenomenon of on-off intermittency in our system of coupled intermittency maps we have chosen a set of parameter values corresponding to a relatively weak coupling and to a situation close to the subcritical period-doubling bifurcation in the individual map. The weak interaction allows the two subsystems to operate nearly independent so that turbulent bursts can occur in one map while the other operates in a laminar phase. The small value of /i causes the laminar phases of the type-Ill intermittency of the individual maps to be relatively long.

Figure 3.26 displays the global dynamics of the on-off intermittency observed for fi = 0.1, 7 = 0.13, b — 0.18697, and e = —0.1. In its temporal behavior the system exhibits long laminar phases in which the excursions in phase plane are relatively small, interrupted by occasional bursts in which the system moves out in phase space, preferably along one of the main axes. As illustrated in the figure, the global dynamics is restricted to taking place within the absorbing area, the boundaries of which are defined by the heavy lines surrounding the attractor.

Figure 3.27(a) provides a close up of the dynamics near the transversely unstable fixed point. Here y, = 0.001, 7 = 0.13, b = 0.18697, and e = -0.0002. For these parameters the longitudinal and transverse eigenvalues for the fixed point are z/y = — 1 — \± = —1.001 and i/± = — 1 — fi — 2e — —1.0006, respectively. Inspection of the figure shows how trajectories reinjected into the vicinity of the fixed point are repelled in all directions, producing the typical structure of a repelling tongue.

If the coupling parameter is changed to e = —0.002, the eigenvalues of the fixed point become v\\ = —1.001 and vL = —0.997. Hence, the fixed point is now a saddle. The bifurcation that has occurred is a subcritical period-doubling, and close to the fixed point we now have a repelling, antisymmetric period-2 cycle. The local dynamics generated in this situation is illustrated in


8

6

4

2

y o

-2

-4

-6

-8 - 8 - 6 - 4 - 2 0 2 4 6 8

x

Fig 3.26. Global picture of the on-off intermittency observed for /J, = 0.1, 7 = 0.13, b = 0.18697, and e = —0.1. The solid lines represent the borders of the absorbing area.

Fig 3.27. Close up of the dynamics near the unstable fixed point (a). After a subcritical period-doubling the local dynamics is controlled by the repelling antisymmetric period-2 cycle (b). The position of the repelling period-2 cycle is marked by the two crosses.


(a) (c) (b)

( a ) — I 1 — | • \i

^ ^i ( b ) ^ ( c )

(b) — | 1 1 • \i

Fig 3.28. Bifurcation sequences in the two main scenarios describing the breakdown of chaotic synchronization.

Fig. 3.27(b). The global dynamics has remained largely unchanged and is still an example of on-off intermittency. In the local dynamics, however, we observe how the repulsion is controlled by the unstable period-2 cycle.

Based on the above results we may distinguish between two main scenarios as sketched in Fig. 3.28. Here, [i^ and (j,^ denote the points for some parameter fi at which the riddling, respectively, the blowout bifurcation occurs. / / c ' represents the point at which the contact bifurcation between the absorbing area and the basin of attraction takes place. The riddling bifurcation is a local bifurcation, and the blowout is quasi-local in the sense that it is controlled by the average conditions along the low-dimensional chaotic attractor. The contact bifurcation, on the other hand, is a global bifurcation that depends on the nonlinear character of the system.

In the first scenario (a), the contact bifurcation occurs between /j,^ and /z^. In this case we can observe a local riddling in the interval between //"' and /z^, global riddling from /z^ to /z^, and an abrupt loss of synchronization in the blowout bifurcation. In the second scenario (b), the contact bifurcation happens after the blowout. In this case we have local riddling in the entire interval from /i(°) to n®, and on-off intermittency between /z^ and fi^. Another distinction is associated with the question whether or not stable synchronous states arise within the absorbing area. Finally, it should be noted that in the case that the riddling bifurcation is soft, there will initially be a mixed absorbing area stretching along the synchronized chaotic state, and additional transformations


must take place before global riddling can be observed.

Bibliography

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[10] T. Kapitaniak, Yu. Maistrenko, A. Stefanski, and J. Brindley, Bifurcations from Locally to Globally Riddled Basins, Phys. Rev. E 57, R6253-R6256 (1998).


[11] S.C. Venkataramani, B.R. Hunt, E. Ott, D.J. Gauthier and J.C. Bienfang, Transitions to Bubbling of Chaotic System, Phys. Rev. Lett. 77, 5361-5364 (1996).



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[16] P. Ashwin and J.R. Terry, On Riddling and Weak Attractors, Physica D 142, 87-100 (2000).

[17] E. Ott, J.C. Sommerer, J.C. Alexander, I. Kan, and J.A. Yorke, Scaling Behavior of Chaotic Systems with Riddled Basins, Phys. Rev. Lett. 71, 4134-4137 (1993).

[18] Yu.L. Maistrenko, V.L. Maistrenko, O. Popovych, and E. Mosekilde, Unfolding of the Riddling Bifurcation, Phys. Lett. A 262, 355-360 (1999).

[19] Yu.L. Maistrenko, V.L. Maistrenko, O. Popovych, and E. Mosekilde, Desynchronization of Chaos in Coupled Logistic Maps, Phys. Rev. E 60, 2817-2830 (1999).

[20] J. Laugesen, E. Mosekilde, Yu.L. Maistrenko, and V.L. Maistrenko, On-Off Intermittency and Riddled Basins of Attraction in a Coupled System, Adv. Complex Syst. 1, 161-180 (1998).

[21] M. Manscher, M. Nordahn, E. Mosekilde and Yu.L. Maistrenko, Riddled Basins of Attraction for Synchronized Type-1 Intermittency, Phys. Lett. A 238, 358-364 (1998).

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[23] C. Mira and T. Narayaninsamy, On Two Behaviors of Two-Dimensional Endomorphisms. Role of the Critical Curves, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3, 187-194 (1993).

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[26] I.U. Bronstein and A.Ya. Kopanskii, Smooth Invariant Manifolds and Normal Forms (World Scientific, Singapore, 1996).

[27] O. Popovych, Asymptotic Behaviour and Stability of Solutions of Coupled Oscillators, PhD Thesis (Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 1999).

[28] C. Mira, C. Rauzy, Yu. Maistrenko, and I. Sushko, Some Properties of a Two-Dimensional Piecewise-Linear Noninvertible Map, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 2299-2319 (1996).

[29] J.C. Alexander, J.A. Yorke, Z. You, and I. Kan, Riddled Basins, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 795-813 (1992).

[30] J.C. Alexander, B.R. Hunt, I. Kan, and J.A. Yorke, Intermingled Basins for the Triangle Map , Erg. Theor. Dyn. Syst. 16, 651-662 (1996).

[31] S.E. Newhouse, Diffeomorphisms with Infinitely Many Sinks, Topology 13, 9-18 (1974).

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[34] G. Iooss and D.D. Joseph, Elementary Stability and Bifurcation Theory, 2nd Ed. (Springer-Verlag, Berlin, 1980).


[35] V. Maistrenko, Yu. Maistrenko, and I. Sushko, Noninvertible Two-Dimensional Maps Arising in Radiophysics, Int. J. Bifurcation Chaos Appl. Sci. Eng. 4, 383-400 (1994).

[36] V.I. Arnol'd (Ed.), Dynamical Systems V, Encyclopedia of Mathematical Sciences (Springer-Verlag, Berlin, 1994).

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[39] J. Laugesen, E. Mosekilde, T. Bountis, and S.P. Kuznetsov, Type-II Intermittency in a Class of Two Coupled One-Dimensional Maps, Discr. Dyn. in Nature and Society 5, 233-245 (2000).

Chapter 4

TIME-CONTINUOUS SYSTEMS

4,1 Two Coupled Rossler Oscillators

As discussed in the preceding chapters, recent studies of chaotic dynamics restricted to low-dimensional manifolds of total phase space have led to the discovery of a variety of new phenomena, including riddled basins of attraction [1], blowout bifurcations and on-off intermittency [2], and attractor bubbling [3]. Full synchronization, in which both the phases and the amplitudes develop precisely in the same manner, can be achieved through the coupling of two (or more) identical chaotic oscillators [4]. Experimental evidence for this type of synchronization in different electronic systems has been obtained by Volkovskii and Rul'kov [5], Pecora and Carroll [6, 7], and Rul'kov et al. [8]. Chaotic synchronization has also been observed in coupled laser systems [9], and del Rio et al. [10] have shown that synchronization of chaotic oscillators can be achieved with unidirectional as well as with bidirectional coupling. Surveys of many of these results may be found in the publications by Rul'kov [11] and by Pecora et al. [12].

When the trajectories of a system of two coupled JV-dimensional oscillators are restricted to an iV-dimensional subset of total phase space (the synchronization manifold), the spectrum of Lyapunov exponents can be divided into N longitudinal exponents that characterize the motion in the synchronization manifold, and N transverse exponents that describe the response to perturbations perpendicular to this manifold. A necessary condition for synchronization

123


is that the largest transverse Lyapunov exponent is negative. In this case, the synchronized state is at least weakly stable [13, 14]. As described in Chapter 2, this implies that the state attracts a finite Lebesgue measure set of points from its neighborhood, but not necessarily the full measure. Related to the existence of particular orbits for which the transverse Lyapunov exponent is positive, there may still exist a positive measure set of points within the neighborhood of the synchronized chaotic state from which the trajectories diverge to infinity or approach some other attracting set. This produces the characteristic riddled basins of attraction with repelling tongues issuing from a dense set of points in the synchronization manifold.

A major question of concern, particularly in connection with the electronic applications, relates to the sensitivity of the synchronized state to a small mismatch in the parameter values between the interacting subsystems. This problem has been considered, for instance, by Rul'kov et al. [15], by Abarbanel et al. [16], and by Johnson et al. [17]. Rosenblum et al. [18] have demonstrated how one can identify and characterize the resulting state of phase synchronization, and Astakhov et al. [19] have examined how the bifurcations that control the loss of synchronization are affected by a parameter mismatch.

Because of inevitable drift of the parameters during operation, passive synchronization methods may not suffice in practice, and a variety of active controls are presently being investigated. In particular, Volkovskii [20] has shown how one can obtain in- and out-of-phase synchronization as well as frequency, phase and phaseshift key-in modulation and demodulation of coupled electronic circuits. Other types of active synchronization methods have been studied by Fradkov and Markov [21] and by Suykens et al. [22]. In-phase and anti-phase synchronization have also been reported by Bohr et al. [23] in a parallel computer simulation of 256 chaotically oscillating kidney nephrons.

In Chapters 2 and 3 we established a detailed description of the processes by which synchronization is lost in a system of two linearly coupled logistic maps [24, 25, 26], emphasizing the distinction between sub- and supercritical riddling bifurcations and the role of the so-called absorbing and mixed absorbing areas [27]. The purpose of the present chapter is to extend this analysis to systems of coupled time-continuous oscillators. The presentation is based on our recent investigations of chaotic synchronization and riddled basins of attraction in a system of two coupled Rossler oscillators [28]. This investigation applied a Lyapunov function approach to obtain a sufficient condition for chaotic synchro-

Time-Continuous Systems 125

nization and a calculation of the largest transverse Lyapunov exponent to establish the necessary condition. We also studied the transverse period-doubling, pitchfork and Hopf bifurcations, by which low-periodic orbits embedded in the synchronized chaotic state become transversely unstable. Finally, we showed how the characteristic riddled basin structure for time-continuous systems can be explained in terms of trajectories that approach an unstable low-periodic orbit.

The above analysis was subsequently extended to the case where there is a mismatch between the two subsystems [29]. With this aim we generalized the approach developed by Johnson et al. [17] for the case of unidirectional coupling in order to show that synchronization in the neighborhood of the synchronization manifold for the identical subsystems can still occur, although the trajectories will exhibit characteristic fluctuations in the directions transverse to this manifold. We determined both the shift of the synchronization manifold and the amplitude of the transverse fluctuations in terms of the coupling strength and the mismatch parameter.

We have also examined the problem of partial synchronization (or clustering) in diffusively coupled arrays of identical Rossler oscillators [30]. As discussed in Chapter 2, the term partial synchronization denotes a dynamic state in which groups of oscillators synchronize with one another, but there is no synchronization among the groups. By combining numerical and analytical methods we proved the existence of partially synchronized states for systems of three and four oscillators and described the dynamics within the clusters. Finally, we analysed the emergence of synchronized states in larger arrays of chaotic oscillators. In Chapter 5 we shall demonstrate how similar phenomena can arise in a system of coupled Sherman oscillators [31] describing the characteristic spiking and bursting behavior of pancreatic /3-cells. There is ample evidence to show both that the behavior of the individual cell model can become chaotic [32, 33] and that neighboring cells synchronize [34, 35].

4.2 Transverse Destabilization of Low-Periodic Orbits

Let us start by establishing the conditions for synchronization of two coupled Rossler systems and examine the mechanisms by which this synchronization is lost. A system of two linearly coupled, identical oscillators may be cast into


the form x= f(x) + C(y - x), V= f(y) + C(x - y), (4.1)

where x, y £ R3 represent the state variables of the individual oscillator and C is the coupling matrix. For simplicity, this matrix will be chosen as C =diag{di, d2, c/3}. This implies that there is no cross coupling between different variables in the two oscillators. In the case of coupled Rossler systems x = (2:1, £2, £3) G R3, and the function /(•) takes the form [36]

( -x2 -x3 \

Xl + ax2 • (4.2) b + x3(x! -c) )

Here, time has been scaled so that the natural (angular) frequency is 1. If the parameters are chosen to be a = 0.42, b — 2.0, and c = 4.0, the uncoupled system x = f(x) can be assumed to generate a chaotic attractor [37].

The synchronization manifold x = y is invariant in the six-dimensional phase space (x, y) of the coupled system (4.1). Complete (or full) synchronization occurs when the interacting oscillators asymptotically exhibit identical chaotic behaviors, i.e., || x(t) — y(t) ||—>• 0 as t —> 00 for any initial condition (a;(0), j/(0)) within some neighborhood of the synchronous state A [12]. Introducing coordinates along and transverse to the synchronization manifold 77 = y + x, and £ = y — x, the linearized equation for the transverse perturbations takes the form

ft=[Df(s(t))-2C}5Z. (4.3) Here, Df denotes the Jacobian matrix evaluated along some solution s(t) C A. Hence, the local stability properties of the synchronized motion can be derived by analyzing the stability of the equilibrium point 5£ = 0 of Eq. (4.3).

Applying the Lyapunov function U =\\ 5£ ||2 to Eq. (4.3) we obtain [29] the domain D\ of transverse stability in the space of coupling parameters as shown in Fig. 4.1. Namely, if (^1,^3) G D\ and d2 > a/2 then the equilibrium point 5£ = 0 is asymptotically stable, which implies synchronization. The Lyapunov function method provides the sufficient conditions for synchronization and, therefore, underestimates the real synchronization region. The blowout bifurcation curve (curve 2 in Fig. 4.1) determines the moment where the largest transverse Lyapunov exponent of the synchronized chaotic attractor changes sign [2]. This curve represents the necessary condition for synchronization without intermittent bursting.


Fig 4.1. Synchronization of two identical Rossler systems. The region of complete synchronization Dx as determined by the Lya-punov function method. Curve 2 is the blowout curve (calculated for rf2 = 1-0). This curve delineates the region of weak transverse stability of the synchronized state.

The exact domain of synchronization in the space of coupling parameters has to be obtained by means of numerical experiments. As in the case of coupled logistic maps, asymptotic stability requires that not only a typical trajectory on the synchronized chaotic set is transversely stable (as used for calculation of the Lyapunov exponent), but that all orbits on the chaotic set are stable in the direction transverse to the synchronization manifold. The normal approach to this problem is to investigate the transverse stability of different low-periodic orbits embedded in the synchronized chaotic state A [38, 39]. As demonstrated by several authors [1, 2, 28, 38], the transverse stability properties of individual periodic orbits do not necessarily coincide with those averaged over the chaotic attractor. Figure 4.2 shows the bifurcation diagram for the synchronized period-1 saddle cycle in the plane (d\,d$), and Fig. 4.3 displays a projection of this cycle and of the synchronized chaotic state for our coupled Rossler oscillators. In Fig. 4.2, region I is the domain of transverse stability for the period-1 cycle. This cycle is transversely unstable in region II. When crossing part P of the bifurcation curve, the period-1 cycle undergoes a pitchfork bifurcation. Similarly, when crossing the parts denoted PD and H, destabilization occurs via a transverse period-doubling or Hopf bifurcation.

As discussed in Chapter 2 and 3, each of these transverse bifurcations may be either sub- or supercritical [26]. Figures 4.4 and 4.5 illustrate how the transverse destabilization of the period-1 saddle cycle occurs in a supercritical period-doubling and in a subcritical pitchfork bifurcation, respectively. In the first case, a period-2 saddle cycle situated symmetrically with respect to the synchronization plane is created in the destabilization process. In the second

4. -

I synchronization I region

2

\

' ' J ,


T

C

.«.—

n

~ P

P D

H

I

stability region for period—1 cycle

P

* " i

Fig 4.2. Bifurcation diagram for the transverse destabilization of the period-1 saddle cycle embedded in the synchronized attrac-tor. Parts P correspond to the loss of transverse stability via a pitchfork bifurcation, PD via period doubling, and H via a Hopf bifurcation.

o.o 1 .o

0.0

4 .0 ; k

I T

i S

" • J" *>K

I 1 I 0.0 4 .0

Fig 4.3. Period-1 saddle cycle embedded in the synchronized chaotic attractor. Its period is T « 2K. S illustrates the Poincare section of the attractor.

case, two mutually symmetric, doubly unstable period-1 cycles existing before the transverse destabilization, approach the symmetric period-1 saddle cycle and destroy its stability as they are annihilated [28].

In Fig. 4.6 we have combined the bifurcation diagrams obtained for period-1, -2, -3, and -4 saddle cycles to determine which is the first to lose its transverse stability. (Here and in the following text we use the term period-i¥ saddle cycle to denote an iV-loop orbit with a period of approximately NT, with T being the period of the period-1 cycle.) The moment of blowout bifurcation is displayed as a bold curve. The rightmost envelope of the destabilization curves for low-periodic orbits gives the moment when the first of the considered saddle


-0.3

Fig 4.4. Antisymmetric cycle of period ss 2T that arises after the supercritical transverse period doubling bifurcation. The synchronization manifold is shown as a projection on the

a)

Fig 4.5. Different projections of the two doubly unstable period-1 cycles (dashed curves) and the transversely stable period-1 saddle cycle before the subcritical pitchfork bifurcation at d, = 1.0, d2 = 1.0, and d3 « -1.75.


cycles becomes transversely unstable. Part I of this envelope corresponds to a period-doubling bifurcation of the period-3 cycle, part II to a period-doubling bifurcation of the period-1 cycle, part III to a period-doubling bifurcation of the period-2 cycle, and part IV to a pitchfork bifurcation of the period-1 cycle.

d*

blowout period-3 period-1 _.._.._.._... period-4

period—2

Fig 4.6. Bifurcation diagrams for the transverse destabilization of low-periodic orbits and the blowout bifurcation curve (solid curve). We consider the rightmost envelope to determine the occurrence of the first transversely unstable (periodic) orbit embedded in the chaotic attrac-tor.

4.3 Riddled Basins

In the previous section we showed how the fully synchronized chaotic state loses its transverse stability via desynchronization of particular orbits in a so-called riddling (or bubbling) bifurcation. Riddled basins of attraction [1, 2] may be observed immediately after the riddling bifurcation where the maximal transverse Lyapunov exponent is still negative for typical trajectories. Under the assumption that the first orbits to become transversely unstable in the coupled Rossler systems are low-periodic cycles [39], the riddling bifurcation curve can be represented by the rightmost envelope in Fig. 4.6.

We conclude that riddled basins of attraction may occur in the domain that is delineated by the riddling bifurcation curve on one side and by the blowout bifurcation curve on the other. This domain is illustrated in Fig. 4.7. Fixing the parameters d\ = 0.1, d2 = 1.0, and d% — —1.86 (point A in Fig. 4.7) we


have calculated the two-dimensional cross-section of the basin of attraction with £2 = £3 = 0,772 = 773 = 1 displayed in Fig. 4.8. The basin shows clear evidence of riddling in the form of a dense set of tongues of initial conditions from which the trajectories are repelled from the synchronized chaotic state. Another cross-section of the basin is presented in Fig. 4.9. Here, the values £3 + 2/3 = % = 0.1, £2 = £3 = 0, and £1 = 1.0 are fixed. Black points are points from which the trajectory diverges to infinity. In addition, in order to clarify the structure of the repelling tongues throughout the attractor, we have investigated the long term behavior of trajectories that start from points 77 6 •A, £ = (0.01,0,0), i.e., with longitudinal coordinates belonging to the attractor and a small transverse perturbation of the form (£1,0,0,0). Figure 4.10 shows the projections of this set onto the (2:1,2:2) and (2:1,2:3) planes, respectively. Black points again correspond to initial values of trajectories that diverge to infinity, and white points to those that converge to the synchronized state.

Fig 4.7. Domain in the (di,d3) plane where the synchronized state is transversely stable on average but there is a transversely unstable periodic orbit embedded in the attractor (da = 1.0).

-0.4 0.0 0.4 0.8 dx

In order to explain the structure of the basins observed in Figs. 4.8, 4.9, and 4.10 we have determined the Poincare section S of the individual Rossler oscillator for fixed 2:1 = 0.44 as illustrated in Fig. 4.3 [28]. The invariant attracting set of the corresponding Poincare map is shown in Fig. 4.11(a). Considering this set to be one-dimensional, we have constructed the one-dimensional map which acts on the set. This is shown in Fig. 4.11(b). As one can see from Fig. 4.6, the period-1 saddle cycle is transversely unstable for the considered parameter values. The fixed point P of the one-dimensional map corresponds


Fig 4.8. Cross-section % = % = 1.0, £2 = £3 = 0 of the basin of attraction for the synchronized state. The coupling parameters are d\ = 0.1, efe = 1.0, and d% = —1.86. This corresponds to point A in Fig. 4.7.

Fig 4.9. The cross-section £1 = 0.1, £2 = £3 = 0, and % = 1.0 of the basin of attraction for the synchronized state. Black points correspond to initial conditions from which the trajectory diverges to infinity. The four orbits starting from the preimages P^ 4 are shown in gray.


Fig 4.10 Here two projections of the synchronized attractor are presented. Adding a transverse perturbation & = 0 01 to the synchronized state on the attractor, the black points correspond to initial conditions of orbits that diverge to infinity. The four orbits starting from the preimages P^4 are shown in gray

to the period-1 saddle cycle, and its preimages P"1, P~ 2 , Pf'f', and P{2 3,4 correspond to orbits that approach the cycle in one, two, three, and four loops, respectively. Considering the various preimages of the 1-D map, we can identify different orbits of the Rossler system that approach the period-1 cycle, namely the orbits which contain the points P, P - 1 , P~ 2 , P - 3 , P - 4 , Plotting further preimages of the fixed point P we can find the "inset" of the period-1 cycle. This set includes the trajectories that share the property of transverse instability with the period-1 cycle, and these orbits form the skeleton of the riddling structures in Figs. 4.8 and 4.9.

The orbits that start from P ^ 4 are shown in Figs. 4.8 and 4.9. We observe that these orbits correspond to the most transversely unstable parts of the synchronized attractor. That the preimages do not coincide with the repelling points in the right part of Fig. 4.9 can be explained by means of Fig. 4.12. Here the projection of the synchronized attractor onto the (xz, x\) plane is presented. The dashed line shows the corresponding projection of the plane from Fig. 4.9. The left part of the attractor is nearly flat, and this allows us to reveal the location of transverse instabilities by investigating the two-dimensional cross-section in Fig. 4.9 "along" the attractor. The same is not the case for the right hand part of Fig. 4.9.


(a)

*3

1.4

1.1

0.8

P4 p ^

P ^ \ P

_t p -4 P"1

1 I

—1

'*1 -1

(b) 1.4

- fM

0.6

/ r 3 P - * \ ~ /

/ 1 *2 \ . /

/-4 / ^ \ P

0.6 0.8 1.2 1.4

Fig 4.11. Invariant set of the Poincare map 5 (cf. Fig. 4.3) for the individual Rossler attractor (a) and 1-D map acting on this invariant set (b). The fixed point P corresponds to the period-1 saddle cycle for the Rossler system. P^' are the (m different) preimages of P of order i.

4.4 Bifurcation Scenarios for Asynchronous Cycles

The next problem to consider is the bifurcation scenarios for the appearance of stable cycles and various types of more complex dynamics outside the synchronization manifold. With this purpose we shall construct a one-dimensional bifurcation diagram along the diagonal line in coupling parameter space passing through the riddling bifurcation points R\ and R% as depicted in Fig. 4.6 (dashed line). This scan was chosen in such a way that both riddling bifurcation points correspond to destabilization of the synchronous period-1 cycle. Accordingly, we shall be mainly interested in solutions arising from this cycle. Let us introduce rfasa new coupling parameter such that its variation corresponds to the motion of a point along the scan. Expressed in terms of this new parameter, our system has the form (4.1) with the coupling matrix

C = diag {d - 0.6,1.0, -3.1d + 0.7} .

The leftmost point of the scan corresponds to d = 0 while the rightmost point is reached for d = 1.0. Figure 4.13 shows the behavior of the numerically largest multipliers /xy and fi± of the synchronous period-1 cycle. The largest longitudinal multiplier is obviously independent of d and numerically larger than 1. The region of transverse stability is located between the points R\ and R2 where the largest transverse multiplier has a modulus less than one. We recall that R\ (d = 0.2408) corresponds to a supercritical period-doubling


Fig 4.12. Projection of the Rossler attractor on the (xi,x3)-plane. The dashed line represents a projection of the cross-section from Fig. 4.9.

bifurcation (fix = — 1) and i?2 (d = 0.7360) to subcritical pitchfork bifurcation

The asynchronous period-2 cycle that arises at point R\ plays an important role in the global dynamics of the coupled system. Figure 4.14 shows the evolution of this cycle in phase space with decreasing d. For d = 0.240, the asynchronous period-2 cycle is still a saddle cycle (shown as a dotted curve in Fig. 4.14). At d = 0.220 the cycle stabilizes via a subcritical pitchfork bifurcation. (Note that although the period-2 cycle does not belong to the synchronization manifold, its symmetry properties allow the pitchfork to occur). The interval of existence for the stable asynchronous period-2 cycle is approximately 0.201 < d < 0.220. At the lower end of this interval the cycle is destabilized in a secondary Hopf (or Neimark-Sacker) bifurcation, producing typically a quasiperiodic motion. Numerically we can observe a stable torus that arises at this bifurcation point. Two different projections of this torus are shown in Fig. 4.15, and the corresponding bifurcation diagram is presented in Fig. 4.16. Finally, this bifurcation branch leads to a stable asynchronous chaotic set via torus destruction at d = 0.197. This is illustrated by the sequence of Poincare sections in Fig. 4.17. This whole development is similar in many respects to the scenario described for two coupled logistic maps in Chapter 3.

Another chaotic set, a chaotic saddle, is born as a result of a period-doubling cascade of the unstable asynchronous orbits associated with the subcritical pitchfork bifurcation at d = 0.220. This is illustrated in Fig. 4.18, where we


_z longitudinal multiplier

transverse multiplier

transverse stability region —*•

-

-

-

-

1

1

0.ioi(NS) 'o.22<KP)

\ ""~V''/<

1

1 '

0.240/

,1/ £ £ ' '

i

-

-

-

-

Fig 4.13. Variation of the modulus of the largest multipliers of the period-1 cycle, p,\\ (longitudinal) and /ij_ (transverse) along the dotted line of Fig. 4.6. The horizontal line in the middle of the graph (around d = 0.4) where two multipliers coincide with one another corresponds to the existence of two complex conjugated multipliers.

Fig 4.14. Phase space representations of the asynchronous period-2 cycle for different values of the coupling parameter d. An unstable cycle is shown by a dashed line while a solid line denotes a stable cycle. (P) and (NS) correspond to the pitchfork and Neimark-Sacker (or secondary Hopf) bifurcation points that occur, respectively, at d ~ 0.220 and d ^ 0.201.

Fig 4.15. Two different projections of the stable invariant torus that exists for d = 0.2. (a) projection onto the (x\,yi) plane, and (b) onto the (xi,Xz) plane.


Fig 4.16. Bifurcation diagram in the parameter region where the stable asynchronous torus appears and is destructed. To the right in the figure we have the stable asynchronous period-2 cycle.

0.2 0.205 d

display both the unstable period-4 cycle (a) and the unstable period-8 cycle (b) observed, respectively, for d = 0.2148 and d = 0.2142. Figure 4.19 provides a schematic overview of the whole bifurcation diagram where two parallel period-doubling cascades can be seen in a parameter range to the left of d = 0.217. Although the chaotic saddle is unstable, it influences the transient behavior of the system. As an example, Fig. 4.20 shows a typical "disordered" transient in the region of the chaotic saddle (d = 0.209) for an orbit approaching the asynchronous period-2 cycle (shown bold).

It is worth noting that all the above bifurcations occur while the synchronous chaotic state is weakly stable, cf. Fig. 4.19. This allows us to expect that before the blowout with decreasing d the coupled Rossler system will reveal riddled basins caused by different asynchronous attracting sets. Indeed, the basin may be riddled with initial conditions approaching a stable limit cycle, a torus, or a chaotic set. Figure 4.21 shows an example of this type of structure for d = 0.204 where the weakly stable synchronous chaotic attractor coexists with a stable asynchronous period-2 cycle. Black points represent initial conditions from which the trajectory approaches the stable period-2 cycle (indicated in the Poincare section by a white point P2). Along the horizontal axis, where the weakly stable synchronous chaotic set is located, we have marked the point Pi of the synchronous period-1 cycle. After the riddling bifurcation this cycle is doubly unstable.

Let us now consider what happens as the coupling parameter d increases. We can then observe the appearance of an asynchronous chaotic saddle while

3

2

1

O

- 1

- 2

"' "'

1 1

^R^^^MaJP'*1'

j&^__. Q Q Q ^ ^ U ^ M --

, 1

----

-

--

1


x3 3.8

1 1

P^r h • • • / • . .

_U**r • &

V - H

S

i

i ' i

-••*. - • ; - ^ , " • - • ^ ! & -

i . i

i ' i

.**>.

v*. • • S ^ V -

0 i i

(A)

2 2.2 2.4 2.6

X,

Fig 4.17. Two-dimensional projections onto the (xi, x3) plane of the invariant set of the Poincare return map at different values of coupling: d = 0.201 (a), d = 0.198 (b), d = 0.1975 (c), and d = 0.197 (d). The figure illustrates the destruction of the torus and the formation of a stable, asynchronous chaotic set.


Fig 4.18. Unstable asynchronous periodic period-4 and period-8 orbits arising through period-doubling bifurcations at d = 0.2148 (left) and d = 0.2142 (right).

weak stability of the synchronous state-stable asynchronous

. chaotic set , stable torus

^Z asynchronous ,. ., ^y chaotic saddle J ;; '

f^ fN r ^ Cj Ci O

complete synchronization"

•«3 X - —

S N ' ;

NS

i 4+-

''%%$' asynchronous w ^z222& c^9}}?, -s,a 41?, J$$%

' asynchronous $, chaotic saddle fli

Fig 4.19. One-dimensional bifurcation diagram for solutions associated with the synchronous period-l cycle. A detailed description may be found in the text. In a region to the left of the pitchfork bifurcation points at d = 0.220 the weakly stable synchronous chaotic set coexists with stable asynchronous attractors.


Fig 4.20. Typical behavior of orbits attracted by the stable asynchronous limit cycle in the presence of asynchronous chaotic saddle for d = 0.209. Note the long chaotic transient.

" " - 4 - 2 0 2 4 6 xi

the synchronous chaotic state is still asymptotically stable. The corresponding bifurcation diagram is shown in the right-hand side of Fig. 4.19. It involves a subcritical pitchfork bifurcation of the synchronous period-1 cycle (P), and simultaneous reverse period-doubling bifurcations (PD), saddle-node bifurcations (SN) and Hopf bifurcations (H) of the symmetric pair of asynchronous period-1 cycles arising in the pitchfork bifurcation. The unstable period-2 cycles produced in the period-doubling bifurcations continue in parallel period-doubling cascades leading finally to the emergence of two symmetric chaotic saddles. Note, however, that we do not observe asymptotically stable limit sets in the phase space for d > 0.736, at least not such which can be associated with the bifurcations described above. In fact, the globally riddled basin in Fig. 4.8 was calculated for parameter values in this region. Here, the basin of attraction for the synchronized chaotic state is riddled with initial conditions from which the trajectory diverges.

4.5 The Role of a Small Parameter Mismatch

When we consider synchronization phenomena in coupled chaotic systems, one of the first questions to arise relates to the influence of a parameter mismatch. An approach to this problem has been outlined by Johnson et al. [17]. In this section we shall extend that approach and describe the interesting effect of a shift of the synchronization manifold which can be observed numerically when the considered parameter mismatch can be expressed in a particular manner. The motion of the coupled system then takes place in the vicinity of some state £ = const 7 0 with occasional chaotic bursts of small amplitude.


Fig 4.21. Globally riddled basin of attraction for d = 0.204. Black points represent initial conditions from which the trajectory is attracted to the stable asynchronous period-2 cycle P2. 71 and 72 are linear combinations of the longitudinal and the transverse coordinates, respectively, chosen such that the plane intersects the asynchronous period-2 cycle.

We start with the following model [29]

x= f(x) + C{y -x) + sg(x),

y=f(y) + c(x-y), (4.4)

which accounts for the difference between the parameters of the two systems via the term eg(x). s is taken as a measure of the magnitude of the parameter difference. If the mismatch is restricted to a single parameter fi, we define

e = fi - no and g(x) = ^ £ ^ In the case that a mismatch exists for more

than one parameter, one can choose, for example, e = \fi\ = max; |/z, — ^oi| a n d g{x) = p | ( / {x, fi) — / (x, Ho))- We shall consider the case where the identical systems, i.e., with e = 0, are synchronized. The approximate solutions of (4.4) may be represented in the following form

xe(t) = x0{t) + e<f>(t), ye(t) = y0(t) + ei/>(t) (4.5)


where we neglect higher order terms in e. xo(t) and yo(t) are the solutions of Eq. (4.4) with e = 0. By substituting (4.5) into (4.4) and comparing terms of order 1 and e we obtain the following equations for the unknown functions <j> and ip:

'cj>= Df (x0 (t)) 4> + C{^-<t>)+g(xv (*)), ( 4 6 )

</>=£/MW + co/.-v). Using Eq. (4.5) the transverse perturbation can be written as £(£) = x(t) —

y(t) = £o(£) + e£iM- Here, £o(i) = xo(t) — yo{t) -> 0 with t -> co because of the assumed synchronization of the identical systems. Therefore,

£ ( t ) ~ e 6 ( t ) = e W * ) - V ( * ) ) . (4.7)

The equation for ^ ( i ) now follows from (4.6)

^ = [Df (x0(t)) - 2C] 6 + 9 (x0(t)) • (4.8)

The non-homogeneous term g {xo(t)) arises due to the parameter mismatch. In the case of identical systems (g — 0), (4.8) has the trivial solution £i(t) = 0. Moreover, due to the assumption that identical systems are synchronized, this solution will be asymptotically stable. Hence, there exists a fundamental matrix solution $(i,io) of

^ ^ 1 = [Df (zo ( i)) _ 2C] $( i , i0), *(*, to) = / := diag {1,1,1} , (4.9)

such that || $(Mo) ||-^ 0 with t ->• oo. In terms of $ the solution of Eq. (4.8) can be expressed as follows

fc(i) = $( i , t„)£i,o + I *(*, r)g ( I 0 ( T ) ) ~ f $(t , r)g (X0(T)) (4.10) •> to *> to

for initial values £i7o from some neighborhood of £i = 0. (Actually, this is the neighborhood that admits the asymptotic synchronization of the identical systems.)

In the case when the identical systems are exponentially synchronized with a rate constant A, i.e., || xo(t) — yo(t) | |~ exp {—At} , A > 0, we have

| | $ ( i , i o ) | | < M e x p { - A ( t - i 0 ) } . (4.H)


where M > 0 is an appropriate constant. Equation (4.10) then implies that the transverse perturbations are of the order e:

£ || < const- 9ix) A

(4.12)

(see also [17]). It can be shown that condition (4.11) is fulfilled provided the criterion for synchronization of the identical systems is expressed in terms of the Lyapunov function method with the function || £ ||2 (see [29]). Indeed, in this case we can estimate —7 > sup^i/max(a;), where fmax.(x) is a maximal eigenvalue of the matrix [Df(x) + DfT(x) - 2 (C + CT)] over the synchronized chaotic attractor A . In the paper by Johnson et al. [17], —7 was chosen as maximal instantaneous transverse Lyapunov exponent for the attractor.

Now consider the case eg(x) = go = const, i.e., the mismatch parameter is included in an additive way. For example, in the Rossler system, the parameter b has this property. Problems involving an external constant force acting on one of the oscillators can also be described by Eq. (4.4) with eg(x) = go. Under this assumption, (4.10) turns into

m / $(i,r)dT ffo- (4.13)

For simplicity, we choose the coupling in the form C = al — diag {a, a, a}. Let us then examine the asymptotic properties of £(£) at large coupling magnitude a.

Consider ty(t,to) — e~2Ct to be the fundamental matrix solution of ^ = - 2 C * . The substitution of $ = ( $ - * ) + * into (4.13) gives

««) = £ + A*(«,r) .J t0

V(t,T))dr 9a- (4.14)

From Eq. (4.9), the behavior of the difference $ — \l/ is governed by the equation

d

dt (* (t, r ) - * (t, r)) = -2C (* (t, r) - * (t, r)) + Df (x0 (t)) $ (*, r ) . (4.15)

Integration of (4.15) yields

f e-2aI(t-T')Df ( a ,Q ^ $ ( r , ) r ) dT,dT $ - * : 90 (4.16)


9o- (4.17)

and, hence

m ~ £ ~ [f f e~2aI{t~T')Df (*« (T')) $ (^i T) dT'dT

We refer to our original publication [29] for the proof of the estimation

II ${T',T) II< M e x p ( - 7 ( r ' - T ) ) ,

with M > 0, and 7 = 2a — m, m > 0. m and M do not depend on a. Assuming the function / to be smooth or to have a bounded derivative,

| | D / (xo (T ' ) ) | | < I with some positive constant I. Therefore, by integrating Eq. (4.17) we obtain

< - ^ U , P , . (4.18) «')-£ 2a(a-iy Here |-| denotes a vector with modules taken of each component. Expres

sion (4.18) shows that for large a (more precisely, for a » I, see [29]), the asymptotic behavior is given by

a*) 9o_

2a <L ho

a2 (4.19)

where the constant L does not depend on a or go. The inequality (4.19) is the main result of this section. For any pair of linearly

coupled oscillators (4.4) with an additive parameter mismatch or a constant external force g0 acting on one of the oscillators, assuming that the corresponding identical oscillators are exponentially synchronized, one can observe the following asymptotic properties with increasing magnitude of the coupling

• there is a parallel shift of the synchronous motion from the synchronization manifold by an amount go/2a;

• the amplitude of chaotic bursts around the shifted state is of the order I go/a2, where I is a measure of the maximal numerical value of the Jacobian determinant in the points of the synchronized chaotic state;

• for large coupling strength a, the bursts becomes small compared with the shift, and one can numerically observe the phenomenon of generalized synchronization where || x[t) — y(t) ||—> const ^ 0.

Let us hereafter illustrate the obtained results in the case of two coupled Rossler systems with C = al. Figure 4.22 shows the projection of the synchronized motion in the system of identical oscillators (go = 0) onto the (x\, y\)


plane for coupling parameter a = 2. As it follows from the previous sections, the system is asymptotically synchronized for this coupling strength. Taking go = (0.1,0,0) and a = 2, numerical calculations demonstrate a behavior of x\{t) — Vi(t) as illustrated in Fig. 4.23. The presence of the shift away from xi — 2/1 = 0 is clearly observed. The dashed line shows the theoretically predicted asymptotic shift by Eq. (4.18). Figure 4.24 shows the same trajectory as a projection onto the (2:1,2/1) plane. The motion now takes place at some distance from the diagonal (which represents the projection of the synchronization manifold). The shifted set is broadened because of nonzero transverse perturbations, again in accordance with Eq. (4.18). With increasing coupling parameter a we can dampen the bursts around the shifted state. Figure 4.25 shows the dependence of x\{t) — y\{t) for three different parameter values. The shift decreases linearly with 1/a while the bursts are damped as 1/a2.

Fig 4.22. Synchronous motion of identical coupled Rossler oscillators with a coupling parameter a = 2.

xry

Fig 4.23. The behavior of &(*) = xi(t) - yi(t) for the parameter mismatch ga = (0.1,0,0) and the coupling a = 2. The dotted line represents our theoretical result (4.19) for the shift of the synchronization manifold.

4.6 Influence of Asymmetries in the Coupled System

Let us now consider the following asymmetr ic coupling scheme

x=f{x)+pC(d)(y-x),

y=f(y) + (2-p)C(d)(x~y), (4.20)


Fig 4.24. (x1,2/1)-projection of the orbit (heavy line) for the parameter mismatch go = (0.1,0,0) and the coupling a = 2.

xry.

' 1 ' ' 1

« = 2

ex=6

^ ct=12

I , I

Fig 4.25. Behavior of £i(t) for the parameter mismatch g0 = (0.1,0,0) with different values of the coupling parameter a. As a is increased, the random bursts away from synchronization nearly disappear.

where parameter p introduces the asymmetry provided that p ^ 1. As discussed in Chapter 3, this type of asymmetric coupling is of interest when studying the transition from one- to two-cluster dynamics in ensembles of globally coupled, identical chaotic oscillators [51]. Under these conditions, our calculations show that the subcritical pitchfork bifurcation (occurring in the parameter point d = 0.736 of Fig. 4.19) transforms into a transcritical bifurcation. Figure 4.26 schematically displays the perturbed bifurcation vis a vis the original unperturbed one (thin lines). For p = 0.9, the transcritical riddling bifurcation occurs at d = 0.7360, and the associated saddle-node bifurcation at d = 0.7382. In the transcritical riddling bifurcation, a doubly unstable asynchronous period-1 solution passes through the synchronization manifold (from below) and exchanges its transverse stability with the synchronized period-1 saddle, leaving this as a transversely unstable solution.

Yet another scenario arises if we perturb the coupled oscillators by introducing a mismatch between their parameters. To investigate this situation let us couple two nonidentical systems in the form

x=fl(x) + C(d)(y-x),

y=My) + c(d)(x-y). (4.21)

In this case, a symmetric period-1 saddle cycle no longer exists. The transverse pitchfork bifurcation scheme is transformed into two isolated cycle branches


Fig 4.26. Perturbation of a subcriti-cal transverse pitchfork bifurcation by a nonsymmetric coupling. The thin line shows the unperturbed case. The figure illustrates the appearance of a trans-critical riddling bifurcation.

with one of them undergoing a saddle-node bifurcation [52]. This is illustrated in Fig. 4.27, where the thin curves represent the transverse pitchfork bifurcation in the fully symmetric case and the dotted curves represent the perturbed diagram in the case of a parameter mismatch. In the fully symmetric case (as shown in Fig. 4.19) we have a synchronous period-1 saddle and two symmetric, asynchronous and doubly unstable period-1 orbits to the left of the bifurcation point. To the right of the bifurcation point we have a transversely unstable synchronous period-1 cycle. To illustrate the effect of a parameter mismatch take fi(x) = f(x,b{) and J2{y) — f{y,b2) with a value of the parameter b<i = kb\, and with b\ = b — 2.0 as defined in Eq. (4.1). Clearly, k = 1 represents the case of identical coupled oscillators. Taking k = 1.001, we obtain the scenario shown schematically in Fig. 4.27 with the bifurcation points dsN — 0.7286, and for the two period-doubling bifurcations dx £* 0.7081, and d2 = 0.7056. With k ^ 1, the two disconnected branches for the period-1 orbits of the coupled system no longer fall in the synchronization manifold, and one of the branches undergoes a saddle-node bifurcation in which the period-1 saddle is transformed into a doubly unstable period-1 cycle. Note that in the presence of symmetry the two period-doubling bifurcations (in accordance with Fig. 4.19) occur at the same value of the parameter d.

4.7 Transverse Stability of the Equilibrium Point

Application of numerical methods to synchronization problems, e.g., calculation of transverse Lyapunov exponents, investigation of the stability of periodic orbits embedded in the attractor, etc., naturally implies some limitations. In particular, this approach restricts the number of control parameters such that the results can be presented in terms of one- or two-dimensional scans in pa-

p=0.9

p=1.0


Fig 4.27. The perturbed subcritical d pitchfork (riddling) bifurcation in the ^ case of nonidentical coupled oscillators.

Fully drawn thin lines illustrate the unperturbed case.

P D *

rameter space. For example, using the results of Sec. 4.2, we are not able to predict what happens if instead of the coupling matrix C = diag {d\, 1, d^} we consider C = diag {di, 0,^3}. Similar problems arise with respect to the local behavior in a neighborhood of the bifurcation branches: Are the considered bifurcations subcritical or supercritical, for instance? This section is meant to investigate the extent to which one can replace detailed analysis of the synchronization of a pair of chaotic oscillators by determining the stability properties of a much simpler solution, such as the equilibrium point, which can be treated analytically.

Let us fix the parameters of the Rossler system to be b = 2.0 and c = 4.0, with a to be varied. It is assumed that if the value of a is about 0.4, the Rossler system in general has a chaotic attractor. As a simple solution to system (4.1), we consider the synchronous equilibrium point O = (xi,x2,xz,x\,x2,x$) = (u,u) ,xi— 2 — 2^/1 — a/2, x2=xi /a , and 2)3= — x2. Figure 4.3 shows the position of this equilibrium point O in a particular projection. For the assumed parameter values we have x\= 0.22 and x2= — x3= —0.53. Although the point O does not belong to the attractor, one may hope that the transverse stability properties of orbits in the chaotic attractor are similar to those for the equilibrium point. In the following analysis we shall consider the coupling C = diag {di, d2, d^} with three free parameters.

Conditions for transverse stability

A stability analysis of the point O can be carried out analytically. For this purpose, we again introduce the transverse and longitudinal coordinates (x — y, x+y) and shift the origin to the equilibrium point. We denote the new variables as a vector u =((ui — u2), (u\— u) + (u2— u)) . f (u,/i) is defined as the right hand side of Eq. (4.1) with respect to the new variables and a new coupling


parameter fi which we shall later define in terms of d;. The linear stability analysis of the equilibrium point 0 gives a characteristic equation of the form

aiA3 + a2A2 + a3A + a4 = 0, (4.22)

where the coefficients depend on the parameters a,d\,d2, and d3. We note that, in general, a six-dimensional system gives a characteristic equation of the sixth order. However, for coupled Rbssler systems the linearization matrix is block-diagonal, with blocks corresponding to the transverse and longitudinal coordinates [37]. This implies that the characteristic polynomial becomes a product of two independent parts. We are interested in the characteristic roots that depend on the coupling parameters and correspond to the transverse directions. In this way, we arrive at the third order equation (4.22). Bifurcations occur when the roots of this equation have vanishing real parts, i.e., either A = 0 or A = ±ioj. This happens when one of the following conditions is fulfilled

a4 = 0, or aicii — a2C3 = 0. (4-23)

In terms of the parameters ai ,di ,d2 , and d3, the first condition of Eq. (4.23) takes the form

a2di (8 + 7) + d27 - 2ad3 - 2ad1d2 (8 + 7) + 4a2did3 - 8adid2d3 - a (4 + 7) = 0 (4.24)

with 7 = — 4 + 2\/4 — 2a. The second condition gives

2d\ (2a - 8 - 7) - 2d\ (7 + 8) + 4ad^ - 8(d?d3 + d\d2 + d\dz + d\di + d\dx+

d\d2) - 16did2d3 + 2a (7 + 8) (dx + d2 + d3) + ( - - 2a2) (di + d3) +

2 (2a - 2 7 - 17) (d : + d2) + 4 (2a - 7 - 8) (did2 + d2d3 + did3) +

( 2a 7 - 2 - 6a2 + 17a - ^ M = 0. (4.25)

Expressions (4.24) and (4.25) define surfaces in parameter space for the boundaries of the regions of transverse stability for the equilibrium point O. In fact, this equilibrium point is of saddle type. So, when speaking about the transverse stability of O we mean the existence of a one- or two-dimensional stable manifold in the transverse direction. To compare the transverse stability curves, defined by Eqs. (4.24) and (4.25), with the numerically computed


Fig 4.28. The critical values of coupling parameters where transverse bifurcations of the equilibrium point O occur. d,2 = 1.0. Solid lines correspond to pitchfork bifurcations, and dashed lines to Hopf bifurcations.

~~" -1 -0.5 O 0.5 1 1.5 2

stability diagram for the chaotic attractor we insert d,2 = 1 and a = 0.42. Figure 4.28 shows the computed stability curves in the (c^efo) parameter plane. As determined by Eq. (4.25), the curve for the transverse pitchfork bifurcation is a hyperbola. The upper right envelope of the stability curves determines the region of transverse stability, like in the case of the chaotic attractor (Fig. 4.6). Its boundary consists of three parts: a Hopf bifurcation curve in the middle (dashed line) and two branches of a pitchfork bifurcation curve (solid lines). Now compare this threshold for the transverse destabilization of the equilibrium point with the destabilization curves for low-periodic orbits embedded in the chaotic attractor as illustrated in Fig. 4.29. Here, the dashed line corresponds to transverse destabilization of the equilibrium point O. As we can see, this analytically obtained curve reproduces the shape of the numerically obtained curves for destabilization of the low-periodic orbits quite well (even though, of course, the equilibrium point cannot undergo a period-doubling).

It is also interesting to observe the shape of the transverse stability region in the case G?2 — 0 as shown in Fig. 4.30. This region is also bounded from the right hand side. Taking cfo = 0 (i.e., the coupling matrix becomes C — diag {d\, 0,0}) and increasing d\ from zero, stabilization for relatively small values of d\ is followed by a destabilization for larger values of the coupling parameter d\. This analytical result is in keeping with the numerical results obtained by Pecora et al. [12, 53]. Moreover, we can assert that with the cosidered coupling scheme considered (i.e., with d2 = 0), for any positive fixed ^3, the desynchronization threshold will persist with increasing d\. For cfo < —1.8, transverse stability cannot be achieved for any values of d\.

3 o

—— pitchfork . — Hopf

Supercr.

C •s

_l 1 L


d 1

o

-1

- 2

1 M

M i

/ — ' 1 /

i - 1

1 1 ' I • 1

-

S^-^ Siibcr. TSupercr. i

i i i . i

-0.4 0.0 0.4 d,

0.8 1.2

Fig 4.29. Comparison of the boundaries of transverse stability for the synchronous chaotic attractor (heavy line), for some of the main low-periodic orbits embedded in the attractor (thin lines), and for the equilibrium point O (dashed line) for the coupling matrix C = diag {di, 1, d$}.

d3 o

1 1 1 1 1

_ Transverse stability region

I . I .

WM 9 -•

i / v

i i ^ -1.5 -1 -0.5 0

pitchfork

Hopf

Fig 4.30. Same as in Fig. 4.28 but for d2 = 0. Note the finite range for the coupling parameter d\ in which transverse stability can be achieved. This implies that for given values of the coupling parameters d2 and d3, increasing dx can lead to destabilization of the coupling.


Criticality of the riddling bifurcations

It was noticed in Sec. 4.2 that the desynchronization bifurcations (transverse period-doubling or pitchfork) for the low-periodic orbits embedded in the chaotic attractor tended to be supercritical in the upper left side of the bifurcation diagram in Fig. 4.6, while they are subcritical along the bottom.

For the equilibrium point O, the type of criticality for the bifurcation can be determined analytically by evaluating the projection onto the one-dimensional subspace spanned by the eigenvector that corresponds to the critical eigenvalue [54]. Define the parameter \i = d%— cfo, where cfo is a critical value of d$ satisfying Eq. (4.25). Let vx be the eigenvector of the Jacobian matrix evaluated in the origin u = 0 with the coupling parameter fixed at the critical value (i = 0. Denote this Jacobian as fu(0). Hence, fu(0)vi = 0. Denote by Wi the corresponding adjoint eigenvector, i.e., fu

T(0)wi = 0, chosen so that it satisfies the normalizing condition (vi, wi) = 1, where (•, •) denotes the scalar product. It is known [54] that locally the bifurcation branches of solutions can be represented by the series

f ! = £ U l f 2U 2 + ; ' ' (4.26)

//(e) = e/ti + e /i2 H , where the amplitude e is the projection on the subspace associated with the adjoint eigenvector wi is

£ = ( u , w i ) . (4.27)

By virtue of the symmetry of the system it can be shown that /ii = 0. Therefore, the sign of /*2 in Eq. (4.26) determines whether the bifurcation is subcritical or supercritical. Substituting (4.26) into the equation u = f (u, //) we find, by identification, a set of equations for the unknowns Ui, U2 and \i<i. Skipping technical details the final expression for \ii is

M2 = ~ ; ^ ( f u u ( 0 | v i | U 2 ) ' W l ) ' ( 4 2 8 )

where U2 is to be determined from the linear system of equations

fu(0)u2 = - ( f u u (0 |v 1 |v i ) > w 1 ) , (4.29)

(u2 ,wi) = 0.

Here the following notations are used: A'(0) = (fu/x(0)vi, wi) determines the derivative of the critical eigenvalue with respect to /z at the point of bifurcation


[54], [fuu(0|u|v)]j = dudu njvk, and fu/i(0) is the component-wise derivative of fu with respect to fi evaluated in the point (0,0).

Fig 4.31. Graph of the function (fuu(0|vi|u2), wi) for fixed d2 = 1 and a = 0.42. This curve determines the sub- or supercritical character of the transverse destabilization of the equilibrium point O. The points T\, T2, and Q are defined in the text.

Our interest concerns the sign of H2- Clearly, A'(0) is positive when the stability region is left by increasing /i and negative in the opposite case. Therefore, to know the type of bifurcation the sign of {fuu(0|v1|u2), Wi) has to be determined. We omit here an analytical expression for this term due to the involved complexity. Figure 4.31 shows the graph of the function for fixed d^ = 1 and a = 0.42. The function changes sign at three points where the coupling parameter di = (a — 1) / [a (a — 2)] = 0.871 (point Q) or

d , = a ± \Ja(a2 + 3 - 4a) (2a - y/4 - 2a) I [2a (a - 2)] S {-0.035, -0.598}

(4.30) (points T2 and TO.

Let us now calculate the sign of /X2- For fixed di = 1 we have

sign(/x2) = - , di >di= {a - 1) / [a {a - 2)] £ 0.871

+, d\ <d\ • (4.31)

Hence, for d\ > d\ the bifurcated branch exists only for \i < 0, i.e., c^ < d$. This implies a supercritical stability loss in the region to the right of Q. On the other hand, for d\ < d\ a subcritical stability loss occurs along the pitchfork bifurcation curve. The transition between the supercritical and subcritical pitchfork bifurcation is indicated on the solid curve in the bottom of Fig. 4.28. Similar arguments can be applied to other values of aV

Returning to the bifurcation diagrams for low-periodic orbits in Fig. 4.6, the above results suggest that as d\ becomes large enough, the riddling bifurcations


for low-periodic orbits tend to become supercritical, as observed for the equilibrium point O. Our numerical calculations confirm this conclusion. Indeed, a threshold d[ = 1.7 has been found such that for d\ > d[ the riddling pitchfork bifurcation of the synchronous period-1 cycle becomes supercritical.

4.8 Partial Synchronization of Coupled Oscillators

If the coupled system involves more than two coupled chaotic oscillators, a whole new range of additional phenomena can occur, including partial synchronization (or clustering) [40, 41, 42, 55] as well as various forms of wave-like dynamics. A state of partial synchronization is said to occur when the interacting oscillators synchronize with one another in different groups, but there is no synchronization among the groups. Interesting questions in this relation concern the types of partial synchronization that can exist with different coupling schemes.

The analysis that we shall present in this section is mainly concerned with partial synchronization phenomena in an array of chaotic oscillators with nearest-neighbor interaction. Another problem of significant interest concerns the behavior of an ensemble of globally (i.e., all-to-all) coupled chaotic oscillators. This problem relates, for instance, to the dynamics of a group of /3-cells that all respond to variations in the glucose concentration of the blood, variations generated at least partly through changes in their own aggregated insulin production. Examples of locally coupled chaotic oscillators are also found in the living world where many cells or functional units, which individually exhibit complicated nonlinear dynamics, interact to produce a coherent behavior at a higher functional level. The basic model for our investigations is a chain of Rossler systems which are coupled in a diffusive way:

Uj^f{uj) + C(uj+1 + uj_1-2uj), j = l,...,N (4.32)

with the boundary condition UJV+I = u\. Here Uj € Rn denote the phase space coordinates of the individual oscillator and C is the coupling matrix. Each of the uncoupled oscillators

XA ( -X2-X3 \ X2 = X = f(X) = Xi + aX2 (4.33) X3J \ 6 + X 3 ( X i - c ) /

is considered to have an invariant attracting chaotic set A for a = 0.42, b — 2.0, and c = 4.0 [36]. It is also evident that the synchronization manifold u\ = u2 =


• • • = ttjv is invariant and contains the invariant chaotic set

A = {ui = • • • = uN, iti G .A}. (4.34)

Let us recall that complete (or full) synchronization takes place when this "synchronous" set As is asymptotically stable. This implies that small deviations from the state (4.34) tend to zero, i.e., ||UJ — itj|| — 0, i ^ j with t —> oo for initial conditions w(0) = (u i (0) , . . . , UJV(O)) from some neighborhood U of As.

A system of the form (4.32) with coupling only via the first component (i.e., C = diagja, 0,0}) was recently considered by Heagy et al. [43]. They discussed an associated size instability that occurs in systems that exhibit a short wavelength bifurcation (e.g., a variant of the Rossler system). This instability limits the number of oscillators capable of sustaining stable synchronous chaos even for large coupling. They also developed a general approach, involving the so-called "master stability function" which makes it possible to investigate different linear coupling schemes [44, 45]. Phase synchronization effects in a nonidentical array of diffusively coupled Rossler oscillators with a coupling matrix C = diag{0,a, 0} were investigated by Osipov et al. [46]. Systems that are coupled by a common internal field (global coupling) were considered numerically by Zanette and Mikhailov [47]. Nakagawa and Komatsu [48] studied coupled tent maps and introduced a Lyapunov exponent that characterizes the dynamical properties of the collective motion. Networks of coupled cells were considered, e.g., by Golubitsky et al. [49] using symmetry arguments.

In contrast to complete synchronization as defined above, in the case of partial synchronization the coupled system splits into clusters of identically oscillating elements. Problems of partial synchronization were studied by Pyragas [40] in connection with the phenomena of generalized synchronization, de Sousa and Lichtenberg [41] showed that partial (in their notation "weak") synchronization does not necessarily precede complete synchronization. Taborov et al. [42] reported on partial synchronization phenomena in a system of three coupled logistic maps, and Belykh et al. [55] have described an effective method to determine the possible states of cluster synchronization and ensure their stability for systems of diffusively coupled, identical chaotic oscillators.

In Section 4.8 we state some useful results for two diffusively coupled systems. Besides the numerical computations we emphasize some analytical results which for certain values of the coupling parameters prove the existence of a trapping region around the synchronous set (4.34). The results of this initial analysis


allow us to obtain the conditions for complete synchronization and riddling for a system of three coupled Rossler systems. We also prove that this system admits partial synchronization for some narrow parameter range. The system of four coupled oscillators can be partially synchronized for a "massive" set of parameters. This problem is considered in Sec. 4.9. Conditions to determine when complete or partial synchronization takes place are also given. Finally, Sec. 4.10 discusses the case of a large number of diffusively coupled systems.

Preliminary results for two coupled systems

The system of two diffusively coupled oscillators has the following form:

ui = f(ui) + C(u2-ui), u2 = f(u2) + C(ui - u2). (4.35)

For simplicity, we shall consider coupling with only one parameter in the form C = a • I where / is the unit matrix. Denote the transverse coordinates by £ = u\ — U2- The synchronization manifold «i = U2 is invariant for the system (4.35).

As discussed in Sec. 4.2 and in accordance with the results of a variety of previous studies [11, 12, 43, 44, 45, 46, 29], one can identify the following qualitatively different values of the coupling parameter a: Case 1: Those values of a for which system (4.35) admits complete synchronization. Let us denote this set as Sc. Case 2: Values for a where (4.35) has a symmetric chaotic attractor u\ = U2 € A such that A is transversely stable on average, but embedded in A there is one (or more) transversely unstable orbit. In this case the largest transverse Lyapunov exponent along any typical trajectory is negative, but A is not asymptotically stable. Denote this set as Sr'. Case 3: Remaining values of a. Let this set be Su. It corresponds to the case when the synchronous chaotic set is unstable.

Note that case 2 may admit two different dynamical behaviors depending on the global dynamics of the system. First, a globally riddled basin of attraction may occur, where the basin of the synchronized attractor is densely riddled by initial conditions from which the trajectory goes to infinity or approaches some other attractor [1]. Second, due to the existence of nonlinear restricting forces, attractor bubbling or local riddling phenomena may occur where intervals of nearly synchronous motion are intermittent with occasional bursts [2].


In order to be able to arrive at conclusions that are independent of the choice of the specific system let us make the following rather general assumption: (A): Suppose, that there exist such constants a\ and a<i so that Sc = {a > a\}, Sr = {ai2 < a < a>i}, and Su = {a < 0:2}. In other words, the attractor loses its asymptotic stability via the transverse destabilization of some nontypical orbit embedded in the attractor at a = a^ (riddling bifurcation) and then, with decreasing coupling parameter, becomes transversely unstable on average at a = a% (blowout bifurcation).

\ ^ X ^ ^ "

\ /NS£

a 2

period-4 period-1

•^ , period—2 jS j . period—3

N?X\ <*,

^ ^ x .

1 \ Xs .

Fig 4.32. Largest transverse Lyapunov exponent versus the coupling parameter a, calculated for the chaotic attractor (bold curve) and for some low-periodic orbits.

Fig 4.33. Behavior of the transverse perturbation ||£|| for some initial value from the neighborhood of the transversely unstable period-1 orbit embedded in the attractor (a = 0.05).

To show that condition (A) is fulfilled for two coupled Rossler systems, we have calculated the largest transverse Lyapunov exponent of the synchronized chaotic attractor versus a. We depict this variation as the bold curve in Fig. 4.32. The intersection of this graph with the horizontal axis (the point a2) determines the moment when the attractor becomes unstable in average. The thin lines in Fig. 4.32 show the same quantity for individual periodic orbits embedded in the attractor. The rightmost point of intersection of these lines with the axis (the point cti) gives us some approximation of the riddling bifurcation point. This corresponds to the transverse destabilization of the period-1 orbit. It follows from the numerical calculations that a\ ~ 0.060 and a2 ~ 0.042.

Figure 4.33 shows the plot of the transverse distance ||£|| versus time for a parameter value a = 0.05 6 Sr. It is evident from this plot how small deviations


from the synchronization manifold are amplified in the neighborhood of the transversely unstable period-1 orbit. Then global restraining forces return the orbit back to the weakly stable attractor.

Let us now state some analytical results [30] that give conditions for the existence of a trapping region around the synchronous chaotic state {wi = u<i e .4} in system (4.35). Here some analogy can be observed to the absorbing area in the case of two coupled one-dimensional maps [24]. The only requirements for the system (4.35) are the following: 1. The form of the coupling is C = a • diag{l, 1,1}. 2. For an individual oscillator X = f(X), X G Rn there exists an invariant attracting set A C Rn. 3. There exists a convex trapping neighborhood UA of A in the phase space of the individual attractor such that orbits never escape from it. This neighborhood should have a "good" local structure (for example, it can be diffeomorphic to a manifold with or without boundary).

Then for any a > 0 system (4.35) will have a trapping region (in fact, this is UA x UA) around the synchronized state {«i = u<i e A} in the phase space R2n

Fig 4.34. Asynchronous stable limit cycle. Projection onto the (uj,^) plane.

^-4 -2 0 1 2 4

The above result implies that the global riddling phenomena with the existence of orbits diverging to infinity can occur only when the coupling matrix C in equation (4.35) does not coincide with al. Taking into account the proof of the theorem, some components of the matrix C must be negative. Such a case was discussed in Sec. 4.3, where riddling was observed, e.g., for C = diag{0.1,1.0, -1 .73} .


su

(transversely unstable) sr

(weak stability) sc

(synchronization)

a 2 =0.042 Olj =0 .06

|« >| ^ 0.028 existence of 0 .0423 a

asynchronous stable period-2 cycle

Fig 4.35. Stability properties of two coupled Rossler system for different values of coupling parameter a.

Returning now to the coupled Rossler systems, the result is that after the symmetric chaotic set loses its transverse stability the orbits will still be confined to some bounded region around the synchronization manifold. Another attractor which appears to exists for these parameter values is a stable asynchronous cycle, cf. Fig. 4.34. This cycle arises in a pitchfork bifurcation at a = 0.0423 and disappears at a = 0.028. Hence, in our case, the regime of weak synchronization is replaced by asynchronous periodic motion. This is illustrated in Fig. 4.35.

Partial synchronization in a system of three coupled oscillators

The symmetry properties of the system (4.32) with three oscillators

ui = / (« i ) + al{—2u\ +u2 + u3), U2 = f{u2) + aI(-'2u2 + u1 + U3), (4.36) "3 — f(u3,) + al(-2u3 + ui + u2)

imply that the synchronous set loses its transverse stability in all transverse directions at the same time. In order to show this, following [44] let us rewrite system (4.36) into the form

u = F(u) + a(G ® I)u. (4.37)

Here, F(u) = ( / (ui) , f(u2), f(u3))T, u = {ui,u2,u3)

T, and the matrix

(4.38)


The variational equation for the synchronized solutions of (4.37) can be reduced to three systems of the form

i={Df{s)+IMaI% (4.39)

where Hi, i = 1, 2,3 are the eigenvalues of G, and s(t) is a trajectory on A. In our case HI = 0 and M2 — M3 = — 3 . A detailed explanation of the reduction procedure can be found in [44]. Each of the above equations corresponds to some "transverse mode" (i.e., it determines the behavior of transverse perturbations restricted to some direction) except for that for fi\ = 0. Hence, the equations for the two transverse modes are the same.

Provided that we consider the system in the small neighborhood of the synchronization manifold it follows that the three oscillators will either all be synchronized or they will be desynchronized. Now we can observe that the transverse variational equation for two coupled systems (4.35) also has the form (4.39) with fi = —2. Hence, taking into account that a^I — —2a ( ^ ) I, we can use the results for the local stability of the synchronous motion for two oscillators and transfer them to a consideration of the stability of the transverse modes, applying the scaling factor — 2 / ^ .

Presuming now that the assumption (A) from the previous section is satisfied we arrive at the conclusions: 1) For coupling strength a > | « i system (4.36) is completely synchronized; 2) For coupling |«2 < ot < | a i either global or local riddling occurs; 3) For a < %a2 system (4.36) can not be fully synchronized.

Using the values of a\ and a2 from the previous section we obtain the thresholds for the coupled triplet: %a\ ~ 0.040 and |tt2 c± 0.028. For the interval 0.028 < a < 0.040 it is difficult to observe the bursts away from the synchronization state numerically. However, when choosing initial conditions in the neighborhood of the symmetric, unstable low-periodic orbit (i.e. the period-1 cycle) we can clearly observe how the desynchronization bursts increase, showing behaviors like those in Fig. 4.33 for the two modes.

In the above considerations we have performed a local analysis for the stability of the synchronization manifold. After the loss of stability for this set, some stable sets may arise outside this manifold. In particular, if such a set is located in one of the hyperplanes:

{ui = u2,u3}, {ui = u 3 , u 2 } , {u2 = u3,ui}, (4.40)


then partial synchronization is observed. In order to investigate the existence of the limit sets for the motion confined to these hyperplanes, we shall consider the following nonsymmetric coupling scheme

X = f(X) + a(Y-X), Y = f(Y) + 2a(X -Y) K ' '

which was obtained by the factorization u\ = ui = X and U3 = Y. Let us define the Poincare return map for system (4.41) at point (0.44, 0,0,

0,0,0) with the normal vector directed along the Xi-axis. Calculations show that this map is defined for all parameter a-values in the considered region as well as for the considered initial values. The bifurcation diagram in Fig. 4.36 shows the evolution of X\ — Y\ for this map after skipping 300 iterations. We may assume that this procedure reveals the limit sets of our map in the X\ — Y\ projection and explains the dynamics inside the hyperplanes (4.40). We clearly observe the loss of synchronization at a ~ 0.028 as predicted by the above linear theory. Some periodic windows are also distinguished.

Fig 4.36. Bifurcation diagram for the Poincare map, defined as a return map for system (4.41).

O 0.02 0.04

d

Our next goal is to determine the properties of transverse stability for the limit sets that are located inside the corresponding hyperplane. In order to estimate this stability numerically, we obtain the variational equation for transverse perturbations. Due to the symmetry it is enough to consider the case U\ = u<i = X, U3 = Y. Denote the transverse coordinates £ — u\ — U2- Then the variational equation for £ takes the form:

8? = [Df(X(t)) - 3a]C (4.42)


where X(t) is a solution of (4.41). The maximal Lyapunov exponent Ac for system (4.42) is shown in Fig. 4.37.

At the point a fa 0.028 we observe a loss of transverse stability in agreement with linear theory.

Fig 4.37. Maximal Lyapunov exponent that determines the stability of the partially synchronous motion in the system of three coupled Rossler oscillators.

O 0,01 0.02 0.03 0.04

It is interesting to note two narrow intervals at the points a « 0.0038 and a « 0.0198, where Ae becomes negative. These parameter values correspond to cases when limit sets located in the hyperplane ui = U2 become stable in the directions transverse to the hyperplane. This implies clustering. We emphasize that this happens when the in-cluster dynamics, i.e., the dynamics within the hyperplane u\ = U2, becomes periodic and "asymmetric" with respect to X\ = Yi (cf. the periodic windows in Fig. 4.36) while the chaotic set is transversely unstable. Figure 4.38 shows the evolution of ||uj — U2II and ||wi — W3{|. The numerical calculations confirm that in this case the partially synchronized state is periodic.

Finally we note that the symmetry of (4.36) implies that similar stable periodic orbits exist in other hyperplanes: ui = U3 and u\ — U3 for the considered parameter values. The different types of partial synchronization are realized for varying initial values.

4.9 Clustering in a System of Four Coupled Oscillators

In this section we shall show that partial synchronization is observed for a more "massive" set of parameters in the case of four coupled oscillators. This is related to the fact that the synchronous set (4.34) first loses its transverse


Fig 4.38. One of the possible partially synchronous motions that can be realized in the system of three coupled Rossler oscillators for a = 0.0198.

stability with respect to some special directions while the other transverse directions remain stable. The system of four coupled oscillators can be written in the form (4.37) with u matrix

(ui, u2, u3, Ui)T, F(u) = ( /(ui) , • • •, f{ui))T, and the

G \

(4.43)

- 2 / The matrix (4.43) has the eigenvalues /xi^ = —2 and /X3 = —4 associ

ated with different transverse modes with different stability properties. To find these modes we calculate the eigenvectors V{ of G corresponding to /x,-: v\ = ( 0 ,1 ,0 , -1 ) , u2 = (1 ,0 , -1 ,0 ) , and u3 = ( - 1 , 1 , - 1 , 1 ) . Therefore, the first two modes involve the coincidence of u<i — 1x4 and u\ = 113, respectively. Their stability properties are described by equation (4.39) with /x = —2. The third mode corresponds to ui + u$ — u2 — 1x4 = 0 and its stability is determined by equation (4.39) with n = —4.

Supposing that assumption (A) is satisfied we can use the same arguments


as in the previous section to obtain the following conclusions: 1) If the coupling satisfies a > a\, then the system of four coupled oscillators of the form (4.32) is completely synchronized. 2) For coupling parameters a2 < a < ai the two least stable transverse modes u2 = u\ and ui = u3 m a Y be unstable for some initial values, at least locally. 3) For ^- < a < «2 the synchronous motion is stable only with respect to the transverse mode with /J3 = —4. Taking into account that in this case neither u2 and U4 nor u\ and 113 can coincide we may generally expect the existence of partial synchronization with the following possible clusterings: M\ = {u\ = u2, U3 = U4} and M2 = {ui = U4, u2 = U3}. Below we shall show that this holds true for Rossler systems. 4) For a < Y the system can not be fully synchronized.

Referring to coupled Rossler systems, the corresponding quantities will be (cf. Sec. 4.8): «i = 0.060,1 a2 = 0.042, a. = 0.030, and f = 0.021. Figure 4.39 shows the largest transverse Lyapunov exponents for a typical orbit in the synchronous set. This figure supports our conclusions. It is clear, that Ai corresponds to the longitudinal behavior confined to the synchronization manifold. This exponent does not depend on a. The next two exponents A2 and A3 have equal values and correspond to the transverse modes determined by the eigenvalues fii^ of matrix G. The most stable mode corresponds to the Lyapunov exponent A4.

**

0.O5

0

-O.05

.'•

-1

~"

«

• •

' • t i

' ' .

" * 4

I . ; . . *

• >

, a 2 / 2

' | T

J ' ' '

•

•

T

"

t

'

"

,

,

" 1

'

" m

" 2

1 A 1 •

1

1

t , 1

•

1

.

• •

-

i i i

*4*

~

T 1

N, clustering

:...y a 2

X O.Ol 0,02 0,03 0.04 0,05 0,06

_. A „ ,_, , T Fig 4.40. Maximal Lyapunov exponent that t i e 4.39. Four largest Lyapunov exponents , , . ., ,.,.. r , , .. ,,

. , , _ , . , .. , , determines the stability of the partially syn-with different value oi the coupling a for the , . ,,

, , , . . „ . . , .. chronous motion ui = u2, "3 = "4 in the system of four coupled chaotic Rossler oscillators.

system of four coupled Rossler oscillators.


With the stability loss of the first transverse mode, some asynchronous at-tractors arise away from the synchronization manifold. As in the case of three coupled systems, consider the stability of this asynchronous attractor with respect to perturbations that drive the system out of the partially synchronous state. We shall look for the following clustering structure:

ui-u2- X, us = ui = Y, (4.44)

(and the symmetric configuration «i = u\ and U2 = 1x3) which comes from the stability analysis of the synchronous set (this corresponds to the least stable mode).

In a standard way we obtain the equations for the "perturbations" for this clustering structure. For this, denote the transverse coordinates £1 = U\ — «2 and £2 = U3 — U4. They measure the deviations of the trajectory from the clustered motion (4.44). The linearized equations admit the form:

5^ = [Df{Y{t))-3a]5^-aS^ l ' }

where X(t) and Y{t) are solutions of two coupled systems. The values of the largest Lyapunov exponents for (4.45) may serve as criteria for the stability of the linearized system (4.45) and, therefore, the stability of the partially synchronous motion (4.44). Figure 4.40 displays these exponents versus a. Inspection of the figure shows that after the loss of complete synchronization at 0J2 we have a wide range 0.02 < a < 0.04 where Ac is negative. This corresponds to the existence of a stable partially synchronous structure (4.44). Owing to the symmetry, both clustering structures u\ = U2,1*3 = U4 and u\ = U4, u-i — uz can be realized, depending on the initial values.

To obtain the equation of the motion of the clusters in this case, we substitute u\ = u<i = X and 113 = U4 = Y (or u\ = 114 — X and u<i = 113 = Y) into system (4.32) of four oscillators. The resulting equation is exactly (4.35). Therefore, the interesting conclusion can be made: the motion of clusters coincides with the motion of two coupled system (4.35). The considered parameter values for the Rossler systems are ^ < a < 0:2, i-e., they correspond to the moment when our system of two coupled Rossler oscillators loses its transverse stability (cf. Sec. 4.8). It follows from Sec. 4.8 that the pair of coupled Rossler oscillators for these parameters will exhibit an asynchronous periodic stable cycle, cf. Fig. 4.35. Hence, we will obtain the periodic asynchronous motion with


u\ — 1*2 and U3 = U4 or with u\ = M4 and u2 = 113, depending on the initial values. Figure 4.41 illustrates both possibilities for a = 0.035.

To clarify the situation, we note that there are actually two stable symmetric cycles in the phase space of the coupled system. One of them is contained in the manifold {«i = ^2,^3 = 114} and the other in {u\ = u±,ui = u{\. It is known that each of the cycles has an open basin of attraction [52]. In our case these basins appear to be strongly mixed. In order to show this we have calculated the basins in a two-dimensional cross-section of phase space. This cross-section was defined as uu = 1.0, ua = 1.0, u2j = 1.0, u^j = 1.0, where i — 1,2,3,4, and j = 2,3,4. The grid was chosen to be 70x70. The behavior of the orbit starting from a center point in a given square was calculated. The result is shown in Fig. 4.42. Black squares correspond to initial conditions from which the system converges to one cycle, leading to the first variant of clustering (cf. Fig. 4.41) and gray points correspond to the second variant of clustering.

4.10 Arrays of Coupled Rossler Oscillators

In the previous sections we have considered the mechanisms of clustering and determined the conditions under which this phenomenon can be observed. The same analysis can be applied to a large system of diffusively coupled oscillators with possible chaotic behavior. Generalizing the results of the previous section, we can prove the existence of a two-cluster symmetric structure in an array of 2N coupled Rossler oscillators (1) with periodic relative motion. This structure is realized when u2k = X{t), k = l,...,N and u2k+i = Y(t), k = 0 , . . . , N - 1. The equation for the relative motion of these clusters assumes of the form (4.35).

The coupling matrix G in the case of N diffusively coupled oscillators will have the eigenvalues [45] fik = —4sin 2 ^7r , k = 1 , . . . , N. If the coupling parameter a is increased, some transverse modes become stable and, as a result, the motion of the system becomes confined to a manifold of lower dimension for some initial values, i.e., a clustering structure arises. Using the above scaling relation, we obtain the condition for the stability of the k-th transverse mode

ax

2 sin2 ^ T T

Therefore the condensation process begins when the first transverse mode with fi — - 4 (k = 1) becomes stable, i.e., for a > ^-. With further increase of


1000 f 2 0 0 0 °

IK-uJ 4

1000 2000

Fig 4.41. (a) and (b) represent the possible asymptotic modes of behavior for a system of four diffusively coupled Rossler systems with a = 0.035. Observe the clustering «i = u2 and u3 = u4

in (a) for one set of initial values and the clustering U\ = 114 and U2 = % f° r the other.


I • ."i •<• •• > ' . „ • ( . • j a r . » j ' > . . * . . • . •

• • v"< 1 1En;^t&m7IZ*'. •" •• -» " • - - • • f 2 - i , w C - * ™ * - • • • - '

&,, - S i '

-! |^t^^<

: ' • • ( • • • • . ™ * . „ " • j * <s

Fig 4.42, Cross-section of the basin of attraction of two stable limit cycles representing different clustering structures for a = 0.035.

. < . • > »

the coupling strength all the transverse modes become stable, and complete synchronization occurs. This happens when a > ° ^ x ~ 4 4 1 .

We have performed numerical simulations for different array sizes of coupled Rossler oscillators (N = 5,6,7,8,9,10,12,15,20,30,50), and for each size we managed to find parameter values for which the system exhibits periodic relative motion. The corresponding a-values belong to those intervals where only a few transverse modes are unstable. Figure 4.43 illustrates the periodic motion of an array of 30 coupled Rossler systems with a — 1. The first components of each oscillator are plotted for some fixed time moments tt. This parameter value corresponds to the case when only two transverse modes are unstable.

-i • r

» « • « » « > «><».»»«

Fig 4.43. Periodic solution in an array of 30 coupled Rossler oscillators. Xi(t,) versus N, (ti = t5)


In the case of a more general coupling scheme, when C = d i a g j d i , ^ , ^ } with positive d^, it is possible to determine the sufficient conditions for synchronization of the coupled array using the Lyapunov function approach.

Denote C = d2C, with C" = diagjc^, 1 , ^ } , d[ = d\/d2, and d'3 = d3/d2. For fixed d[ and d'3 the equation

djt={J-d2\iik\C')^ (4.46)

determines the stability of the A;-th transverse mode. The mode is stable for sufficiently large values of a because of positiveness of all elements di, i — 1,2, 3. This fact reveals the existence of a stability threshold function a'(d[,d'3) such that for any C IM&I > a'{d\,d'3) the solution £ = 0 of (4.46) is asymptotically stable. We construct this function in [30] using the Lyapunov function method. Given the function a'(di,d3) we can write the condition for the vanishing the fc-th transverse mode:

d2\iik\>a'{d[,d'2). (4.47)

Hence, the sufficient conditions for synchronizat ion of N oscillators become

d2rnin |7fc| = 4d2 sin2 ^ > a ( ^ , ^ ) . (4.48) MO W \d2 d2J

It follows from (4.48) that for given positive constants d\,d2, and d3, the maximal number N = Nmax of oscillators that can be synchronized can be estimated as the integer part of some function

AL.X = int (4.49)

Figures 4.44a and 4.44b illustrate the variation of Nmax with the coupling parameter d2 for fixed d\ = 1.0 and d3 = 3.0, and with d3 for fixed d\ = 1.0 and d2 = 1.0, respectively.


(a)

N

(b)

N

0,12 0,14 0,16 0,18 0,2

Fig 4.44. The maximal number Nmin of oscillators that can be fully synchronized for the given coupling parameters. Figures (a) and (b) show the dependence on di (with fixed d\ = 1.0 and d$ = 3.0) and d3 (with fixed di = 1.0 and di = 1.0) respectively.

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Chapter 5

COUPLED PANCREATIC CELLS

5.1 The Insulin Producing Beta-Cells

By virtue of the far-from-equilibrium conditions in which they are maintained through the continuous action of ion pumps, many biological cells display an excitable electrical activity, or the membrane potential exhibits complicated patterns of slow and fast oscillations associated with variations in the ionic currents across the membrane. This dynamics plays an essential role for the function of the cell as well as for its communication with neighboring cells. It is well-known, for instance, that pancreatic /3-cells under normal circumstances display a bursting behavior, with alternations between an active (spiking) state and a silent state [1, 2, 3, 4, 5]. It has also been established [6, 8, 7] that the secretion of insulin depends on the fraction of time that the cells spend in the active state, and that this fraction increases with the concentration of glucose in the extracellular environment. The bursting dynamics controls the influx of Ca2+-ions into the cells, and calcium is considered as an essential trigger for the release of insulin. In this way, the bursting dynamics organizes the response of the /3-cells to varying glucose concentrations. At glucose concentrations below 5 mM, the cells do not burst at all. For high glucose concentrations (> 22 mM), on the other hand, the cells spike continuously, and the secretion of insulin saturates [9].

A number of experimental studies have shown that neighboring /3-cells in an islet of Langerhans tend to synchronize their membrane activity [10, 11],

177


and that cytoplasmic Ca2+-oscillations can propagate across clusters of /3-cells in the presence of glucose [12, 13, 14]. The precise mechanisms underlying this interaction are not known. It is generally considered, however, that the exchange of ions via low impedance gap junctions between the cells plays a significant role [15]. Such synchronization phenomena are important because not only do they influence the activity of the individual cell, but they also affect the overall insulin secretion. Actually, it appears that the isolated /3-cell does not burst but shows disorganized spiking behavior as a result of the random opening and closing of potassium channels [11, 16, 17]. A single /3-cell may have of the order of a few hundred such channels. However, with typical opening probabilities as low as 5-10%, only a few tens will open during a particular spike. Organized bursting behavior arises for clusters of 20 or more closely coupled cells that share a sufficiently large number of ion channels for stochastic effects to be negligible.

Models of pancreatic /3-cells are usually based on the standard Hodgkin-Huxley formalism [18, 19] with elaborations to account, for instance, for the intracellular storage of Ca2+, for aspects of the glucose metabolism, or for the influence of ATP and other hormonal regulators. Over the years many such models have been proposed with varying degrees of detail [20, 21, 22, 23, 24]. At the minimum, a three-dimensional model with two fast variables and one slow variable is required to generate a realistic bursting behavior. In the earliest models, the slow dynamics was often considered to be associated with changes in the intracellular Ca2+-concentration. It appears, however, that the correct biophysical interpretation of the slow variable remains unclear. The fast variables are usually the membrane potential V and the opening probability n of the potassium channels. More elaborate models with a couple of additional variables have also been proposed [17].

Although the different models have been around for quite some time, their bifurcation structure is so complicated that it is not yet fully understood. Conventional analyses [11, 19, 25] are usually based on a separation of the dynamics into a fast and a slow subsystem, whereafter the slow variable is treated as a bifurcation parameter for the fast dynamics. However, compelling such an analysis may appear, particularly when one considers the large ratio of the involved time constants, it fails to account for the more interesting dynamics of the models. Simulations with typical /3-cell models display period-doubling bifurcations and chaotic dynamics for biologically interesting parameter values [26, 27], and

Coupled Pancreatic Cells 179

for obvious reasons such phenomena cannot occur in a two-dimensional, time-continuous system such as the fast subsystem. Simplified analyses can also provide an incorrect account of the so-called period-adding transitions in which the system changes from a bursting behavior with n spikes per burst into a behavior with n + 1 spikes per burst. Finally, the simplified analyses lead to a number of misperceptions with respect to the nature of the homoclinic bifurcation that controls the onset of bursting.

Wang [28] has proposed a combination of two different mechanisms to explain the emergence of chaotic bursting. First the continuous spiking state undergoes a period-doubling cascade to a state of chaotic firing, and this state is destabilized in a boundary crisis. Bursting then arises through the realization of a homoclinic connection that serves as a reinjection mechanism for the chaotic saddle created in the boundary crisis. In this picture, the bursting oscillations are described as a form of intermittency, with the silent state corresponding to the reinjection phase and the firing state to the normal laminar phase. Wang supports his analysis by a calculation of the escape rate from the chaotic saddle, and he outlines a symbolic dynamics to characterize the various bursting states. However, the question of how the homoclinic connection arises is left unanswered.

Terman [29, 30] has performed a more detailed analysis of the onset of bursting. He has obtained a two-dimensional flow-defined map for the particularly complicated case where the equilibrium point of the full system falls close to a saddle point of the fast subsystem that has a homoclinic orbit. By means of this map, Terman proved the existence of a hyperbolic structure (a chaotic saddle) similar in many ways to a Smale horseshoe. This represents an essential step forward in understanding the complexity involved in the emergence of bursting. However, since Terman's set is non-attracting it cannot be related directly to the observed stable chaotic bursting phenomena.

More recently, Belykh et al. [31] have presented a qualitative analysis of a generic model structure that can reproduce the bursting and spiking dynamics of pancreatic /3-cells. They consider four main scenarios for the onset of bursting. It is emphasized that each of these scenarios involves the formation of a homoclinic orbit that travels along the route of the bursting oscillations and, hence, cannot be explained in terms of bifurcations in the fast subsystem. In one of the scenarios, the bursting oscillations arise in a homoclinic bifurcation in which the one-dimensional stable manifold of a saddle point becomes attract-


ing to its whole two-dimensional unstable manifold. This type of homoclinic bifurcation, and the complex behavior that it can produce, do not appear to have been examined before.

Most recently Lading et al. [32] have studied chaotic synchronization (and the related phenomena of riddled basins of attraction, attractor bubbling, and on-off intermittency) for a model of two coupled, identical /3-cells, and Yanchuk et al. [33] have investigated the effects of a small parameter mismatch between the coupled chaotic oscillators. In the limit of strong interaction it was found that such a mismatch gives rise to a shift of the synchronized state away from the symmetric synchronization manifold, combined with occasional bursts out of synchrony. This is precisely the same result we obtained for coupled Rossler oscillators in Chapter 4, and many of the phenomena that arise in connection with the transverse destabilization of the synchronized chaotic state are also the same. As emphasized in Chapter 1, this whole class of phenomena is of general interest to the biological sciences [34] where one often encounters the situation that a large number of cells (or functional units), each performing complicated nonlinear oscillations, interact to produce a coordinated function on a higher organizational level.

The purpose of the present chapter is to give a somewhat simplified account of the bifurcation structure of the individual /3-cell [35]. Our analysis reveals the existence of a squid-formed area of chaotic dynamics in parameter plane with period-doubling cascades along one side of the arms and saddle-node bifurcations along the other. The transition from this structure to the so-called period-adding structure involves a subcritical period-doubling bifurcation and the emergence of type-Ill intermittency. The period-adding transition itself is found to be non-smooth and to consist of a saddle-node bifurcation in which stable (n+l)-spike behavior is born, overlapping slightly with a subcritical period-doubling bifurcation in which stable n-spike behavior ceases to exist. The two types of behavior follow each other closely in phase space over a major part of the orbit to depart suddenly and allow one of the solutions to perform an extra spike.

Bursting behavior similar to the dynamics that we have described for pancreatic /3-cells is known to occur in a variety of other cell types as well. Plant and Kim [36], for instance, have developed a mathematical model to account for experimentally observed burst patterns in pacemaker neurons, and Morris and Lecar [37] have modelled the complex firing patterns in barnacle giant muscle


fibers. Braun et al. [38] have investigated bursting patterns in discharging cold fibers of the cat, and Braun et al. [39] have studied the effect of noise on signal transduction in shark sensory cells. Although the biophysical mechanisms underlying the bursting behavior may vary significantly from cell type to cell type, we expect many of the basic bifurcation phenomena to remain the same.

5.2 The Bursting Cell Model

As basis for the present analysis we shall use the following simplified model suggested by Sherman et al. [11]:

Tltt = ~Ica(y) ~ IK{V' n) ~ 9sSiy ~ VK) ( 5 1 )

T^ = v(n00(V)-n) (5.2)

rs^ = S00(V)-S (5.3)

with

ICa(V) = gcam^V^V - VCa) (5.4)

IK(V,n) = gKn(V-VK) (5.5)

and

woo(^0 — [1 + e x P -^7; ] _ 1 for w = m,n, and S. (5.6)

Here, V represents the membrane potential, n may be interpreted as the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. As previously noted, the correct biophysical interpretation of this variable remains uncertain. Ica and IK are the calcium and potassium currents, gca = 3.6 and gx = 10.0 the associated conductances, and Vca = 25 mV and VK = — 75 mV the respective Nernst (or reversal) potentials [19]. T/TS defines the ratio of the fast (V and n) and the slow (5) time scales. The time constant for the membrane potential is determined by the capacitance and the typical total conductance of the cell membrane. With


T = 0.02 s and TS = 35 s, the ratio ks = T/TS is quite small, and the cell model is numerically stiff.

The calcium current Ica is assumed to adjust instantaneously to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards their voltage dependent steady state values n^iy) and Soo(V). Together with the ratio ks of the fast to the slow time constant, Vs will be used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady state value for the gating variable S attains half its maximum value. The other parameters are g$ = 4.0, Vm = - 2 0 mV, Vn = - 1 6 mV, 6m = 12 mV, 6n = 5.6 mV, 9S = 10 mV, and a = 0.85. These values are all adjusted to simulate experimentally observed time series. In accordance with the formulation used by Sherman et al. [11], all the conductances have been scaled with a typical conductance. Hence, we may consider the model as a model of a cluster of closely coupled /3-cells that share the combined capacity and conductance of the entire membrane area.

Figure 5.1 shows an example of the temporal variations of the variables V and S as obtained by simulating the cell model under conditions where it exhibits continuous chaotic spiking. Here, ks = 0.57 • 10~3 and Vs = —38.34 mV. We notice the extremely rapid opening and closing of the potassium channels. The opening probability n changes from nearly nothing to about 10% at the peak of each spike. We also notice how the slow variable increases during the bursting phase to reach a value just below 310, whereafter the cell switches into the silent phase and S gradually relaxes back. If the slow variable is assumed to represent the intracellular Ca2+-concentration, this concentration is seen to increase during each spike until it reaches a threshold, and the bursting phase is switched off. S hereafter decreases as Ca2+ is continuously pumped out of the cell.

Let us start our bifurcation analysis with a few comments concerning the equilibrium points of the /3-cell model. The zero points of the vector field in Eqs. (5.1)-(5.3) are given by

gcamx{V){V - Vca) + gKn(V - VK) + gsS(V - VK) = 0, (5.7)

Vn-V n -noo(V) =

and

1 + exp @n

(5.8)


S. -40 >

u u liMj W u 10 15 20 25

Time (s)

Fig 5.1. Example of the temporal variations of the membrane potential V(t) and slow variable S(t) that controls the switching between the active and the silent phases, ks = 0.57 • 10~3 and V$ = —38.34 mV. The model exhibits continuous chaotic spiking. Here, and in the following figures, S(t) has been multiplied by a factor TS/T.

s = s^v) = 1 + exp Vs-V Os

- l

(5.9)

so that the equilibrium values of n and S are uniquely determined by V. Substituting Eqs. (5.8) and (5.9) into (5.7), the equation for the equilibrium potential becomes

f(V) = gcamoolVXV - VCa) + 9Knoo(V){V - VK) + gsS^V^V -VK) = 0 (5.10)

with rriooCV) as given by Eq. (5.6). Assuming Vca > VK and considering the conductances gca, 9K and g$ to be positive by definition, we observe that any equilibrium point of the /3-cell model must have a membrane potential in the interval VK < V < Vca, and that there must be at least one such point. This assertion follows directly from the continuity of f{V) as defined from Eq. (5.10). For V < VK, all terms in the expression for f(V) are negative, and f(V) < 0. For V > Vca, all terms in f(V) are positive. Hence, there is at least one root of Eq. (5.10) in the interval between VK and Vca, and no roots outside this interval.

Evaluated at such an equilibrium point, the Jacobian matrix for the /3-cell model has the form


with

^11 J\2 ^13 J = { J2l J22 0 } (5.11)

• 31 0 J33

Jn = ~9Ca—gy—(V - Vca) - gCamooiV) v _ v (5.12)

J12 = ~9K(V - VK) (5.13)

Ji3 = -9sks(V-VK) (5.14)

_ dnjy) J21 = cr d v (5.15)

J22 = -a (5.16)

1/31 _ ~dv~ (5-17)

and

J33 = - * 5 - (5-18)

Applying Hurwitz' theorem (which gives the conditions for all solutions of the characteristic equation to have negative real parts) we obtain the following criteria for the equilibrium point to be asymptotically stable

Jn + J22 + J33 < 0 (5.19)

and

(Jll + J22 + J33)(^11^33 + ^11^22 + ^22^33) + ^31^22^13 + ^21^12^33 < 0. (5.20)

(For a characteristic equation of third order a-ox3 + a\x2 + aix + 03 = 0 with do > 0, we have the Hurwitz conditions


ai > 0 , ai

a3 0-2 > 0, and a3 > 0. (5.21)

In order to derive conditions (5.19) and (5.20) we have only to substitute the coefficients of our characteristic equation into (5.21), noting that the condition 03 > 0 will always be satisfied with the assumed parameter values.)

Fig 5.2. Bifurcation diagram for the bursting cell model as obtained by means of one-dimensional continuation methods. The equilibrium point undergoes a Hopf bifurcation for Vs = -42 mV and ks = 0.1.

For a Hopf bifurcation to occur, the second condition in (5.21) must be violated. As shown in the bifurcation diagram of Fig. 5.2, this may happen as Vs is increased. Here, we have plotted the equilibrium membrane potential V as a function of Vs for ks = 0.1. All other parameters assume their standard values. For low values of Vs (fully drawn curve in the bifurcation diagram), the equilibrium point is asymptotically stable. At Vs = - 4 2 mV, however, the model undergoes a Hopf bifurcation, and the equilibrium point turns into an unstable focus. The stable and the unstable branches of the bifurcation curve were followed by means of standard continuation methods. It should be noted, however, that due to the stiffness of the model, such methods are not easy to apply. In the next section we shall investigate the main organization of the subsequent bifurcations. Here, we shall base the analysis mainly on brute force bifurcation diagrams [35].

5.3 Bifurcation Diagrams for the Cell Model

Figure 5.3 shows a one-dimensional bifurcation diagram for the cell model with Vs as the control parameter. Here, ks = 0.57 • 10~3. The figure resembles figures that one can find in early papers by Chay [26] and by Fan and Chay


[27]. The diagram was constructed from a Poincare section in phase space with n = 0.04. With this secant we can ensure that all spikes performed by the model are recorded. For Vs > —37.8 mV, the model exhibits continuous periodic spiking, and the diagram shows only a single branch. As Vs is reduced, the spiking state undergoes a usual period-doubling cascade to chaos with periodic windows. Each window starts with a saddle-node bifurcation (to the right) and is terminated by a period-doubling cascade (to the left).

Fig 5.3. One-dimensional bifurcation diagram for ks = 0.57 • 10~3. The model displays chaotic dynamics in the transition intervals between continuous periodic spiking and bursting and between the main states of periodic bursting.

-40.0 -39.0 -38.0 V s (mV)

Around Vs = —38.3 mV, a dramatic change in the size of the chaotic at-tractor takes place. This marks the transition to bursting dynamics through the formation of a homoclinic connection in three-dimensional phase space [31]. Below Vs = —38.5 mV, the bifurcation scenario is turned around, and for Vs — —38.9 mV a reverse period-doubling cascade leads the system into a state of periodic bursting with five spikes per burst. The interval of periodic bursting ends near Vs = —39.7 mV in a saddle-node bifurcation leading to chaos in the form of type-I intermittency [40]. With further reduction of Vs, the chaotic dynamics develops via a new reverse period-doubling cascade into periodic bursting with four spikes per burst. It is clear from this description that chaotic dynamics tends to arise in the transitions between continuous spiking and bursting and between the different bursting states.

To establish a more complete picture of the bifurcation structure, we have applied a large number of such one-dimensional scans to identify the main periodic solutions (up to period-10) and to locate and classify the associated bifurcations. The results of this investigation are displayed in the two-dimensional


0.006

0.005

0.004

0.003 -

0.002 -

0.001 -

!

-

-

-

Hopf

\ i

FP£L~~-~~'

i i i i

1 - 2 / \

2-4 ^, ^

•^"^^^-i^^MfJ/ -

0 -45 -44 -43 -42 -41 -40 -39 -38

Vs(mV) -37

Fig 5.4. Two-dimensional bifurcation diagram outlining the main bifurcation structure in the (Vs, ks) parameter plane. Note the squid-formed black region with chaotic dynamics.

-40.5 -40.0

Vs(mV)

Fig 5.5. MagniEcation of part of the bifurcation diagram in Fig. 5.4. Note how the chaotic region in each squid arm narrows down as the bifurcation curves on both sides approach one another and intersect.


bifurcation diagram of Fig. 5.4. To the left in this figure we observe the Hopf bifurcation curve discussed in Sec. 5.2. Below this curve, the model has one or more stable equilibrium points. Above the curve we find a region of complex behavior delineated by the period-doubling curve PDl~2. Along this curve, the first period-doubling of the continuous spiking behavior takes place. In the heart of the region surrounded by PDl~2 we find an interesting squid-formed structure with arms of chaotic behavior (indicated in black) stretching down towards the Hopf bifurcation curve.

Each of the arms of the squid-formed structure separates a region of periodic bursting behavior with n spikes per burst from a region with regular (n + 1)-spikes per burst behavior. Each arm has a period-doubling cascade leading to chaos on one side and a saddle-node bifurcation on the other. It is easy to see that the number of spikes per burst becomes large as k$ approaches zero. Figure 5.5 is a magnification of part of the two-dimensional bifurcation diagram. Here, we observe how the chaotic region in the arms narrows down as the bifurcation curves on both sides approach one another with decreasing values of Vs- This leads to the so-called period-adding structure [26]. Along the curves of this structure a periodic bursting state with n spikes per burst appears to be directly transformed into a state with n + 1 spikes per burst.

Fig 5.6. One-dimensional bifurcation diagram obtained by scanning from the two spikes per burst regime into the three spikes per burst regime for ks = 1.0 • 10~3. Note the subcritical period-doubling and the associated transition to type-Ill intermittency for Vs ^ -40.827 mV.

-40.86 -40.84 -40.82 -40.80 -40.78 -40.76 Vs(mV)

To illustrate what happens in this transition, Figs. 5.6 and 5.7 show one-dimensional bifurcation diagrams obtained by scanning from the two spikes per burst regime into the three spikes per burst regime for ks = 1.0 • 10 - 3

and ks = 0.82 • 10~3, respectively [35]. To the left in Fig. 5.6 we find the


two spikes per burst behavior, and to the right we have periodic bursting with three spikes per burst. As Vs is gradually increased from -40.86 mV, the two spikes per burst behavior remains stable all the way up to Vs = —40.827 mV where it undergoes a subcritical period-doubling. In the absence of another attracting state in the neighborhood, the system explodes into a state of type-Ill intermittency [40].

If we go backwards in the bifurcation diagram of Fig. 5.6, the unstable period-4 solution generated in the subcritical period-doubling bifurcation stabilizes in a saddle-node bifurcation near Vs = -40.851 mV, and with increasing values of Vs the stable period-4 solution undergoes a period-doubling cascade to chaos. Around Vs — -40.841 mV, the chaotic attractor disappears in a boundary crisis as it collides with the inset to the unstable period-4 solution. This process is likely to leave a chaotic saddle which can influence the dynamics in the intermittency regime for Vs > -40.827 mV. For higher values of Vs, the chaotic state (with periodic windows) continues to exist until the saddle-node bifurcation at Vs = -40.765 mV where periodic bursting with three spikes per burst emerges.

Fig 5.7. One-dimensional bifurcation diagram obtained by scanning Vs in both directions across the PA 2 - 3 period-adding curve in Fig. 5.4. ks = 0.84 • 1(T3. Note the interval of coexisting two-spike and three-spike solutions.

-41.48 -41.46 -41.44 -41.42 41.40 -41.38 Vs (mV)

As ks is gradually reduced, the subcritical period-doubling bifurcation in which the two-spike solution is destabilized gradually approaches the saddle-node bifurcation that gives birth to the three-spike solution. In bifurcation diagrams like Fig. 5.6 one can see how the period-doubling bifurcation point gradually moves to the right relative to the rest of the diagram. Whenever the period-doubling bifurcation falls in a periodic window, the corresponding


periodic solution in general coexists with the two-spike solution. However, as ks is further reduced, and the period-doubling bifurcation point again enters a chaotic regime, the chaotic attractor soon disappears in a crisis as it collides with the unstable period-4 solution. Finally, Fig. 5.7 shows a brute force bifurcation diagram obtained by scanning Vs in both directions across the PA2~3 period-adding curve for ks — 0.84 • 10 - 3 . Inspection of this figure clearly reveals the narrow interval around Vs = —41.43 mV where the two-spike and three-spike solutions coexist. In full accordance with the above discussion, evaluation of the eigenvalues shows that the two-spike solution disappears in a (subcritical) period-doubling bifurcation and that the stable three-spike solution arises in a saddle-node bifurcation. In the next period-adding transition (PA3"4) the three-spike solution undergoes a subcritical period-doubling, and a four-spike solution emerges in a saddle-node bifurcation. Again there is a small interval of coexistence between the two solutions. This is a very different scenario from the continuous transition from n-spike to (n + l)-spike behavior described by Terman [29].

Fig 5.8. Phase space projection of the coexisting two-spike and three-spike solutions for Vs = -42.0 mV and ks = 0.699 • 10~3. Note the sharp point of departure between the two solutions.

V[V]

Figure 5.8 shows a phase space projection of the coexisting two-spike and three-spike solutions that one can observe for Vs = —42.0 mV and ks = 0.0069 • 10~3. Note how these solutions follow one another very closely for part of the cycle to depart sharply at a point near V = —57 mV and S = 264. Hence, with a slightly reduced numerical accuracy, it may appear as if the two solutions smoothly transform one into the other. Figure 5.9 displays the basins of attraction for the two coexisting solutions. Here, initial conditions attracted to the two-spike solution are marked gray, and initial conditions from which


the trajectory approaches the three-spike solution are left blank. The figure was constructed for initial values of the fast gate variable of n = 0.04. Finally, Fig. 5.10 shows a magnification of part of the basin boundary in Fig. 5.9 around V = -50 mV and 5 = 210.2. Inspection of this magnification clearly reveals the fractal structure of the basin boundary with the characteristic set of bands of rapidly decreasing width.

Fig S.9. Basins of attraction for the coexisting two-spike and three-spike solutions Fig 5.10. Magnification of part of the basin for ks = 0.84 • 10-*. Initial conditions boundary in Fig. 5.9 illustrating the charac-from which the trajectory approaches the two teristic fractal nature of this boundary, spikes per burst solution are marked gray.

In this section we have presented a simplified bifurcation analysis of a three-variable model that can produce the characteristic bursting and spiking behavior of pancreatic /3-cells. (A more mathematically oriented description of the homoclinic bifurcations leading to bursting was given elsewhere [31].) Our main observations were,

(i) a squid-formed regime of chaotic dynamics may exist in parameter plane inside the region surrounded by the first period-doubling curve for the periodic spiking behavior. The arms of this squid separate regions of different number of spikes per burst,

(ii) each arm has a structure with a period-doubling cascade on one side and a saddle-node bifurcation on the other,

(iii) towards the end of the arms the first period-doubling bifurcation tends to become subcritical. In a certain parameter region this gives rise to a chaotic boundary crisis followed by a transition to type-Ill intermittency,


(iv) the so-called period-adding structure arises when the subcritical period-doubling curve intersects the saddle-node bifurcation curve on the other side of the arm. This leads to a region of coexistence of stable n- and (n + l)-spike behavior.

These results are at odds with the bifurcation structures usually presented in the literature [29]. It is obvious that different results may be obtained with different models and different parameter settings. However, the consistency in our bifurcation scenarios seems to imply that the same scenarios may be found in other bursting cell models as well.

5.4 Coupled Chaotically Spiking Cells

As discussed in the introduction to this chapter, a variety of experimental studies have shown that neighboring /3-cells tend to synchronize their electrical activity [10, 11] and that cytoplasmic Ca2+-oscillations can propagate across clusters of /3-cells in the presence of glucose [12, 13]. The precise mechanisms underlying these phenomena are not yet known. It is generally considered, however, that the exchange of ions via low-impedance gap junctions between the cells plays a dominant role [15]. Other possible coupling mechanisms involve interaction via hormonal signals or adjustments in the behavior of a particular cell in response to the intercellular Ca2+-concentration produced by the neighboring cells. The /3-cells are arranged along small capillaries, into which they secrete their insulin. Hence, there is the possibility that a given cell reacts to the release of insulin from cells that are upstream to it along the same capillary. These coupling mechanisms are all of a local nature. However, there are also more global controls of the cells via nerve signals and via the cells reaction to the glucose concentration in the blood, a concentration which to a large extent is controlled by the overall release of insulin from the same cells (i.e., a type of global coupling). All of these coupling phenomena are of significant interest, because not only do they influence the activity of the individual cell, but they may also give rise to new types of oscillations in the overall insulin secretion.

Synchronization of the bursting behavior between two interacting /3-cells was studied in detail by Sherman [25] who showed how various forms of in-phase, antiphase and quasiperiodic behaviors can arise, depending on the strength of the intercellular coupling. More recently, de Vries et al. [41] have extended the investigation to allow for cell inhomogeneities (or a parameter mismatch


between the cells). Both of these investigations were devoted to studying synchronization between periodically bursting cells, and the particular phenomena that one expects to observe in connection with chaotic synchronization were not considered. As we have seen, chaotic spiking behavior typically arises in the transition region between bursting and continuous periodic spiking. In practice, this type of behavior is observed at relatively high extracellular glucose concentrations (11-16 mM), where the release of insulin approaches saturation. This is also the regime where the interaction between the /3-cells is assumed to be most pronounced [13]. Hence, it is of interest to consider the case of chaotically oscillating /3-cells as well.

A system of two coupled, identical /3-cells may be defined through the equations

x= f(x) + C{y - x) (5.22)

V= f(y) + C(x - y) (5.23)

where x= f{x) with x = (Vi, n\, S\) and V= f(y) with y = (V2, n2, S2) represent the dynamics of the individual cells in accordance with Eqs. (5.1-5.6). C is the coupling matrix, for which we shall assume the form C = diag (d\, 0, d3), indicating that coupling takes place via the first and the third variables. The membrane potentials are coupled resistively via electric currents that flow between the cells, and (to the extent that it represents the Ca+2-concentration in the cells) the third variable is coupled via the diffusive exchange of calcium. In the rest of this chapter we shall assume that Vs = —38.34 mV and use the coupling constants d\ and d$ as bifurcation parameters.

To examine the stability of the synchronized chaotic state to desynchronizing perturbations we proceed in precisely the same way as for the coupled Rossler systems in Chapter 4. Hence, we introduce a set of longitudinal and transverse coordinates, 77; and £; with i = 1,2,3. A necessary condition for the synchronized chaotic state to be transversely stable is that the largest transverse Lyapunov exponent is negative. If this condition is satisfied, the state is at least weakly stable [42]. For the synchronized chaotic state to be asymptotically stable, not only the typical trajectory but all trajectories on the chaotic set must be transversely stable. This leads us again to consider the transverse stability of the low-periodic orbits embedded in the chaotic state [43].


Figure 5.11 shows a bifurcation diagram for the coupled cell system in the two-parameter coupling plane. Here, the curves marked period-1, period-2, period-3, and period-4 delineate the regions in which the synchronized period-1, period-2, etc., saddle cycles are transversely stable. The regions of stability are to the upper right, where the coupling is positive and sufficiently strong.

o.i

0.08

0.06

0.04

0.02

0

-0.02

-0.04

! ! ' ! ; periodeM J ;

i I

: I I ! " / / •

... ^ 4 H . H . ^ ^

;J

1 / PV 1 1

H

1 perimfc-1'

pwiW?-2'

I

PD

: r*%" i i i i i

1

: PF

i

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Fig 5.11. Phase diagram showing the curves in two-parameter coupling space where some of the low-periodic orbits embedded in the synchronized chaotic state lose their transverse stability. PF denotes a pitchfork bifurcation, PD a period-doubling bifurcation, and H a Hopf bifurcation.

Transverse destabilization occurs via a pitchfork bifurcation along those parts of the bifurcation curves that are denoted PF, via a period-doubling along those parts that are indicated PD, and via a transverse Hopf bifurcation where the curves are denoted by an H. We notice that all these types of transverse destabilization of low-periodic orbits can be observed in the region where d\ and d% are both positive, and that the order in which the various cycles destabilize changes along the bifurcation curves. In no case, however, is a transverse Hopf bifurcation the first bifurcation to occur. Hence, we cannot investigate riddling phenomena associated with this type of bifurcation. With the considered coupling structure, the pitchfork bifurcations occur along the sides of the stability region, and the period-doubling and Hopf bifurcations at the lower left corner of this region, where both coupling constants are numeri-


cally small. A Hopf bifurcation always appears to occur between the pitchfork and the period-doubling bifurcation.

The parameter region of interest in connection with a study of locally and globally riddled basins of attraction is the region where the first low-periodic cycle has become transversely unstable while the synchronized chaotic state is still stable on the average. Figure 5.12 shows the same bifurcation diagram as Fig. 5.11, but the curve that marks the blowout bifurcation has been included. At this curve (heavy line), the largest transverse Lyapunov exponent becomes positive, and the synchronized chaotic state loses its average attraction. By comparing the positions of the various curves, we conclude that the riddling transition takes place via a pitchfork bifurcation of the period-1 saddle cycle along the branches I and III and via a period-doubling bifurcation of the period-4 saddle at branch II.

o.i

0.08

0.06

0.04

d3

0.02

0

-0.02

-0.04 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Fig 5.12. Same bifurcation diagram as in Fig. 5.11, but the curve that marks the blowout bifurcation has also been drawn. Riddling of the basin of attraction for the synchronized chaotic state occurs in the region between the first transverse destabilization of a periodic orbit and the blowout bifurcation.

As discussed for the coupled Rossler oscillators in Chapter 4, each of the transverse bifurcations may be either sub- or supercritical [44, 45]. In a sub-critical pitchfork bifurcation, two mutually symmetric doubly unstable cycles will be situated on either side of the symmetric saddle cycle before the bifur-

1

Blowout

- i"\S

/

x j

1 r\

*^

i

. . . .

^

j

,11 :

^ - H ^ - ^ V . ; ^

i i i

i

gff^r1-^

i

i

• -

III

i


cation. As the point of destabilization is approached, the asynchronous cycles move closer and closer to the saddle cycle to finally be annihilated with it and leave a doubly unstable cycle in the synchronization manifold. The unstable manifold of this cycle often stretches all the way to infinity, allowing trajectories starting near the synchronization manifold to diverge if they happen to pass close to the transversely unstable periodic orbit. By virtue of the transitivity of the chaotic state, many trajectories will do so, and this produces an observable riddling of the basin of attraction for the synchronized chaotic state. An analogous description applies for a subcritical period-doubling bifurcation.

In a supercritical transverse pitchfork bifurcation, two mutually symmetric saddle cycles will exist around the destabilized low-periodic orbit immediately after the bifurcation. In analogy with the formation of a mixed absorbing area for two coupled one-dimensional maps, as discussed in Chapters 2 and 3, the stable and unstable manifolds of these cycles are likely to wrap around the synchronized chaotic state, and in this way trajectories starting near the synchronization manifold may be restrained from reaching other asymptotic states or diverging to infinity. Hence, we can expect to observe the phenomena of local riddling, attractor bubbling, and on-off intermittency. Unfortunately, for coupled time-continuous oscillators we are not able to determine the various manifolds of the asynchronous orbits or to directly verify the existence of a trapping zone.

5.5 Locally Riddled Basins of Attraction

In order to investigate the different phenomena in more detail we have performed a series of experiments in which trajectories have been started near a destabilized periodic cycle, but with a small displacement 5 transverse to the synchronization manifold in the unstable direction [32]. When following trajectories starting near the unstable period-1 cycle for the coupling parameters d\ — 0.067 and d$ = 0.030, we typically obtain phase plots of the form illustrated in Figs. 5.13(a) and (b). For some values of the transverse perturbation (Fig. 5.13(a)), trajectories never move far away from the synchronization manifold and are soon attracted to the synchronized chaotic state. For other values of 6 (Fig. 5.13(b)), the trajectories make a much larger excursion in phase space. At the end, however, they are almost all attracted to the synchronized chaotic state. We take this as an indication of the existence of a trapping region around


the synchronized chaotic state. Hence, the basin of attraction is only locally riddled, and the observed phenomenon is an example of attractor bubbling.

-i -20.0 I 1 1 . 1

-30.0 • ift/ A/W// '

-4o.o • I' BM //Jyx^^ -

-60.0 - / /

-,„.„ -70.0 I ' ' ' ' -70.0 -60.0 -50.0 -40.0 -30.0 -20.0 -70.0 -60.0 -50.0 -40.0 -30.0 -20.0

V, (mV) Vi (mV)

Fig 5.13. Attractor bubbling, (a) and (b) display typical trajectories observed for d\ = 0.067 and dz = 0.030 (i.e., near branch I in Fig. 5.12). In (a) the trajectory is almost immediately absorbed by the synchronized chaotic state. In (b) the trajectory first performs a major excursion in phase space. In the presence of noise, one can observe randomly excited excursions of the form (b).

If the coupling parameter di is reduced to d\ = 0.063 while cfa is maintained at G?3 = 0.030, we have crossed the blowout bifurcation curve in the bifurcation diagram of Fig. 5.12. Hence, the synchronized chaotic state is no longer attracting on the average. However, a trapping regime still exists, and as a result we can observe the phenomenon of on-off intermittency. This is illustrated by the phase space trajectory of Fig. 5.14(a) and the corresponding temporal variation in Fig. 5.14(b). Here, we can observe how the system exhibits laminar (or off) phases of varying lengths in which the membrane potentials of the two /3-cells exhibit the normal chaotic oscillations, interrupted by relatively short turbulent (or on) phases where the cells move out of synchrony. With decreasing values of the coupling parameter d\, the average length of the laminar phases is found to decrease as we move further away from the blowout bifurcation. It appears, however, that on-off intermittency still occurs for a coupling parameter as small as d\ = 0.04. For this value of d\, all the considered low-periodic saddle cycles have become transversely unstable.

In order to visualize the structure of the locally riddled basins of attraction we have introduced a distinction between trajectories that are attracted almost im-

^ -40.0 • > B

> -50.0 •


-» ! Il l K

t i

liiii !KH!

120 160 200 240 Time(s)

Fig 5.14. On-off intermittency for rfi = 0.063 and d3 = 0.030. The synchronized chaotic state has lost its average attraction, (a) Phase space trajectory and (b) corresponding temporal variation of the membrane potential V\.

Fig 5.15. Structure of the locally riddled basin of attraction for di = 0.070 and d3 = 0.030. % = V2 + V\ and & = V2 — V\. Initial conditions from which the trajectories are almost immediately attracted to the synchronized chaotic state are shown in gray.


mediately to the synchronized chaotic state (like the trajectory in Fig. 5.13(a)) and trajectories that, while finally being attracted to the synchronized state, first perform a significant burst into phase space (like in Fig. 5.13(b)). Figure 5.15 shows a cross section of the basin of attraction for d\ = 0.070 and d$ = 0.030. With these two values of the coupling parameters, only the synchronized period-1 cycle is transversely unstable. Initial conditions from which the trajectories are immediately attracted to the synchronized chaotic state are shown in gray, and initial conditions from which the trajectories first make a major excursion into phase space are left blank. Although the tongues are extremely narrow, we notice the dense set of locally repelling tongues emanating from the chaotic set in the synchronization manifold (£i = 0).

Fig 5.16. Temporal variation (a) and phase space projection (b) of a trajectory that starts in the white area of Fig. 5.15. We observe how the trajectory approaches the unstable period-1 orbit in the synchronization manifold and then bursts away.

Figures 5.16(a) and (b) illustrate how a bubbling excursion is initiated. Here we have plotted the temporal variation and the corresponding phase plot for a trajectory that starts in the white area of Fig. 5.15. (More precisely, the initial conditions in the considered cross section are 771 = —125 mV and £1 = 0.78 mV). The temporal behavior shows how the membrane potentials after a short transient, reach a motion of near synchrony. After approximately 12 s, this motion has approached the unstable period-1 cycle, and the transverse perturbation starts to grow, leading the coupled cells into the very different, asynchronous bubbling phase. The phase plot illustrates the relatively long period of time that the system spends in the neighborhood of the period-1 cycle before it bursts away from synchrony. With the considered coupling parameters, the cells will finally synchronize in the chaotic state.


-120 -100 -80 -60 -40 -120 -100 -80 -60 -40

Fig 5.17. Variation in the structure of the locally riddled basin of attraction as the coupling parameter d\ is reduced, (a) d\ = 0.069 and (b) dx = 0.067. We observe how the riddled structure becomes more and more open.

Figures 5.17(a) and (b) illustrate the changes that occur in the structure of the locally riddled basin of attraction as the system approaches the point of blowout. Here, d$ is maintained constant at d% — 0.030 while d\ is gradually reduced. In Fig. 5.17(a), d\ = 0.069, and like the synchronized period-1 cycle, the period-2 cycle has now become transversely unstable. If one follows the trajectory starting, for instance, from r/i = —62 mV and £i = 0.98 mV one can observe how it approaches the unstable period-2 orbit and moves in the neighborhood of this cycle for a while before the transverse perturbation grows large enough for the cells to move out of synchrony in a bubbling dynamics.

As compared with Fig. 5.15, the riddled basin structure in Fig. 5.17(a) is considerably more pronounced. Figure 5.17(b) shows the structure of the locally riddled basin for d\ — 0.067. For d\ — 0.066, i.e., right before the blowout bifurcation, the measure of initial conditions in the neighborhood of the synchronized chaotic state for which the trajectories are immediately attracted is quite small.

5.6 Globally Riddled Basins of Attraction

In accordance with our discussion in Sec. 5.3, the presence of a locally riddled basin of attraction is likely to be associated with a supercritical riddling bifurcation. Globally riddled basins of attraction will arise either directly after a subcritical riddling bifurcation or after a supercritical riddling bifurcation followed by a global bifurcation in which the locally riddled basin of attraction is


transformed into a globally riddled basin. Both of these scenarios were considered in detail for a system of two coupled logistic maps in Chapters 2 and 3, and similar scenarios also occured in our model of two coupled Rossler oscillators in Chapter 4.

0 10 20 30 40 50 60 70 Time (s)

Fig 5.18. Temporal variation of the membrane potential V\ for two different initial conditions with d\ = 0.090 and d3 = 0.0025, i.e., near branch II in Fig. 5.12 where the symmetric period-4 cycle has undergone a transverse period-doubling bifurcation.

0 20O0 4000 6000 8000 100O0 12000 14000

Time (units)

0 2000 4000 6000 8000 10000 12000 14000

Time (units)

Fig 5.19. Temporal variation of the membrane potentials for the two coupled cells for dx = 0.090 and d3 = 0.00125. The basin of attraction is globally riddled, and many trajectories escape synchronization and approach an asynchronous chaotic state.

Let us start by considering the structure near branch II in Fig. 5.12 where the synchronized period-4 cycle is the first to lose its transverse stability, and where the destabilization occurs via a period-doubling bifurcation. For d\ — 0.090 and c?3 = 0.0025 we again observe a locally riddled basin of attraction. In spite of the dense set of repelling tongues issuing from the synchronized chaotic


state, almost all trajectories starting in the neighborhood of this state sooner or later end up being attracted to it. However, the trajectories can perform several bursts away from synchrony before the attraction materializes. This is illustrated in Fig. 5.18, where we have plotted the temporal variation of the membrane potential for two different initial conditions near to the synchronized chaotic state.

-120 -100 -80 -60 -40

T^mV)

Fig 5.20. Cross section of the globally riddled basin that exists for di = 0.104 and d3 = —0.0065. Here, black points denote initial conditions from which the two cells synchronize. Trajectories starting in white points diverge to infinity.

In the left panel of Fig. 5.18, the chaotically spiking cells synchronize almost immediately. In the right panel, a couple of excursions into phase space must be completed before synchronization is achieved. For some initial conditions, five or more such bubbling excursions are observed. As d$ is reduced, we observe a crisis-like metamorphosis of the basin of attraction from a locally riddled into a globally riddled structure. This implies that the nonlinear restraining mechanisms responsible for the existence of a trapping region have ceased to function and that routes have been opened for trajectories starting near the synchronized chaotic state to reach other limiting states [46]. For d\ = 0.090 and c?3 = 0.00125, many trajectories are found to approach an asynchronous chaotic state with a typical temporal variation as illustrated in Fig. 5.19. The transition from locally to globally riddled basins of attraction occurs even before the point of blowout is reached. Under these conditions, on-off intermittency


cannot be observed. Along branch III of the riddling bifurcation curve in Fig. 5.12, the trans

verse pitchfork bifurcation of the symmetric period-1 orbit is subcritical, and the riddling bifurcation directly leads to a globally riddled basin of attraction. Figure 5.20 shows a cross section of this basin for d\ — 0.104 and d$ = —0.0065. Here, black points denote initial conditions from which the two cells attain complete synchronization, and white points represent initial conditions from which the trajectories diverge to infinity. The cross section was obtained for & = & = 0, m = 0.02, and % = 616.

5.7 Effects of Cell Inhomogeneities

Let us finally consider our system of two coupled /3-cells in the presence of a small parameter mismatch between the cells [33]. A simple way to introduce such an inhomogeneity is to assume that one of the cells beside the currents specified in Eq. (5.1) has an additional small and constant leak current Iieak-

Figures 5.21(a), (b) and (c) show the variation in the potential difference V\{t) — V2(t) for a pair of coupled /3-cell models with a constant leak current of magnitude 7/eafc = 2.0 • 10~4 applied to one of the cells. The coupling parameter a = di = di = d% is increased from a = 0.2 in Fig. 5.21(a) to a = 0.5 in Fig 5.21(b) and a = 1.0 in Fig. 5.21(c). In all cases we observe that the parameter mismatch shifts the region of operation away from the symmetric situation V\{t) — V^i) = 0. At the same time it produces a nearly periodic bursting behavior where (relatively small) differences in the spiking potentials arise. As the coupling parameter is increased, both the shift away from the symmetric manifold and the magnitude of the bursts decrease in full agreement (quantitatively as well as qualitatively) with the predictions of Chapter 4.

As discussed above, the bursting behavior of pancreatic /3-cells is considered to play an essential role in organizing the secretion of insulin as a function of the extracellular glucose concentration. It is also clear that interactions between the cells (via low-impedance gap junctions and other mechanisms) are important, both because they can shift the cell community between different states of synchronization and because the interactions can give rise to waves that propagate across the cells. Unfortunately, at the present very little is understood about such phenomena. Our investigations have revealed that a model of two coupled, chaotically spiking and identical /3-cells can exhibit a


„ 0.15

120 130

Time (s)

(a)

120 130

Time (s)

00

' j

120 130

Time (s)

W

Fig 5.21. Variation in the potential difference V\(t) - Vi[t) for a system of two coupled /?-cells with a parameter mismatch. As the coupling parameter a is increased, both the shift of the average working conditions and the occasional bursts are reduced, (a) a = 0.2, (b) a = 0.5, and (c) a = 1.0. Note, that in this simulation d<i = a / 0.


variety of different sub- and supercritical riddling bifurcations, involving transverse pitchfork or period-doubling bifurcations. With the assumed coupling structure, destabilization of low-periodic orbits embedded in the synchronized chaotic state tends to occur via pitchfork bifurcations along the sides of the stability regime, and via period-doubling or Hopf bifurcations in the lower left corner, where both coupling constants are relatively small. On one side of the stability region, the pitchfork bifurcations are subcritical, and along the other side they are supercritical. These results are surprisingly similar to the results we found for coupled Rossler systems in Chapter 4.

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[28] X.-J. Wang, Genesis of Bursting Oscillations in the Hindmarsh-Rose Model and Homoclinicity to a Chaotic Saddle, Physica D 62, 263-274 (1993).

[29] D. Terman, Chaotic Spikes Arising from a Model of Bursting in Excitable Membranes, SIAM J. Appl. Math. 51, 1418-1450 (1991).

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[32] B. Lading, E. Mosekilde, S. Yanchuk, and Yu. Maistrenko, Chaotic Synchronization between Coupled Pancreatic (i-Cells, Prog. Theor. Phys. Suppl. 139, 164-177 (2000).


[33] S. Yanchuk, Yu. Maistrenko, B. Lading, and E. Mosekilde, Effects of a Parameter Mismatch on the Synchronization of Two Coupled Chaotic Oscillators, Int. J. Bifurcation and Chaos 10, 2629-2648 (2000).

[34] K. Kaneko, Relevance of Dynamic Clustering to Biological Networks, Phys-ica D 75, 55-73 (1994).

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[36] R.E. Plant and M. Kim, Mathematical Descriptions of a Bursting Pacemaker Neuron by a Modification of the Hodgkin-Huxley Equations, Biophys. J. 16, 227-244 (1976).

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Chapter 6

CHAOTIC PHASE SYNCHRONIZATION

6.1 Signatures of Phase Synchronization

In the first part of this book we have been concerned mostly with the interaction of identical chaotic oscillators, or with situations where there was only a small mismatch between the parameters of the interacting subsystems. This has led us to discuss a variety of interesting phenomena, including weak attractors [1], riddled basins of attraction [2, 3], and on-off intermittency [4]. Besides the significance that these phenomena may have from a mathematical point of view, they are also of interest in connection with ongoing research aiming at the use of chaotic synchronization for new types of communication [5, 6]. We have illustrated how on-off intermittency and attractor bubbling can arise in a model of interacting pancreatic cells [7], and we have shown how different forms of trapping zones play an important role for the behavior of the coupled oscillators once the synchronization breaks down [8].

In general, we do not expect interacting chaotic oscillators to be identical, particularly not in the living world. It turns out, however, that even when the parameters or the nature of the interacting systems are different, chaotic oscillators can show evidence of synchronization. Typically, one observes that while the amplitudes of the interacting subsystems vary quite differently, the average frequencies (or the mean return times to some Poincare secant) are commensurate (i.e., they attain a rational ratio). This phenomenon, which is

211


similar in many respects to the synchronization phenomenon we know for regular oscillators [9] is termed chaotic phase synchronization. It was observed in experiments with coupled electronic circuits in the early 1990's [10, 11, 12], but has subsequently been found for a wide variety of different systems, including the synchronization of the respiratory cycle with the heart beat for healthy young subjects [13, 14].

In Chapter 9 we shall discuss an example of chaotic phase synchronization in the pressure and flow regulation of neighboring functional units of the kidney. The purpose of the present chapter is to examine the phenomenon of chaotic phase synchronization from a somewhat more theoretical point of view. The idea in this presentation is to emphasize the similarity between the transition from non-synchronous to synchronous chaos and the transition from quasiperi-odic to periodic behavior as described by the classical theory [9]. We shall follow the changes that occur in the spectral distributions and Lyapunov exponents during the transition to synchronized chaos, we shall consider the bifurcation structure of this transition [15] and the role of multistability [16], and we shall examine the transition to synchronized chaos via suppression of the natural dynamics [17]. Finally, at the end of the chapter, we shall discuss a new mechanism for the realization of a so-called chaotic hierarchy [18].

Most of the published work on chaotic phase synchronization refers to the case where the chaotic dynamics has developed through a cascade of period-doubling bifurcations. This type of behavior is usually characterized by the rotation of the trajectory around some center point and by the presence of a fundamental frequency, representing the mean return time to some Poincare section.

Some of the early works [11, 12] considered coupled Anishchenko-Astakhov generators in order to compare experimental results with theoretical considerations. Recent theoretical work [19, 20, 21, 22] has largely focused on coupled Rossler systems, but there have also been efforts to study various types of electronic, physical, and biological systems, including coupled Chua circuits [23, 24], Lur'e systems [25], and magnetoencephalographic signals [26].

In the present chapter we shall base most of our analyses on the Rossler system [27]:

x = — uy — z

y = ux + ay (6.1)

Z = fi + z(x - jti),

Chaotic Phase Synchronization 213

where a, 8, and fi are control parameters, and w determines the characteristic (angular) frequency of the oscillations. As discussed in Chapter 1, the chaotic dynamics of this model is well-studied. There are two possible types of chaotic attractor. A cascade of period-doubling bifurcations leads to the emergence of chaos. This is referred to as "spiral chaos". If /x is further increased, a more complicated type of chaos referred to as "screw chaos" can be observed [12].

(b)

-60.0 i i ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ M 1.0 0.0 0.5 1.0 1.5 2.0

CO

Fig 6.1. The chaotic attractor of the Rossler model demonstrates the typical features of period doubling chaos. One can observe how the phase trajectory rotates around some center point in the two-dimensional projection (a), and one can see the corresponding peak in the power spectrum (b).

As illustrated in Fig. 6.1(b), spiral (or weakly-developed) chaos in the Rossler model is characterized by the presence of a strong peak in the power spectrum. In order to examine some of the basic aspects of the transition from non-synchronous to synchronous chaos we have previously considered a forced version of the Rossler system:

x = —uiy — z + Ksinujft

y = OJX + ay

z = /3 + z(x — fi).

(6.2)

For finite values of K we showed how one can find an interval of u> where the peak frequency of the spectrum is in a 1:1 relation with the forcing frequency Uf. Let us try now to pursue this synchronization phenomenon a little further.


Mean return time and mean frequency

We define the characteristic period of a chaotic oscillator as the mean return time < T > of the phase trajectory to a properly chosen secant surface. In this way, the mean frequency of the chaotic oscillations is related to the characteristic period by

< ujm > = 2-KJ < T > (6.3)

By analogy with a forced regular oscillator, the ratio between the mean return time and the period 7 / = 2n/u)f of the external forcing can be considered as a mean rotation number < 8 >. The main region of chaotic synchronization then corresponds to < 6 > = 1 : 1, and higher order regions correspond to < 6 >= p : q, with p and q being intergers.

Numerical calculation of the mean return time to a secant surface allows us to easily plot the mean rotation number as a function of the parameters of a system and, hence, to determine the boundary of the various synchronization regions. With a forcing frequency UJJ = 1, the mean rotation number is given by

< 9 >=< T > /2TT. (6.4)

1.004

1.002

A ® 1.000 V

0.998

0.996

(a)

aoooooooooooooq - *j .000

- 0.998

_ i _

0.915 0.920 0.925 CO

1.004

1.002

0.996 0.920 0.925 0.930

(0

0.996 0.925 0.930 0.935 0.940

(0

Fig 6.2. Rotation number vs. the endogenous frequency parameter u (a) for regular dynamics (/j, = 3.5), (b) for weak chaos (/i = 4.0) and (c) for fully developed chaos (n = 6.5). The forcing frequency u)j = 1. Other parameters for the forced Rossler oscillator are fixed at a = 0.2, fi = 0.2, and K = 0.02. < 9 > is computed over 104 quasi-periods.

Figure 6.2 displays the mean rotation number for the forced Rossler oscillator Eq. (6.2) as a function of w in an interval around the main synchronization


region for regular dynamics (a), for weak chaos (b), and for fully developed chaos (c). Here, \i = 3.5, 4.0, and 6.5, respectively. As a secant surface we have used the plane x = 0. Figure 6.2(a) presents a single step of a "devil's staircase" with its well-known fractal structure of synchronization intervals [9]. In Fig. 6.2(b), where the dynamics of the Rossler oscillator is weakly chaotic, one can still distinguish the region of 1 : 1 synchronization. (Note, that when the external forcing is close to being resonant, the chaotic dynamics in the forced Rossler system (6.2) sets in at lower values of (x than in the autonomous Rossler system.) For fully developed chaos (Fig. 6.2(c)), on the other hand, the region with constant rotation number is nearly destroyed.

0.05

0.04

0.03

L

0.02

0.01

0.0

(a)

jj

•

0.05

0.04

0.03

0.02

0.0

' (b)

uy -

5.0 5.5 6.0 6.5 7.0 5.0

T

6.0 6.5 T

Fig 6.3. Histograms for the return time T to a Poincare secant for synchronous (a) and asynchronous (b) chaos. The parameter values are the same as in Fig. 6.2(b). The mean return time < T > = 2%/ujm.

The mean return time < T > is not necessary the most probable one. In Fig. 6.3 histograms of return times are given for synchronous (a) and asynchronous (b) chaos. It is evident that cum = 2n/ < T > does not correspond to a specific maximum of the plots. Note also that the transition from (a) to (b) is not associated with obvious qualitative changes in the distribution of the return times. In the synchronization process, the chaotic attractor changes its internal rhythm as an integrated structure.

Momentary amplitude and phase

To introduce a momentary (or instantaneous) amplitude and phase of an irregular (e.g., chaotic) signal x(t) in a unique manner one can use the following


representation:

x{t) = A(t)cos${t)

H{x{t)} = A(t)sin*(*). (6.5)

Here, A(t) and &(t) are the momentary amplitude and phase of the signal and H{...} denotes Hilbert transformation [32]. This approach to the investigation of chaotic dynamics was first introduced by Rosenblum et al. [19]. In Chapter 9 we shall illustrate how it can be put to use in practice. We emphasize, however, that numerical determination of the variables A(i) and $(i) may sometimes be fairly complicated.

In the case when the dynamical variables x(t) and y(t) are connected in a linear way (as for the Rossler system), it is easier to apply the following substitution:

x(t) = A{t)cos$(t)

y(t) = A(t)sin$(<), (6.6)

where A(t) and $(£) denote the polar coordinates of the point (x(t), y(t)) in the (xj/)-plane. A disadvantage of this simple approach is that the momentary amplitude and phase can be determined in different ways, because they depend on the choice of the two variables x{t) and y(t) in the phase space projection. However, for the Rossler system the procedure is generally applicable. Substituting (6.6) into (6.2) and setting cuj = 1 it is easy to recast the system into the new variables:

A = aA + (K sin t — a A cos 5> - z) cos $

<t> = LO - (1/A)(Ksini - aAcos<E> - z ) s i n $

z = a + z(Acos$- n). (6.7)

The behavior of the phase difference between the internal oscillations and the external forcing A$(i) = $(<) - < on both sides of the boundary of the synchronization region is shown in Figs. 6.4(a) and (b) for weak and fully developed chaos, respectively. Inside the synchronization region (curve 1), the phase difference for weak chaos oscillates slightly near the constant mean level. Outside the synchronization region (curve 2) a continuous increase of the phase difference with time can be observed. The boundary of phase locking for chaotic oscillations determined by this procedure coincides with the boundary obtained


40.0 -.

0 2000 6000

<

100

-100

-200 10000 0 2000 6000 10000

Fig 6.4. Momentary phase difference as function of time for the forced Rossler system: (a) for weak chaos (fi = 4.0) inside the synchronization region (curve 1 at u = 0.927) and outside this region (curve 2 at to = 0.930); (b) for fully developed chaos (/j = 6.5), curve 1 corresponding to u = 0.933 and curve 2 to ui = 0.937. Note the difference in vertical scales between the two figures, (a = 0.2, P = 0.2, and K = 0.02).

by calculation of mean return times to a secant plane. For fully developed chaos we observe a stepwise variation of A<3>, implying that the phase difference remains constant, but occasionally undergoes a slip of ±2n [28].

6.2 Bifurcational Analysis

With the above introductory remarks, let us now consider the results of a more detailed bifurcation analysis of the forced Rossler system [15]. Figure 6.5 displays a segment of the phase diagram in the vicinity of the main synchronization region in the (CJ, /x) parameter plane. Other parameters are fixed at a = 0.2, /3 = 0.2, and K = 0.02. The following regions can be distinguished in the figure:

1 - region where a stable cycle with period 4Tf exists; 2 - region where a stable cycle with period 8T/ exists; 3 - region of synchronous chaos; 4 - region of four-band torus, corresponding to quasiperiodic oscillations

with two incommensurable periods, one of which is the period of the external forcing T/ and the other 4To is related to the internal period of the autonomous limit cycle oscillations;

5 - region of eight-band torus (the periods of oscillation are Tj and 8TQ) ;


6 - region of non-synchronous chaos with the spectrum of Lyapunov exponents " + , 0 , - , - " ;

7 - region of non-synchronous chaos with the spectrum of Lyapunov exponents " + , 0 , 0 , - " .

The figure also shows the period-3 and period-5 windows with some of their main subregions. For the bifurcation curves we have used the following notation:

h is a curve of tangent bifurcation of limit cycles; h is a curve of period-doubling bifurcation; Z3 is a curve of doubling of one of the torus periods; Icri is a critical curve, corresponding to the accumulation of period-doubling

bifurcations of cycles inside the synchronization region; la-i is a boundary for the appearance of chaos outside the synchronization

region; Z4 is a boundary of the synchronization region; Z5 is a boundary between two types of chaos outside the synchronization

region;

Fig 6.5. Bifurcation diagram for the forced Rossler system (6.2) (a = 0.2, j3 = 0.2, and K = 0.02). The figure shows how 1:1 synchronization tongue continues up through the period-doubling cascade into the region of chaos. In region 3 the dynamics is synchronized chaos, whereas regions 6 and 7 correspond to two different forms of non-synchronous chaos.

"6.914 0.916 0.918 0.920 0.922 0.924 0.926 0.928

(0

The destruction of quasiperiodic oscillations precedes the appearance of chaos along the curve Z„.2- The bifurcational structure associated with curve 1$


between the two types of asynchronous chaos is not clear. Inspection of figure 6.5 shows how the synchronization tongue (regions 1 and

2) continues through the period-doubling cascade into the region of chaotic synchronization. We can also see how the critical bifurcation curve Z4 separates the region of chaotic synchronization 3 from the regions of asynchronous behavior. Finally, we note how the synchronization tongue is shifted to lower values of u as the system (under variation of /i) enters the period-3 window.

To understand the dynamics of system (6.2) when the nonlinearity parameter \i and the forcing amplitude K are changed, let us consider the part of the bifurcation diagram displayed in Fig. 6.6. The various regimes in this diagram correspond to the behavior in Fig. 6.5 for w = 0.925. In accordance with the above discussion, the bifurcation diagram verifies the existence of a cut-off for phase synchronization of chaos in an externally forced system (the dashed line to the right in Fig. 6.6). For K > 0.015, there is a critical value of the parameter fi = 6.37 above which phase synchronization of chaos is no longer observed. The natural frequency of oscillation depends on the nonlinearity parameter fi. As long as the coupling is relatively small, this dependence gives rise to the existence of well-separated tongues of synchronization in the parameter plane. However, when the forcing amplitude is increased, the natural frequency becomes less sensitive to the changes of the bifurcation parameter. Hence, the


tongues start to overlap.

Let us finally consider the structure of chaos inside the synchronization region (direction A in Fig. 6.5). The chaotic attractor that appears via a sequence of period-doubling bifurcations has a multiband structure. When the parameter fi is increased, band-merging bifurcations take place. In this direction, the chaotic synchronization is lost only after the last band-merging bifurcation has occurred. The same phenomenon is seen for the period-3 window. Along direction B, chaos with 3n-band structure is observed at the critical curve la-i. Inside the synchronization region these bands disappear one by one as we approach the boundary until only chaos with 3-band structure exists. Thereafter, the transition to non-synchronous behavior takes place. Thus, the chaotic attractor stays synchronized as long as it has a multiband structure. After the last band-merging, the phase oscillations become stronger, and this leads to a destruction of the synchronization.

Lyapunov exponents

It is obvious that a bifurcation of the chaotic set takes place at the boundary of synchronization. But what is the nature of this bifurcation? Anishchenko et al. [11] have shown that the boundary is located in an area where there is an accumulation of lines of tangent bifurcations of different saddle cycles existing in the attracting basin of the chaotic attractor. Inside the synchronization region the chaotic attractor has a structure which is similar to the structure of chaos in the autonomous system. This structure is determined by saddle cycles and their manifolds. These cycles do not exist outside the synchronization region. Each saddle cycle has to disappear at the boundary via a tangent bifurcation. Numerical experiments show that the above bifurcations for saddle cycles embedded in the chaotic attractor do not occur simultaneously along the parameter axis. As usually, information concerning the structure and qualitative transformations of the various regimes can be obtained by following the variation of the Lyapunov exponents. Let us investigate therefore the behavior of these exponents in the vicinity of the synchronization boundary for the chaotic oscillations of system (6.2) in different chaotic regimes.

Figure 6.7 shows the two main Lyapunov exponents as functions of the parameter LU. Note, that a third Lyapunov exponent, relating to the external forcing, is always equal to zero (horizontal line in Fig. 6.7). The largest Lya-


U.04

0.924 0.925

CO 0.926

0.02

CO

3 0.00

-0.02

-0.04

-

\

J

i i

i

-

i i i i i

0.926 0.927

CO 0.928

Fig 6.7. The largest Lyapunov exponents as functions of the parameter w (a = 0.2, /} = 0.2, and K = 0.02): (a) n = 3.71, transition from region 3 to 6 in Fig. 6.5; (b) n = 3.8 , transition from region 3 to 7 in Fig. 6.5.

punov exponent is positive in most of the considered interval and does not indicate a transition at the boundary of synchronization (dotted vertical line) in any obvious way. Inside the synchronization region this Lyapunov exponent becomes negative in the windows of periodicity. The second Lyapunov exponent, on the other hand, being negative everywhere inside the synchronization region, demonstrates two different kinds of behavior:

1) At the boundary it becomes equal to zero and then immediately turns negative again (Fig. 6.7(a)), or

2) At the boundary it becomes equal to zero (within the accuracy of calculations) and it maintains this value for some interval of the control parameter to (Fig. 6.7(b)).

In both cases, the second Lyapunov exponent becomes zero at the boundary of chaotic synchronization. This is similar to the behavior of the first Lyapunov exponent of a synchronous periodic attractor when a tangent bifurcation takes place. Let us conclude this investigation by noting that all our numerical results (the devil's staircase for the rotation number, the dynamics of the phase difference, and the behavior of the Lyapunov exponents) allow us to draw analogies between chaotic synchronization and the tangent bifurcation on a torus that leads to the appearance of the resonant structure of a limit cycle.

When the parameter /u is increased inside the synchronization region, a cascade of period-doubling bifurcations of synchronous limit cycles (stable and


saddle) takes place. Hence, non-attractive sets of chaotic trajectories can arise together with chaotic attractors. At the boundary of chaotic synchronization attractive and non-attractive chaotic sets disappear via the analogy of a tangent bifurcation. At this moment different characteristics of attractive chaotic sets behave similarly to the characteristics of regular oscillations. From this point of view, the function of the vanishing second Lyapunov exponent as measure of another direction of stability is clear. Recently, it has been shown for a model map [20] that the bifurcation mechanism of chaotic synchronization consists in a merging of a chaotic attractor and a chaotic repeller. For continuous-time systems, however, non-attractive chaotic sets in general have at least one stable direction (i.e., they are not repellers). Hence, they can not be determined by means of time inversion.

6.3 Role of Multistability

In the previous sections we considered some of the main phenomena associated with the transition between synchronous and asynchronous chaos. To simplify the problem, the model of a periodically forced chaotic oscillator was used. This choice allowed us to ignore an important phenomenon related to the synchronization of complicated oscillations. Namely, the so-called phase multistability.

Multistability, i.e., coexistence of a set of attractors in the phase space of a dynamical system, is a typical nonlinear phenomenon. The development of different families of regular and chaotic attractors for coupled oscillators has been investigated by several authors [33, 34, 23]. As shown by Astakhov et al. [33], for instance, two dissipatively coupled, identical oscillators following the period-doubling route to chaos will exhibit a hierarchy of bifurcations in which different families of attractors emerge. For two coupled Rossler systems, Ras-mussen et al. [34] found the replacement of some of the period-doubling bifurcations by torus bifurcations leading to quasiperiodicity, frequency-locking and the emergence of new non-symmetric families of attractors, and Anishchenko et al. [23] have shown that this multistability is structurally stable with respect to a mismatch between the basic frequencies.

The purpose of the present section is to examine the role of phase multistability in connection with the transition to chaotic phase synchronization by performing a two-parameter bifurcation analysis for a pair of coupled non-identical Rossler oscillators [16]. In particular we shall show that the phenomenon of


phase multistability, while originating in the regime of high-periodic orbits, remains valid for multiband chaotic attractors as well. In the next section we shall show that the main features of this phenomenon can be simulated by means of a specially constructed model map. Here, we shall see how the presence of phase multistability manifests itself in the form of a nested structure of Arnol'd tongues on the frequency-mismatch vs. coupling-strength parameter plane.

Let us consider the pair of coupled Rossler oscillators

z'i = -uiyi-zi + K(x2-xi), 2/i = UiXt + ayi,

ix = 0 + Zl{xi-n), (6.8) X2 = -W2V2- Z2 + K(xi -X2),

2/2 = u2x2 + ay2,

z2 = (3 + z2(x2- n).

as introduced, e.g., by Rosenblum et al. [19]. As before, the parameters a, (3, and \i govern the dynamics of each subsystem. K is the coupling parameter, LJI = tu + A and u>2 = UJ — A determine the frequencies of the uncoupled oscillators (we hereafter suppose that w = 1), and A is the mismatch between these frequencies.

Because synchronization involves phase relations between the interacting systems, it is useful to rewrite (6.8) in terms of the phases and amplitudes of the two oscillators. Instantaneous amplitudes A\$ and phases $it2 can be introduced in accordance with (6.5). This allows us to recast (6.8) into the form:

Aii2 = aA l j 2 + {KA2>i cos $2,i - Ai>2(a + K) cos $1,2 - 21,2) cos <&i]2

$1,2 = wij2 - {KA2,i cos $2,1 - Ai>2(a + K) cos $ l i 2 - 21,2) sin $i )2/Ai )2

ii,2 = /3 + -zi,2(Ai,2cos$i i2-/z). (6.9)

As in Eq. (6.8), // controls the bifurcations of the individual Rossler systems, and A determines the detuning between the two interacting systems. Hence, these two aspects can be separately considered. By numerically integrating (6.9), it is easy to follow the phase difference 5$(t) = $2(*) — $ i ( 0 f° r a n y oscillating solution. With /x and A as active parameters we can now perform a detailed bifurcation analysis of the coupled Rossler systems and follow the periodic and chaotic trajectories of different attractor families. For each such family, a phase-locking region appears.


Figure 6.8 shows a segment of the bifurcation diagram for the synchronous solution on the (A,/z) parameter plane, while u = 1, K, a, and /3 remain fixed. Because the number of synchronous regimes depends on the period of oscillations, new synchronous regimes appear above each line of period-doubling and there exists an infinite but denumerable set of such regimes. Hence, it is difficult to analyze and display all of them in the diagram. Let us consider attractors from only two families: "in-phase" attractors for which the phase difference of x\(t) and x2(t) is zero at A = 0 and "out-of-phase" attractors for which the phase difference is 2TT at A — 0. These families have the largest basins of attraction. Denote the attractors as 2sCo and 2!Ci, respectively, where i — 1,2,3,... and 2l is the cycle period, normalized in terms of the period of cycle Co- For the various bifurcation curves we have used the following notation:

/_i is a curve of period-doubling bifurcation of cycles 2lCo,i; Z+i is a curve of tangent bifurcation of cycles 28Coii; l^. is a critical curve, corresponding to the accumulation of period-doubling

bifurcations of 2JCb cycles; and Z . is a critical curve of family 2lC\.

Fig 6.8. Bifurcation diagram for two coupled Rossler systems (a = 0.15, p = 0.2, and K = 0.02). The solid curves are bifurcation lines of "in-phase" attractors. The dashed curves show-bifurcations of "out-of-phase" attractors. A determines the frequency mismatch between the oscillators.

0.0090 0.0095 0.0100

A

7.5

7.0

M-6.5 -

. CAX, ^ _ -

B A

-"' ^ ^

i

I

J i CAt

cAo^fCAi; 1° ^ < ^ / °

- " 4Q

c

/ <-crs '{ / / 1

C -' A 8C,/ i

4 c . /*«

' 4T2

J, :


Above the critical curves there exist two families of chaotic attractors, 21CAQ

and TCA\. As \i is increased, band-merging bifurcations in each of these families take place. We omit the corresponding bifurcation curves and denote the regions of chaos as CAQ and CAi, respectively. CA-% arises from a merging of the chaotic attractors CAQ and CA\. The figure shows that the cycles TC\ remain stable for larger values of the detuning parameter A than the cycles 2*Co. Outside the synchronization region, quasiperiodic oscillations 4T2 and asynchronous chaos CAt are found.

Coexistence of a set of attractors characterized by different phase shifts is a universal phenomenon for coupled systems with period-doublings, and their main features are generally independent of the particular properties of the model.

e to

0.0093 0.0095 0.0097

A 0.0O99

Fig 6.9. One-parameter bifurcation diagram for two coupled Rossler systems (a = 0.15, j5 — 0.2, A" = 0.02 , and ^ = 6.7).

The number of coexisting attractors inside a synchronization region for weak coupling tends to infinity close to the threshold of chaos. When the detuning parameter A is increased, the synchronous chaotic regimes sequentially lose their stability (direction A in Fig. 6.8). We can construct a bifurcation diagram (Fig. 6.9) for the different families of chaotic attractors CA§ and CA\. It is easy to see that the number of possible synchronous chaotic solutions decreases at A = 0.0095. The chaotic attractor CA\ is stable in a wider range of detuning parameters, and its bifurcation curve forms the boundary of the synchronization region.

Along the direction B in Fig. 6.8 the number of possible synchronous so-


lutions also decreases, but in a rather different way. As \i is increased within the chaotic region, a sequence of crises of chaotic attractors takes place. Each crisis reduces the number of possible synchronous regimes by two. Finally, a single chaotic attractor is formed by merging of chaotic trajectories of all families. The merging process leads to a number of new properties for the chaotic solution and can be diagnosed in different ways.

To illustrate some of the signatures of the merging process, let us consider the distribution of phase differences p for chaotic attractors of the various families. Fig. 6.10 (a) shows the corresponding plots for the coexisting chaotic attractors CAQ and CA\ (curves 1 and 2, respectively) and for the attractor that arises as a result of the merging of these attractors (curve 3). It is easy to see that the phase distribution of the resulting chaos is rather different from the structure of the chaotic regimes before merging. The observed Gaussian-like phase distribution can not be found for chaos that arises via a period-doubling scenario.

0.00' • • — ^ - = ^ • ' . 0 0 2 -1 1 I ' 1 ' 1 0.0 0.1 0.2 0.3 0.4 6 6 6 8 7 0 7 2

Fig 6.10. Two coupled Rossler systems (a = 0.15, /3 = 0.2, c = 0.02, and A = 0.0093): (a) Distribution of momentary phase differences for the synchronous chaotic attractors CAo and CA\ (curve 1 and 2, respectively) at /x = 6.6 and for the merged chaos CAs (curve 3) at /i = 7.2; (b) two largest Lyapunov exponents vs. the parameter p, for the synchronous regimes of the two families (dark points correspond to CAo and open circles to CAi).

Further investigation shows that the merging of the two chaotic attractors leads to hyperchaos. Each attractor (CAQ or CA\) is characterized by only one positive Lyapunov exponent. But when the transition to merged chaos occurs, a second direction of exponential instability and, hence, a second positive Lyapunov exponent appear. Figure 6.10 (b) shows the two largest Lyapunov exponents for the attractors of two different families as a function of the pa-


rameter [i. The dark points correspond to CAQ, and the open circles to CA\. For A = 0.0093 at \i = 6.97, a crisis of CA0 and CA\ takes place. This leads to the appearance of a new chaotic attractor CA-£. The attractor CAY, contains the trajectories of CAQ and CA\ and is characterized by two positive Lyapunov exponents. Hence, emergence of hyperchaos is observed. The mechanism described in Sec. 6.6 for the formation of a chaotic hierarchy utilizes a similar process.

Increasing the coupling coefficient leads to the destruction of multistability because the "out-of-phase" solutions die out [33, 34]. Thus, when the coupling becomes sufficiently strong, only attractors corresponding to zero phase difference for the partial oscillations (i.e., 2lC0, 21CAQ) exist. Following the variation of the Lyapunov exponents when the coupling is increased, it is easy to find the transition from hyperchaos to chaos CAQ with one positive Lyapunov exponent. In the paper by Rosenblum et al. [35], this phenomenon was termed "lag synchronization".

6.4 Mapping Approach to Multistability

To construct a model of the emergence of phase multistability let us introduce an analytical description of a high-periodic signal in the form [16]:

x(t) = A{<f>{t))sin(ujt) (6.10)

Here, 4> = ut is the phase of the oscillations, and A((j>) = rL=iU —°i s m ( F + * f ) ) represents the momentary amplitude, u is the natural (or fundamental) frequency of oscillation, N defines the period of the considered signal T/v = 2N(2n/w), and <7j specifies the amplitude of each of the subharmonic components. The term i | is introduced to obtain a more obvious phase portrait of each period-doubling in our model. The temporal variation described by Eq. (6.10) is illustrated in Fig. 6.11(a). As N increases, x(t) provides a qualitative representation of a sequence of high-periodic cycles, leading in the limit to the birth of chaos via a cascade of period-doublings.

For two synchronized oscillators coupled via the variables x\(t) and x%(t), each described by an expression like (6.10) , the phase difference can attain 2N

different values, i.e. 0 — <f>\—<j>2 = 2nm, m = 0,1,2, . . .2^-1 . Hence, coexistence of a large number of periodic attractors will occur. When approaching the boundary of the synchronization region, these attractors disappear one by one


0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8

en/8Tc

Fig 6.11. (a) Time series x(t) for the periodic orbits with period 4T0 simulated from the expression (6.10). (b) The model map (6.13) for the case of N = 2.

except for a single family whose bifurcations determine the transition to the non-synchronous regime. In order to understand the structure of this boundary in more detail we shall investigate a sequence of model maps.

For quasiperiodic oscillators, the phase difference is known to develop according to an equation of the form [36]:

e = A - 7 / ( A l ! J 4 2 ) s i n 6 , (6.11)

where /(•) is a function of the amplitudes Ai and A-i that is defined by the type of interaction. A represents the mismatch between the basic frequencies and 7 is the coupling strength.

In our case the oscillators have different momentary phases </>i and fo while their amplitudes, as specified above, depend on the phases in the following way:

A2 = A{<k - 6) = f[(l - a, sin A - § + if)). t = i

(6.12)

It is not possible to obtain an explicit relation for the phase difference of two chaotic oscillators. However, qualitatively we can consider the oscillators as high-periodic cycles of periods Tjf = 2N2-K/LO, where u is the natural frequency


of the partial system (LJ\, for example). To obtain a discrete model, Eq.(6.11) is integrated over the characteristic time T of the system. This gives:

e»+1 = e» + n-KFN(e») mod 2*2*. (6.13) where 9* + 1 = eN{t0 + nTN) and 6 * G [0,2*2*]. Q = TNA, and K is a measure of the strength of interaction. We have not succeeded in determining the function FN analytically. We may suppose, however, that the interaction strength depends on the phase differences in the same way as the amplitude of the individual subsystem depends on its phase. As a simple approach we shall therefore assume an expression of the form

FN{@Nn) = s in(e^) f [ ( l - ^ s i n ( ^ + i\)). (6.14)

Equations (6.13) and (6.14) may be viewed as a generalized form of the well-known circle map for simple oscillators [37]. Varying N = 1, 2,3, . . . , we obtain a family of maps, each being a model of synchronization for 2JV-periodic cycles. The case of N — 2 is illustrated in Fig. 6.11(b). The above equations are not normalized on the same scale because they are taken to the modulus 2*271", which is changed with each period doubling. This allows us to preserve the values of Q and K and to compare the results for different N. A similar approach to constructing a model map in the non-autonomous case was suggested by Pikovsky et al. [38].

With these preliminaries let us now investigate the structure of the boundary of the synchronization region for the main resonance 0:1 (or 1:1 for time-continuous systems). In terms of the map, the transition at that boundary corresponds to a tangent bifurcation. The condition for such a bifurcation to occur is:

e f + a - KFN(e?) = of d(GN + n-KF»(GN))

^QN lew=e? = !> (6-15)

where G* is the fixed point. Equation (6.15) immediately gives:

KFN(<d?) = Q

230 Chaotic Synchronization' Applications to Living Systems

dFN(eN)i

deN le«=ef = 0. (6.16)

Hence, it is easy to see that for any value of O^, the set of points corresponding to the tangent bifurcation forms a straight line in the (0, K) parameter plane. The number of roots of Eq.(6.16) defines the number of possible synchronous regimes. For the case of small N, Eq.(6.16) can be solved analytically. For larger N, the solution can be obtained numerically. Figure 6.12 shows the results for N — 1 (fully drawn lines) and N = 2 (dotted lines). Each line corresponds to a tangent bifurcation for one of the fixed points of the map. Under variation of Q, a pair of stable and unstable fixed points arises at each line. For larger K, the stable fixed point can subsequently lose its stability through a period-doubling bifurcation. To find the corresponding parameter values, one only has to replace the zero on the right side of Eq.(6.16) by 2/K. However, in the present context we shall not consider the further bifurcations of the stable periodic solutions.

i - i

0 75 -

K 05

Fig 6.12. Phase-locking iegions for different families of atti actors foi the model map (6.13) with S — 0.45. The solid lines correspond to N = 1 (two cycles of peiiod-two coexist). The dashed lines correspond to N = 2 (four cycles of period-four coexist)

Fig 6.13. One parameter bifurcation dia-giam foi the model map (A* = 0.5, a = 0.45, B = 1.2, and N = 3). The fig-ure shows how the coexisting noise inflicted periodic orbits one by one lose their synchronization. Compare with Fig. 6.9 show mg a similar phenomenon for two coupled Rossler systems.

Thus, for small enough K there are 2^ stable (and a similar number of unstable) fixed points near the center of the synchronization region. In terms of


continuous-time dynamical systems, a set of stable fixed points corresponds to a set of possible synchronization regimes for the coupled oscillators. A two-dimensional torus exists both outside (where it is ergodic) and inside the synchronization region (where it is resonant). Entering into the synchronization region corresponds to the birth of a pair of stable and saddle cycles, both lying on the torus surface. In these terms, the appearance and coexistence of other fixed points of the map represent the birth of additional pairs of stable and saddle cycles on the torus surface which do not intersect each other.

On this background we can draw the following conclusions concerning synchronization of high-periodic oscillations in coupled period-doubling systems: (i) There are 2^ coexisting synchronous solutions which differ from one another by phase shifts; and (ii) the boundary of synchronization for these solutions consists of a set of tongues inserted one into the other.

The question is now how the results listed here manifest themselves in the case of two interacting chaotic oscillators. We restrict our considerations to highly dissipative systems. Such systems can usually be characterized by a few specific time scales. The first of these is the return time to a surface of section (quasi-period of oscillations), and the second is the time constant characterizing the transient approach to some attractor. Thus, highly dissipative dynamical systems cannot distinguish an extremely high-periodic regime from a weakly-chaotic one if the envelops of their Fourier spectra are nearly the same. From another point of view, this type of chaotic motion may be considered as a regular behavior with an applied random excitation.

It is well-known that for the period-doubling route to chaos the chaotic attractor has an ./V-band structure (N = 1,2,4...) within a range of control parameters. This structure is geometrically similar to the structure for the N-periodic cycles. Thus let us simulate an ./V-band chaotic attractor by means of the model map (6.13) with an added noise term. The logistic map may be used as the source of such random excitations:

0^ + 1 = Q% + to-KFN{G%) + Bxn mod 2N2v, (6.17)

Xn+\ = AXny±. %n)i

where the value of A is fixed at 3.99. Note, that we introduce the source of noise in the above way (not a Gaussian noise, for example) to maintain the multi-band structure of the chaotic attractor.


Within some range of the noise amplitude B, the attractors produced by this equation become irregular but they still coexist in the phase space of the system and their basins of attraction differ. When B is further increased, merging of the attractors starts to take place [39].

Figure 6.13 shows a one-parameter bifurcation diagram for the case of an 8-band chaotic attractor. There are eight different synchronous chaotic regimes which coexist at small 0 . When the phase shift parameter 0 increases, the coexisting chaotic attractors one by one disappear at the edges of their respective synchronization regions. At Q, > 0.535 a single synchronous solution is still stable. Note, how the "ghosts" of all eight synchronous solutions remain distinguishable inside the region of merged chaos at fi > 0.6. The number of possible synchronous regimes decreases in the same way as we observed for coupled Rossler systems in Sec. 6.3 (Fig. 6.9).

0 .03-

K 0 .02-

0.01 -

-0.0 j -0.005

6.5

6.0

5.5

5.0

4.5

(b)

'

- 1 C, o -

. • •**"

• - ^ » -

s,

.»•"§• -

f -s, 4

4 : c,

4.0 4.5 5.0 5.5 6.0 6.5 7.0

yi

Fig 6.14. (a) Nested structure of Amol'd tongues for the coupled Rossler oscillators with a = 0.15, P = 0.2, and \i = 6.1. The solid and the dashed lines correspond to the different coexisting families of regimes [16]. (b) Poincare' section of the resonant torus for two coupled Rossler models. The secant was chosen as xi = 0. Control parameters are a = 0.15, 0 = 0.2, JX = 5.0, and K = 0.0273. Points Cii2 denote the stable limit cycles while 5i)2 are the saddle cycles. Arrows indicate the stable directions along the resonant torus surface.

Hence, our conclusions with respect to synchronization of high-periodic orbits also apply for weakly-chaotic solutions. Moreover, we may expect the nested structure of synchronization tongues to be preserved in the case of an /V-band chaotic attractor and to remain similar to the structure for an N-


periodic cycle. To verify the conclusions based on the model map dynamics, consider again

the dynamics of coupled Rossler oscillators. In Fig. 6.14(a) the numerically obtained structure of four Arnol'd tongues is depicted. The control parameter H was set fj. — 6.1 while the detuning A was scanned. This figure clearly demonstrates good agreement with the results for our model map, at least for K < 0.01.

Another interesting question is: Do the coexisting synchronous solutions actually lie on the same torus surface? Note, that this is not necessarily the case for time-continuous systems. In Fig. 6.14(b) the numerically obtained Poincare section for the resonant torus surface is given. The parameters of two coupled Rossler systems correspond to the period-two limit cycle. Two stable coexisting solutions are observed in the plot, each paired with a corresponding saddle cycle. Moreover, inspection of the figure clearly shows that all the solutions belong to the same closed curve, formed by the unstable manifold of the saddle cycles.

6.5 Suppression of the Natural Dynamics

The basic question to be addressed in this section is: How does the classical mechanism of synchronization via suppression of the natural dynamics as discussed in Chapter 1 manifest itself when applied to chaotic oscillations? In many studies [11, 23], even if a synchronized chaotic regime can be found for significant frequency mismatch and strong coupling, the transition to synchronization occurs via regular oscillations. In this case, the mutual coupling (or periodic forcing) leads to the suppression of the internal chaoticity of the coupled (driven) system. In contrast to the above cases, we consider a direct transition from an asynchronous chaotic state to synchronized chaotic behavior via the suppression of the natural dynamics [17]. We show that this transition is accompanied by changes in the power spectrum and the spectrum of Lyapunov exponents.

All the mechanisms we are going to describe apply to systems of interacting chaotic oscillators as well. However, to make the analysis as clear as possible we shall consider a Rossler system interacting with a Van der Pol oscillator. This combination represents either periodic dynamics driven by a chaotic signal or chaotic dynamics subjected to a periodic forcing. Both versions represent a wide class of dynamical systems and may serve as archetypes in the study of


synchronization. The equations are given by:

x = —y — z + Kr(u — x),

y = x + ay,

z = 0 + z{x-fi), (6.18)

u — v + Kv(x — u),

v = - e ( l - u2)v - u>\ (6.19)

where fj, = 6.5, a = 0.2, (3 — 0.15, and e = 0.1. The parameters Kr and Kv determine the strength of the mutual interactions and control the coupling configuration, w is the natural frequency of the Van der Pol oscillator, while the fundamental frequency of the Rossler system now is 1.

Periodic forcing (Kv = 0,Kr > 0). Let us first consider how the external forcing can suppress the chaotic dynamics in the driven system. In this case, the Rossler system is subjected to a periodic signal from the Van der Pol oscillator. To avoid the regions of strong resonance, we choose ui = 1.4. As illustrated in Fig. 6.15, we are interested in the evolution of non-resonant quasiperiodic and chaotic regimes when the coupling parameter is varied. As the coupling strength is increased, the chaotic behavior is regularized into quasiperiodic motion (that is, in the opposite direction with decreasing coupling, chaos appears via the loss of torus smoothness [12]). The figure also shows how the two-frequency attractor undergoes a sequence of inverse torus-doubling bifurcations. This type of truncated torus-doubling cascade was previously observed, for instance, by Arneodo et al. [40] and by Kaneko [41]. With further increase of Kr, the torus collapses and a stable synchronous cycle appears.

Similar behavior has been identified for diffusively coupled chaotic systems and was discussed in terms of the suppression of the natural chaotic dynamics [11, 23]. Thus, neither periodic forcing nor mutual interaction can exhibit the direct transition from asynchronous to synchronous chaos through suppression.

Chaotic forcing (Kv > 0,Kr — 0). Let us now turn to the case when the Van der Pol oscillator, showing a stable limit cycle, is subjected to an external chaotic force from the Rossler system. Since the control parameters of the Rossler system are fixed, the system remains chaotic at any coupling strength. Hence, all observed transitions are related to bifurcations and crises of the chaotic attractor.


Fig 6.15. Doe-parameter bifurcation diagram for the system (6.18)-(6.19) with Kv = 0. The diagram is constructed as a double Poincare section specified by the conditions (u = 0, x = 0).

0.06-

0.04-

0.02-

0 - , • 0.6 0.8 1.0

CO 1.2 1.4

Fig 6.16. Bifurcation diagram for the chaotically forced Van der Pol oscillator. Inserts present phase portraits for the synchronous and asynchronous chaotic regimes.

In accordance with our discussion in the introduction to this chapter, a variety of different diagnostic tools can be used to characterize the system's behavior in the plane of parameters that control the mismatch between the fundamental frequencies of the two subsystems and the coupling strength. When so doing, as illustrated in Fig. 6.16, a region of chaotic synchronization similar to an Arnol'd tongue is found. Near the tip of the tongue, along the direction A, the previously studied case of phase locking for the chaotic regime via a set of saddle-node bifurcations is realized [11, 28, 20, 38].

We focus our study on the development of chaotic oscillations within the range of large mismatch and coupling values (route B in Fig. 6.16). In the classical synchronization theory this corresponds to the case where a self-sustained system is forced across a Hopf bifurcation [42, 43]. For the chaotically forced Van der Pol oscillator we observe how the chaotic oscillations outside the synchronized region (asynchronous chaos) differ topologically from the behavior inside this region (synchronous chaos). This is illustrated by the inserts of Fig. 6.16.

Figure 6.17 shows how the power spectrum of the Van der Pol oscillator (heavy curve) is transformed under the influence of the chaotic forcing from the Rossler system (fine curve). As long as the coupling is relatively weak (Figs. 6.17(a) and (b)), the two fundamental frequencies and their subharmonics


Fig 6.17. Evolution of the power spectrum Sx illustrating the forced chaotic synchronization via suppression of the fundamental frequency along the direction B in Fig. 6.16. (a) Kv = 0.0, (b) Kv = 0.01, (c) Kv = 0.017, (d) Kv = 0.8.

are clearly distinguishable. However, with increasing Kv, the basic frequency of the Van der Pol oscillator is gradually suppressed (Fig. 6.17(c)) until, finally, the spectrum of the Van der Pol oscillator is no longer distinguishable from that of the chaotic drive.

The evolution of the Poincare section in Fig. 6.18 supports the above results. Inspection of the figure clearly shows how the chaotic set, originally concentrated around a circle, is transformed into a patch. This resembles the regression of an invariant closed curve into a point in the classical approach to synchronization of periodic oscillations when a torus vanishes via an inverse torus birth bifurcation.


- 3 . 0 - 1 . 5 0.0 1.5 3.0 U

- 3 . 0 - 1 . 5 0.0 1.5 3.0 U

- 3 . 0 - 1 . 5 0.0 1.5 3.0 U

Fig 6.18. Evolution of the Poincare section (x Kv = 0.01, (b) Kv = 0.014, (c) Kv = 0.02.

0) along the direction B in Fig. 6.16: (a)

The spectrum of Lyapunov exponents also reflects the boundary of synchronization. Two different regions are clearly seen in the plot of Fig. 6.19(a): At the transition, one of the two vanishing Lyapunov exponents becomes negative, while the positive exponent remains nearly unaffected. Note, that except for the presence of a positive Lyapunov exponent, this transition looks similar to the transition from quasiperiodic to periodic motion (i.e., the inverse of a torus birth bifurcation).

As shown by Anishchenko et al. [11] and by Pikovsky et al. [38], the phase locking transition for period-doubling chaos is closely related to bifurcations of the unstable orbits that are embedded in the chaotic attractor. Following this idea, we have calculated the Floquet multipliers for the unstable orbits of period-1, -2, -4, -8, and -16. As illustrated in Fig. 6.19(b), each saddle orbit undergoes a Hopf bifurcation, and these bifurcations occur at slightly different values of the coupling strength Kv ~ 0.0175. It is clear, however, that the region of accumulation for the bifurcation points corresponds to the final "death" of the natural dynamics for the forced system as characterized through its power spectrum, Poincare section, and Lyapunov exponents.

With decreasing Kv, the third Lyapunov exponent becomes equal to zero at Kv « 0.01575 (Fig. 6.19(a)), and close to the same value of the coupling strength a Hopf bifurcation of the period-1 cycle takes place. This observation fits well with the hypothesis that the transition to asynchronous behavior occurs when the last saddle cycle embedded in the chaotic set loses it stability in an additional direction. Since in our case a one-band attractor of the Rossler


Fig 6.19. (a) The largest Lyapunov exponents and (b) Floquet multipliers of periodic orbits embedded in the chaotic attractor as functions of the coupling strength Kv along the direction B in Fig. 6.16. In (b), the curves denoted 1, 2, 4, 8, and 16 represent the modulus of two complex conjugated Floquet multipliers for the period-1, -2, -4, -8, and -16 cycles, respectively.

system forces a period-1 cycle in the slave system, one can envisage that with decreasing coupling strength the period-1 saddle cycle embedded in the chaotic attractor will be the last to undergo a Hopf bifurcation. This is confirmed in our calculations.

The numerical results that have been presented in this section allow us to draw an analogy between the suppression of natural dynamics by a regular signal and by a chaotic forcing. In the former case, quasiperiodic dynamics collapses into periodic dynamics via an inverse torus birth bifurcation. In the second case, however, a direct transition from asynchronous to synchronous chaotic behavior takes place. To investigate the bifurcation mechanism of this transition we have considered the simplest case, where the driven system shows a periodic behavior. A full characterization of the synchronization of two interacting chaotic systems in terms of periodic orbits is rather complicated. Both systems possess an infinite number of saddle cycles that act as a skeleton of the attractor and are involved in the process of synchronization. Hence, to simplify the analysis we have suggested the above model, which describes the interaction between single cycles from one side and a complex set from the other side. We believe that our results may be generalized to the case of chaotic behavior of the slave system.

The mechanism of chaotic synchronization via suppression of the natural dynamics is complementary to the previously studied scenarios for chaotic phase locking. This mechanism may play an important role for biological oscillators


that often exhibit strong interaction and significant differences of their natural frequencies. Similar transitions have been observed, for instance, for the cascaded bacteria-virus population system to be discussed in Chapter 7.

We can conclude the above discussion by noting that the transition from high-periodic regular oscillations to chaotic oscillations does not affect the appearance of the synchronization phenomena in any decisive manner, at least not for period-doubling chaos. The effects we have discussed in the present chapter mainly appear at the stage when the form of the oscillations becomes complex via a period-doubling cascade. A number of important issues are left for future investigation. First, the loss of lag synchronization inside the region of phase synchronization must be studied in more detail. Second, phase-locking regions for each family of chaotic attractors should be investigated in terms of periodic orbits embedded in the chaotic set. Finally, the role of multistability in phase synchronization should be generalized to systems demonstrating other routes to chaos. Let us complete the chapter by presenting a new mechanism for the formation of a so-called chaotic hierarchy. This mechanism is based on the merging of chaotic attractors with chaotic saddles involving different directions in phase space.

6.6 Chaotic Hierarchy in High Dimensions

The interest in higher-order chaos was partly initiated by Rossler [44], who noted that as the dimension N of the available phase space for a time-continuous system increases, the number of directions in which exponential divergence can take place grows as N — 2. Hence, in the limit N —> oo, the fraction of unstable directions can, in principle, approach 1. Rossler continued to develop a systematic approach to the construction of dynamic systems with the maximum number of unstable directions, and he introduced the term "chaotic hierarchy" to denote a succession of such systems. In general, however, it was only possible to realize the maximum order of chaos for a very restricted set parameters. In practical cases, the dimension of the most complicated attractor tends to saturate as N increases [45].

The concept of a chaotic hierarchy is of interest, for instance, for ecological and epidemiological systems, i.e., population dynamical systems where species interact via their mutual predation and competition for resources. The general opinion among ecologists appears to be that increasing the dimension of an


ecological system by introducing more and more species adds to its stability [46]. With more species around, a predator can substitute a more abundant prey for a less abundant one, thus introducing a negative feedback regulation on the predation rates. At the same time, competition for a diverse set of prey species may introduce balancing interactions among the predator populations.

In an attempt to address this type of question, Baier et al. [47] examined the dynamics of a microbiological population model with interacting bacteria and viruses in a single homogeneous habitat as considered experimentally by Levin et al. [48]. It was shown that with increasing number of bacterial strains (each sensitive to attacks from a specific phage variant), a chaotic hierarchy could be realized. Each time a new bacterial variant was added, the most complicated dynamics exhibited by the model attained an additional positive Lyapunov exponent. Moreover, these results were found to be robust to significant changes both in the parameters and in the structure of the model. The results were explained in terms of the specific coupling structure of the system in which a number microbiological predator-prey oscillators interact with a common source of nutrients. As illustrated in Fig. 6.20, this structure can be reduced to that of an ensemble of self-sustained oscillators coupled via a mean field M. Similar dynamical phenomena have subsequently been observed for systems of coupled Rossler oscillators and explained in terms of nonlinear modes enriched by linear modes [49].

(a) (b) (V)

Fig 6.20. The system of (a) interacting bacterial and viral populations coupled via a shared flow of nutrients (a) can be reduced to the model (b) of a system of oscillators coupled via a mean field M.

As discussed in Chapter 7, we have also considered the complex dynamic phenomena that can arise in a cascaded system of microbiological population models where the coupling is brought about through a unidirectional flow of primary resources [50]. In this way the individual habitat is modulated by the


varying nutrient concentration in the overflow from the upstream habitat, and a variety of different synchronization phenomena can be observed. In particular, we have discovered a homoclinic mechanism of chaotic phase synchronization [50]. With this mechanism the transition from asynchronous to synchronous chaos is associated with the collision of the asynchronous chaotic attractor with an unstable periodic orbit.

The purpose of the present section is to examine a new mechanism for the development of higher-order chaos. A discrete-time model is proposed in order to demonstrate how the creation of coexisting chaotic attractors combined with attractor metamorphosis through boundary crises can give rise to a continued growth of the Lyapunov dimension for the chaotic regime [18].

The model we shall consider describes the phase dynamics of an ensemble of interacting time-continuous oscillators in terms of an Af-dimensional system of coupled one-dimensional maps. Since the presence of a global (i.e., all-to-all) coupling seems to be essential for a system of coupled oscillators to attain the maximum degree of chaos (i.e., the highest possible Lyapunov dimension), we have introduced the coupling via a mean field form. As discussed in Chapter 8, this type of coupling is of interest in connection with many biological applications. Models of competing populations typically demonstrate anti-phase synchronization, i.e., the presence of high densities of one species tends to be associated with reduced densities of competing species. In our analysis we shall therefore allow for the coupling parameter K to assume negative values. With these remarks, our set of equations are given by

JV

pj, + w' + tf sin (2TT £ > * - ¥ £ ) ) m o d i , (6.20)

1,...,N.

Herej if1 represents the phase of the i-th oscillator, and ul is the frequency of its free-running dynamics. To maintain a certain generality, we shall assume that these frequencies distribute uniformly over a certain interval, i.e., UJ1 = 1.00+ d((i — 1)/-/V) with d = 0.001. n is the time index, and N is the dimension of the system, i.e., the number of coupled oscillators.

The above model can be derived if N single oscillators with phases (jf and angular velocities Q,1 are coupled via a mean field term Mn with a strength k,

fn+\ =


provided that the following substitutions are performed:

M» = | X > » > < = % wi = f; K = w (6-21) t=i

A similar approach is widely used to describe the behavior of globally coupled oscillators [51, 52, 53]. In contrast to the model suggested by Kaneko [51], however, we introduce the coupling term in the form sin (2?r X)»=i(^n — $i)) (*n~ stead of j - ^ Ylf=i sia(2n((p3

n — #,))), and we also assume a frequency mismatch between the interacting units.

Pig 6.21. Synchronization regions (q = 1,2) for the system (1) for N = 3.

Let us consider the model from the viewpoint of the phase relations between the interacting oscillators. For N = 1, (6.20) describes a monotonical growth of the phase with rate u>. For N = 2, the model reduces to the well-known circle map [37] demonstrating multistability of periodic and chaotic solutions. Let us focus now on the case N = 3. The interaction in the system is characterized by three independent control parameters K, wi/u^, and 0*2/ 3. u\ = i is assumed to be fixed throughout the paper. To analyze the collective behavior we define the phase rotation numbers as the growth rates of the phases:

— u mn-»oo(yn ~ fo) _ ^•lan-*oo{(Pn ~ Vp) /g 29^ lim^oo^-y^)' 2 limn-Kx, (<££-¥?§) '


In this formula the condition mod 1 in Eq. (6.20) is not applied. According to the values of 7-1,2, the observed dynamics (being either periodic or chaotic) can be classified as synchronous (ri^ = p/q is rational) or non-synchronous. Let us limit our considerations to the case of strong resonances q = 1,2,3. As illustrated in Fig. 6.21, the central region of synchronization corresponds to the main resonance r\ — 1 : 1, r2 = 1 : 1, while the nearby regions are specified by the following combinations of r\ and r2: ( j , 5), (j, 2)1 (2"' i)> (2' 2)' (h 5)' (§' §)• N°*e that the conditions of partial synchronization (geometrically, this corresponds to a two-dimensional quasiperiodic motion on the surface of a three-dimensional torus) is fulfilled within narrow regions near oj\ — L02 = 1. They are clearly seen in the indicated if-planes (K = 0.5 and 1.0), but to simplify the representation they are not identified in the full parameter space of the diagram.

Fig 6.22. Spectrum of Lyapunov exponents as a function of coupling strength. The figure demonstrates the transitions to hyper-chaotic solutions.

""-1.0 -0.5 0.0 0.5 1.0

K

Hence, for N = 3 the model (6.20) reveals a structure that is similar to Arnol'd tongues for a three-frequency quasiperiodic motion. The location of the resonance regions is symmetric with respect to oj\ and u>2. By analogy, N — 4 corresponds to four-frequency quasiperiodic oscillations, and so on. As we extend this construction to high-dimensional systems, the complexity and the number of coexisting solutions will increase. Moreover, the motion on torus Tm where m > 3 is structurally unstable [54]. In the following discussion we shall focus our investigations on a limited parameter region, namely, on the development of attractors along the route A in Fig. 6.21 when the dimension

2.0

1.0

0.0

-1.0

-2.0

-3.0

-A n

++0 ^ L ! Xf^/^++Q

'

^


of the system is increased. Figure 6.22 shows the variation of the spectrum of Lyapunov exponents

versus the coupling strength K for the case N = 3. Depending on K, the plot reveals the occurrence of different types of behavior ranging from regular (0, —, —) to hyperchaotic dynamics (+, +, 0) both for positive and for negative coupling strength. Let us consider this evolution in more detail:

Compe t i t ive coupling (K < 0). In this range of coupling there is a tendency for the oscillators to attain anti-phase synchronization, whereas in-phase solutions are unstable. The main stages of the evolution of coexisting attractors are schematically shown in Fig. 6.23. At weak negative coupling, six period-3 solutions coexist in the phase space of the system. The number six corresponds to the number of possible combinations of the set <pi$,3- Let us follow the evolution of one such solution by considering the thrice-iterated map (i.e., a single fixed point represents each of the period-3 cycles).

® - penod-doubliog bifmcatson

^ - Mddle-iiode bjfiucauoa

/ . -crises of chaotic sets

Stage 1. The fixed point evolves into an attracting invariant closed curve that contains a pair of stable and saddle solutions with double period. Figure 6.24 shows how the original stable fixed point (a) undergoes a normal period-doubling bifurcation at K = —0.212 (b) and then a transverse period-doubling bifurcation at K — —0.2.15 (c). As a result, stable and saddle cycles with double period appear while the original cycle becomes unstable. These bifurcational transitions lead to the formation an invariant curve which is equivalent to a resonant 1:2 torus for time-continuous systems.

Stage 2. With numerically increasing K, a pair of saddle-node bifurcations at K =- —0.23276 and K = —0.23295 multiply the number of coexisting attractors. Now the stable manifolds of the saddle cycles form the basin boundaries for the stable period-2 cycles labeled as I, II, and III in Fig. 6.25(b). The total number


(a)

K=-0.195

(b)

'K=-0.212

(C)

K=-0.240

Fig 6.24. Sequence of bifurcations leading to the formation of an invariant curve: (a) fixed point, (b) a normal period-doubling bifurcation, (c) a transverse period-doubling bifurcation. The phase projections are given on the (4>2 — 4>i, <fe — <f>i) plane.


of attractors approaches 18. Note, that the above saddle-node bifurcations will happen simultaneously in the case of identical systems d — 0. With further increase of K, a sequence of bifurcations of the coexisting periodic solutions gives rise to chaotic regimes (Fig. 6.23). Figure 6.26(a) shows 3 attractors, developed from the respective period-3 solutions, together with their basins of attraction. Two of the attractors are "ordinary" chaotic attractors with a spectrum of Lyapunov exponents (+,0, —). The third happens to fall in a periodic window.

Fig 6.25. (<p2 - 0i , 4>3 ~ 4>i) phase projections of a periodic solution (a) that is undergoing two saddle-node bifurcations (K = —0.23276 and K S -0.23295). As a result the number of coexisting attractors is increased (b).

Stage 3. With varying K, one of chaotic attractor touches the boundary of its basin of attraction, loses its stability, and becomes a "chaotic saddle" [55]. The trajectory escapes from this region and approaches another stable set (Fig. 6.26(b)), and the number of coexisting attractors is reduced to 12. With a slight variation of the coupling, the next chaotic attractor undergoes a similar crisis (Fig. 6.26(c)). Now, the number of chaotic attractors is reduced to 6. However, two nonattractive fractal sets (chaotic saddles) exist in the vicinity of each chaotic attractor. Finally, at K = —0.28508, the last chaotic attractor undergoes a boundary crisis. The trajectory remains located within the same region of phase space, but all chaotic saddles are now involved in the motion. Note that the chaotic saddles arise from families of attractors with different orientation in phase space. Hence, the new, merged attractor involves additional directions of instability, and the spectrum of Lyapunov exponents is specified as "+,+ ,0" . This is the transition to hyperchaos.

We have considered the above scenario for one of six families of coexisting attractors. For the other families, the same evolution is observed. In the coupling range K € [—0.286; —0.328] six coexisting hyperchaotic attractors can be detected. This is illustrated in Fig. 6.27. Here, the numbers 1,...,6 denote


Fig 6.26. (02 - 01, 03 - 0i) phase space projections of the basins of attraction and solutions when the coupling strength is increased, (a) Three chaotic attractors coexist (K = -0.279); (b) Boundary crisis for the first chaotic solution occurs (K = -0.284); (c) Boundary crisis for the second chaotic solution occurs (K = -0.285); (d) Boundary crisis for the last chaotic attractor occurs (K = -0.28508).

Fig 6.27. (02 - 01. 03 - <t>\) Phase space projections of (a) six coexisting hyperchaotic attractors that undergo merging crisis (b) at K = —0.328.


Fig 6.28. {4>2 - <t>u 4>s - <t>i) phase space projections of (a) coexisting local chaotic attractors (A" — 0.2050) that proceed through crises to hyperchaotic attractors (b) (K = 0.2054). (c) the coexisting high-order solutions have merged into a single hyperchaotic set (K = 0.244).

the different segments of the six attractors. At K fa —0.328, they all merge into a single hyperchaotic set (Fig. 6.27(b)). Because all possible directions are already involved in the chaotic motion, no qualitative change in the spectrum of Lyapunov exponents is associated with this final crisis.

Diffusive coupling (K > 0). Similar transitions and mechanisms are found to occur with a diffusive interaction. In this case, the coupled systems tend to become synchronized in phase. This defines a different geometry of the coexisting attractors (Fig. 6.28). However, the same mechanism of hyperchaos development as seen for negative coupling can be observed. In Fig. 6.28, three illustrative examples are given. A set of coexisting chaotic attractors (a) undergoes a sequence of crises at K € [0.200; 0.2060] and forms the hyperchaotic attractors (b) which merge into the global hyperchaotic regime at K « 0.340 (c).

To summarize the results for N = 3, we can conclude that the appearance of hyperchaos is preceded by the evolution of a number of solutions which involve only one direction of instability. Then the joining of chaotic saddles takes place. According to this mechanism, the transition to high-dimensional chaos occurs suddenly rather than through a gradual development of complexity. Note, how-


ever, that the spectrum of Lyapunov exponents as a function of the coupling parameter demonstrates a smoothing effect in the diagnostics of this transition. Since the Lyapunov exponent is an average measure and the residence time on each chaotic saddle changes smoothly with varying parameter, the plot in Fig. 6.22 indicates only "zero-crossing" instead of a "jump" of the exponent.

We note that for N — 3, the final merging of the hyperchaotic attractors does not increase the chaotic dimension, since all possible directions of instability are already included in the motion.

6 .7 A R o u t e t o H i g h - O r d e r C h a o s

With increasing dimension N we expect the number of coexisting solutions to grow significantly. For N = 5, the coupled map model (6.20) allows us to observe both a huge number of coexisting solutions and a variety of different types of transition to high-order chaos. Figure 6.29 illustrates the complicated pattern of basins of attraction for K = 0.1. We estimate the number of coexisting solutions as N\, which gives 120 attractors as the basic set including all possible configurations in phase space.

1.2

Fig 6.29. (fa - 4>u 4>s - <k) phase projection of the basins of attraction of coexisting solutions for N = 5 and K = 0.1.

-12 -12 12

Certainly, the merging crisis does not lead to an increasing attractor dimension for every group of attractors. This process depends strongly on the mutual orientation of the attractors inside the specified group. Actually, two different scenarios are possible:


1) The merging low-dimensional attractors together involve all possible directions of instability in the given system. In this case the dimension of the joint attractor may abruptly increase up to highest possible value, as we observed for N = 3.

2) The group of attractors that are undergoing a merging crisis are localized in a similar way in the phase space. Thus, the joining process can add only a single (or even no) direction of instability.

Hence, the evolution of the attractor dimension with varying coupling strength depends strongly on the features of the particular system. As illustrated in Fig. 6.30, our calculations of the Lyapunov exponents for the coupled map system (6.20) with N = 5 reveal a variety of different behaviors for positive as well as for negative K.

Fig 6.30. The spectrum of Lyapunov exponents versus coupling parameter for N = 5.

"-0.50 -0.25 0.00 0.25 0.50

K

K > 0. Abrupt transition to high-order chaos. Figure 6.31 displays the evolution of one family of coexisting attractors from regular behavior (a) to hyperchaotic dynamics (b). An enlargement of the small square in Fig. 6.31 (a) reveals the complex development of the coexisting attractors which is illustrated in Fig. 6.32. When the coupling strength is increased, a set of quasiperiodic attractors (Fig. 6.32(a)) undergo bifurcations that lead to the formation of local chaotic attractors. The different stages of development for the attractors are clearly observable due to the weak asymmetry of our system (d = 0.001). The most developed solution among the coexisting attractors will be first both in the transition to chaos and in the formation of a chaotic saddle. Correspondingly, the least developed attractor will be the last to remain stable before the

<<

o.u

2.0

1.0

0.0

-1.0

-2.0

-3.0

. . . . . . . . . ! . . . . . . .

++++0 ^<h«^§k ' I \ ++++0

M *


crisis when the coupling strength is increased (Fig. 6.32(b)). At K = 0.2170, this attractor undergoes the boundary crisis and a joined chaotic attractor is born (Fig. 6.32(c)). In contrast to the case N = 3, this event does not lead to an increase in the number of instability directions. The signature of the spectrum of Lyapunov exponents remains (+,0, ). Finally, the merging crisis of all families of local attractors produces the abrupt transition to a global hyperchaotic attractor specified by (+,+,+,+,0) .

Fig 6.31. (c/>2-< i, 4>3 — 4>i) phase projections of (a) a periodic solution (K = 0.213) that evolves into (b) a hyperchaotic set (K = 0.218).

K < 0. Gradual development of high-order chaos. Fig. 6.33 shows the transformation of the phase space projections of at

tractors when only one additional positive Lyapunov exponent appears. The evolution of attractors looks quite similar to the case of K > 0. The coexisting quasiperiodic regimes (a) are transformed into chaotic behaviors (b), each characterized by the spectrum of Lyapunov exponents (+, 0, —, —, —). However, the joined local attractor (c) at K = —0.08970, specified by ( + , + , 0 , —, —), is already hyperchaotic. There is no qualitative difference of this transition relative to the transition described for the K > 0 (Fig. 6.32). The only difference is that the negative value of the coupling strength supports anti-phase regimes rather then in-phase dynamics. Thus, the coexisting attractors of the same family should be more widely spread in the phase space, involving more directions of instability.

A further change of K leads to the merging crisis of all local attractors at K = —0.936. At K = —0.097 the spectrum of Lyapunov exponents has the structure ( + , + , + , 0 , - ) (Fig. 6.33(d)), and at K = -0.1125 fully developed hyperchaotic dynamics specified by (+, +, +, +, 0) is observed.

From the above description we conclude that the abrupt transition from "or-


Fig 6.32. Enlargements of the small square in Fig. 6.31. (02 — <f>i, 4>3 — 0i) phase projections of (a) quasiperiodic solution (K = 0.213) that develops into the chaotic sets (b) and (c) at (K = 0.215) and (K = 0.218), respectively.

0.15 0.15|

Fig 6.33. (fa — <f>i, 03 — 4>i) phase space projections of (a) quasiperiodic solutions specified as (0,0, - , - , - ) at K = -0.088; (b) local chaotic attractors (+, 0, - , - , - ) at K = -0.08856; (c) hyperchaotic solution identified as (+, +, 0, —, —) at K = —0.08970 and (d) high-order chaotic attractor (+, +, +, 0, - ) at K = -0.097.


Fig 6.34. Variation of the Lya-punov dimension DL of attractors that occur in system (1) as a function of the coupling parameter K and the number N of interacting oscillators. For large enough K, Di is seen to increase linearly with N (at least up to N = 8).

dinary" chaos with (+, 0, —,...,—) to the high-order chaos with ( + , . . . , + , 0) is structurally stable. Our results were derived for systems with broken symmetry (d y£ 0). For the symmetric case, the boundary crises happen simultaneously, but the main features of the considered mechanism will be preserved. We have also shown that depending on the mutual orientation of the coexisting attractors, the merged attr actor may or may not involve a new instability direction into its motion. This problem clearly needs further investigation.

For sufficiently large values of the coupling strength, all possible instability directions of the model become involved in the motion on the global chaotic attr actor. This fact establishes the connection to the concept of a chaotic hierarchy as discussed in Sec. 6.6. In Fig. 6.34 the Lyapunov dimension Di of the attractors is depicted versus the coupling strength and the system dimension. Inspection of the figure clearly shows that Di is proportional to N, at least up to N = 8. In the frame of the discussed mechanism for the development of high-order chaos, there are no reasons to expect the saturation of D/, at higher N.

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Chapter 7

POPULATION DYNAMIC SYSTEMS

7.1 A System of Cascaded Microbiological Reactors

Spatiotemporal dynamics play an essential role for many ecological, epidemiological and microbiological systems where species interact with one another and with a supply of primary resources [1, 2]. Typical examples are the emergence of patches of algae blooms [3], the development of complex growth patterns in bacterial colonies [4], and the propagation of swarms of insects [5]. For aquatic ecosystems, the spatiotemporal dynamics may also be influenced by flows that carry individuals of the biological species from one place to another [6]. In this chapter we consider a specific, and to a certain extent simplified, example of such a system. As illustrated in Fig. 7.1, our system consists of a chain of population pools connected through the flow of primary resources. Each pool is the habitat for a three-variable predator-prey system. The first (upstream) population pool is assumed to receive nutrients at a rate pa\ where p denotes the overall flow (or dilution rate) along the chain, and a\ the afferent nutrient concentration. For low values of <T\, the system exhibits a stable equilibrium point. However, as the supply of nutrients increases, the system passes through a Hopf bifurcation into a regime of self-sustained oscillations.

Depending on the population sizes attained in the first habitat, different degrees of depletion of the nutrient concentration will occur, and as the surplus resources continue to flow into the next habitat, this population pool will be modulated by a temporal nutrient supply that depends on the type of dynamics

259


* 1 , < I > 1 B

1

P . S

1

<M>2

__ B 2

P 2

s 2

<M>3

__ B 3

P 3

s 3

Fig 7.1. Cascaded system of interacting bacterial and viral populations. Bi,Pi, and S; denote the concentrations of bacteria, phages and nutrients in the various pools along the chain. Besides nutrients in the overflow from the upstream pool, each habitat receives a local supply of primary resources of concentration CT*. <j> represents a weak source of virus contamination, and p is the dilution rate.

realized in the former pool. Along the chain there will be a net consumption of resources. Hence, there is a tendency for the complex behavior gradually to die out. To compensate for this we have allowed for the possibility that each habitat, besides the resource overflow from the previous habitat, also receives a local nutrient supply at a rate pai, with i = 1,2,3... denoting the pool number in the chain.

The aim of our study is to examine such questions as: Under what conditions will the complex behavior spread along the entire chain? With sufficient supply of nutrients, can the complexity continue to grow from pool to pool so that we observe a kind of chaotic hierarchy [7] with an increasing number of positive Lyapunov exponents? What are the conditions for the population pools to synchronize? And, can we observe domains of synchronized behavior moving along the chain? Problems of this type are often discussed in terms of coupled map lattices [8, 9, 10] or of chains of coupled oscillators [11]. However, as we shall see, the specific form of our population dynamic system and of the coupling between the pools introduce a number of new phenomena.

To provide realism and specificity to the analysis we shall consider a microbiological system of interacting bacteria and viruses (bacteriophages or simply phages) first studied by Levin et al. [12]. Systems of this type play an important role in many biotechnological industries. The homogeneous, well-controlled bacterial cultures used in modern cheese production, for instance, are often quite sensitive to phage attack, and considerable effort is devoted to the search for more resistant cultures [13]. The relatively short time scales involved, and the fact that microbiological systems can be prepared under many different, well-controlled conditions also render systems of this type particularly amenable to

Population Dynamic Systems 261

experimental studies of complex population dynamic phenomena.

Baier et al. [14] have examined various aspects of the bacteria-virus model for a single homogeneous habitat. It was shown that with increasing number of bacterial strains (each sensitive to attack from a specific phage variant), a chaotic hierarchy could be realized. The dynamics of two diffusively coupled habitats was considered by Mosekilde et al. [15]. In that work emphasis was given to co-evolutionary phenomena associated with the processes that allow bacterial cells to develop resistivity towards specific phages and the competing processes through which the viruses become capable of overcoming this resistivity. Such phenomena will not be considered in this chapter, and we shall assume that the properties of both cells and viruses remain unchanged.

Our system bears certain similarities to coupled chemical reactors as studied, for instance, by Schreiber and Marek [16]. However, where our model has a unidirectional flow of nutrients along the chain, studies of chemical oscillators usually involve a diffusive coupling that allows one (or more) reactant to pass between the reactors. Studies of bifurcation phenomena in periodically modulated chemical systems [17, 18, 19] have revealed a number of interesting structures that can arise when a system is forced across a Hopf bifurcation. Similar structures have been observed in connection with forced insulin secretion in man [20] and for forced microwave diodes [21]. Various aspects of coupled nonlinear oscillator systems have also been studied, for instance, by Badola et al. [22], by Anishchenko et al. [23, 24], by Osipov et al. [25], by Rogers and Wille [26], and by Mattews and Strogatz [27]. Instabilities in chemical reaction-diffusion systems with a unidirectional flow have been studied by Kuznetsov et al. [28] and by Andresen et al. [29]. The results that we shall present in this chapter are primarily based on our recent analyses of phase synchronization in cascaded population dynamical systems [30, 31]. A more detailed description of the microbiological oscillator may be found in Topics in Nonlinear Dynamics [32].

The bifurcation structures that we observe in the bacteria-virus model differ significantly from previous results [17, 18, 19]. In particular we find that global bifurcations seem to occur right down to the smallest forcing amplitudes. This is clearly at odds with standard synchronization theory as established originally by Arnol'd [33] and according to which the edges of the resonance tongues at low forcing amplitudes consist of saddle-node bifurcation curves. The peculiar bifurcation structure observed when forcing a single population pool is also


found for coupled habitats (i.e., without external forcing). Having discussed these new bifurcation phenomena at some length, we shall consider the local and global synchronization phenomena that can arise for a chain of 20 population pools.

7.2 The Microbiological Oscillator

For each of our population pools (all assumed to have the same volume) we consider a culture of bacterial cells Bi, with i denoting the pool number along the chain. In the absence of viruses, the bacterial populations grow in accordance with standard Monod kinetics, i.e.

dBj | _ vBjSj , . dt lsrawth~ Si + K' ( >

where Si is the concentration of nutrients in pool i. v is the maximal bacterial growth rate as obtained under conditions of ample resources, and the Michaelis-Menten constant K represents the concentration of nutrients at which the growth rate is reduced to half its maximal value. Each cell division is assumed to be associated with a resource consumption 7.

Besides bacteria, each pool is considered to contain a population of viruses Pi. These populations originate in a homogeneous and constant contamination of the habitats at the very low rate <j>. As the Brownian motion carries a virus particle into the proximity of a cell, its tail fibers will sweep along the cell surface and affix to specific protein receptors inserted in the membrane. This adsorption process is assumed to proceed at a rate which is proportional to the product of the two concentrations and to be characterized by a rate constant a. Following the adsorption, the virus drives its DNA (or RNA) into the cell, where it takes control of the reproductive system and starts to replicate. After a latent period r of the order of 30 min, the infected cell bursts and releases an average of /3 new viruses. This is known as a lytic response to the virus attack [12, 15], and /3 is referred to as the bursting size.

Phages may also adsorb to already infected cells, in which case the virus particle is considered to remain bound to the cell surface and to be unable to infect other cells. Reactor fluid containing surplus resources, waste products from the bacterial growth processes, cells and phages is assumed to be washed out at a rate p. According to our assumptions, however, only nutrients will be


transmitted to the next pool. In this way each subsystem receives a net supply of resources proportional to p(Sj-i + &i — Si) where Si denotes the resource concentration in pool i. Altogether this leads to the following set of coupled differential equations

dBi vBA „ , „ , , ,

-w = s-ti-B^+a^ ^ -± = aujBiPi - pit - Ii/r (7.3)

^ . = lf,-Pi(p + a{Bi + Ii)) + fiIi/T (7.4)

-#=p&-i + «-s*)-s^rK (7-5)

where 1, denotes the concentration of infected cells, w is the probability that a virus particle successfully infects a cell, once it has affixed to its surface. Coupling between the pools is obtained only through the flow of nutrients. For the first habitat, 5,_i = SQ is assumed to be zero.

The parameter values that we have applied in the present analysis

v = 0.024/mm w = 0.8

K = 10 fig/ml 7 = 0.01 ng

T = 30 mm /3 = 100

correspond to the values used in our previous studies [14, 15]. These values are also in general agreement with the experimental values obtained by Levin et al. [12] for particular strains of bacteria and viruses. The concentrations Bj, Ii, and Pi will be specified in units of 106/mL Hence, we have used a value of a = 10~3ml/min as compared with the value a = 10~9ml/min applied by Baier et al. [14].

Like many other ecological models, our system involves positive feedback mechanisms related to the replication of bacteria and viruses. There are nonlinear constraints associated both with the bacterial growth rate and with the infection rate, and there is a delay associated with replication of the phages. The rate of dilution is a major determinant of dissipation in the system. In the absence of phages, the single pool model displays an equilibrium point


BQ -pK

Sn = pK

v — p (7.6)

v — Py in which the rate of bacterial growth balances the wash out. For dilution rates p> pc = avj (K + a), only the trivial equilibrium point B\ = 0, S\ = a exists.

As p is reduced below pc, the equilibrium population of bacteria starts to increase. At the beginning, the cell concentration is still too small for an effective replication of viruses to take place, and the phage population remains nearly negligible. As the dilution rate continues to decrease, however, the virus population grows significantly. The model then undergoes a Hopf bifurcation, and the system starts to perform self-sustained oscillations.

0.0

8.0

6.0

4.0

r>k' =_J>> .

• „ ? M S i /

stable equilibrium (focus)

I I I !

(a) .

-

• --

OH

35.0

25.0

15 0

50 I P,

P

\ *

(b) -

-

~

J

0.0 0.002 0.004 0.006

P

Fig 7.2. Self-sustained oscillations in a single-pool bacteria-phage model. Two-parameter bifurcation diagram (a) and phase projection of the limit cycle obtained for p = Q.Q05/mm and o = 12.0 fig/ml (b). P0 and Pi denote the unstable focus and the saddle equilibrium point, respectively.

The two-parameter bifurcation diagram and the phase projection of a limit cycle are depicted in Fig. 7.2. In these diagrams the dilution rate p is specified in /min, the resource concentration a in pg/ml and the population sizes P in 108/T7II. The limit cycle was obtained for p — 0.005/min and a — 12.0 pg/m.l.

Note, that the saddle point at (B i , / i ,P i , 5 i ) = (0,0,0, a) is located near the limit cycle. The stable and unstable manifolds of the saddle point are determined by (B — 0, P ^ 0, / = 0) and (B ^ 0, P = 0, / = 0), respectively, and the presence of this point provides the conditions for a strong modulation of the limit cycle period under applied forcing. In Chapter 9 we shall see


how a similar situation in a model of nephron pressure and flow regulation is responsible for a dephasing effect close to a homoclinic bifurcation.

7.3 Nonautonomous Single-Pool System

Let us start by considering a single homogeneous pool of interacting bacteria and viruses (a continuously stirred microbiological tank reactor), subjected to an external forcing in the supply of primary resources. The rationale for introducing this forcing is that in the cascaded population model each habitat will be modulated by variations in the nutrient supply from the upstream habitat. Obviously, one can also envisage experimental conditions in which the supply of nutrients to a microbiological reactor is periodically modulated. As we shall see, the external forcing introduces a variety of different bifurcation phenomena through which the qualitative behavior of the model changes.

The modulation of the afferent nutrient concentration is specified through the relation

< r i ( t ) = < 7 o ( l - y ( l + s innt ) ) (7.7)

where fi is the modulation frequency and m the amplitude of the forcing. <JQ denotes the maximum nutrient concentration in the inflow to the reactor. Our formulation implies that, as the forcing amplitude increases, the average nutrient supply decreases, and the system becomes more stable. At the same time, as m becomes large enough, we will be forcing the system across the Hopf bifurcation, i.e., there will be periods during a forcing cycle in which the system operates with a nutrient supply that produces stable behavior in the stationary case and periods in which the system operates in a region of self-sustained oscillations.

The phase diagram of Fig. 7.3 provides an overview of the stationary solutions obtained for different forcing amplitudes and modulation frequencies. The flow rate is taken to be p — 0.005/mm, and the maximum nutrient concentration in the inflow is Co = 12.0 fig/ml. In the crudest interpretation, the diagram shows quasiperiodic behavior on a two-torus with characteristic synchronization zones for m < 0.4. The quasiperiodic (or high-periodic) behavior is indicated by a light gray shading. For forcing amplitudes m > 0.4, the internal oscillation is quenched, the two-torus no longer exists, and the system displays a folded structure of tilted resonance peaks. Finally, as the forcing amplitude becomes


larger than about 0.65, period-doubling bifurcations leading to chaos occur on the resonant (i.e., large amplitude) side of the overlaying resonance tongues.

•• 1/ \ I II

0.0 0,002 0.004 0.C

Q

i 0.010

Pig 7.3. Phase diagram delineating regions with qualitatively different behaviors of the periodically forced single-population system. Q is the forcing frequency (given in units of /rain) and m the forcing amplitude. The figure demonstrates the characteristic structure of overlaying resonance regions. Light gray zones indicate regions with quasiperiodic solutions. Crosshatched zones are regions with period-doubling cascades, and tiled zones are regions with chaotic dynamics.

0.004 0,005 0,006

Q

Fig 7.4. Bifurcation curves defining the main structure of a typical resonance region. This structure differs from previous results by the fact that the global bifurcation curves G seem to proceed all the way down to the smallest forcing amplitudes. Bn denote Takens-Bogdanov bifurcation points. SNn are saddle-node bifurcation curves, SSNn saddle-double saddle bifurcation curves, and Tn torus bifurcation curves.

The above interpretation combines a number of features that are known from previous studies. However, as we shall discuss in the following, the phase diagram in Fig. 7.3 exhibits a peculiar structure of the resonance zones. The light gray regions that fall inside the synchronization zones represent the coexistence of an ergodic torus with a stable periodic orbit. (Figure 7.6 displays an example of such a pair of coexisting periodic and quasiperiodic solutions.) The structure of the 1:1 resonant tongue is formed by the bifurcations of the stable limit cycles C*:1 and C\-2 (see Fig. 7.3), the saddle cycle (7*:1, and the unstable cycle Cu

that exists inside the two-torus. This is illustrated in more detail in Fig. 7.4, where we have drawn the related bifurcation curves. Here, T\ and T2 denote the


branches of the torus bifurcation curves on the two sides of the main resonance zone. SNi,SN2 and SN3 are saddle-node bifurcation curves, delineating the main boundaries of the tongue. If we start above the torus bifurcation curve 7\ and increase the modulation frequency, we can follow the stable cycle C| : 2

up to the saddle-node bifurcation curve SN3 where it collides with the saddle cycle C]'x. This saddle cycle can then be followed back to SNi where the stable cycle C\A is born.

(a)

(c)

4

c1i:1*'^c;:i (b)

l s < i " i « | ) j ' ) ^

^-^ (d)

n1:1 " \ 1 / \

Fig 7.5. Poincare sections illustrating the structural changes that occur along the arrow A in Fig. 7.4. For (b), (c), and (d) the unstable manifold of the saddle cycle is plotted using the method suggested by Kevrekidis et al. [17, 18].

B\ and J32 are so-called Takens-Bogdanov points [34], where two complex conjugated eigenvalues of Cs

1:1 and C\l meet at Ai = A2 = 1, and SSN\ and SSN2 are bifurcation curves along which the saddle cycle C]'1 merges with a double saddle Cu (i.e., a periodic solution with two unstable directions). The curves denoted G are global bifurcation curves at which the two-torus collides with the inset (the stable manifold) of the saddle cycle C]'A. The unusual aspect of the phase diagram is the fact that these global bifurcation curves seem to proceed all the way to the tip of the tongue. Previous studies [19] have typically found that the global bifurcation curves terminate at finite forcing amplitudes, and that the edges of the resonance zones at low forcing amplitudes are made up by saddle-node bifurcation curves with no further structure.

Let us illustrate by means of a series of Poincare sections what happens


0.1 0.2 0.3 0.4

B

Fig 7.6. Coexisting periodic cycle and quasiperiodic attractor for f2 = 0.00390/min and m = 0.30. By contrast to previous studies, coexistence of the two attractors is observed all the way down to zero forcing amplitude.

Fig 7.7. Persistent chaotic transient close to the heteroclinic bifurcation (crisis) along the edge of the dark grey zones in Fig. 7.8. One observes how the system finally settles down in the periodic orbit indicated by small diamonds. Parameter values correspond to point A in Fig. 7.8.

with the two-torus at the left hand side of the 1 : 1 resonant tongue. In the regions immediately outside the saddle-node bifurcation curves, the system displays a single quasiperiodic attractor. This is illustrated in Fig. 7.5(a). At the saddle-node bifurcation curves, a pair of periodic orbits, one stable and the other unstable, are born outside the torus. As shown in Fig. 7.5(b) the unstable manifold of the saddle cycle connects to the stable cycle on one side and approaches the torus on the other. The stable manifold of the saddle cycle (not shown) separates the basins of attraction for the two asymptotic solutions. Along the homoclinic bifurcation curves, the torus makes contact with the saddle cycle and disappears in a boundary crisis (Fig. 7.5 (c)), and between the two homoclinic bifurcations the system displays a stable and an unstable periodic orbit in a heteroclinic structure that is topologically equivalent to a resonant torus (Fig. 7.5 (d)). More precisely, when the torus approaches the saddle cycle, a folding occurs. This process does not directly involve the manifolds of the saddle cycle but destroys the torus via the loss of smoothness right before the homoclinic event.


c!' \ ci'

\ O.fi I 1 i 1 1 i J is. 1 ' 0 002 0.00J (1004 OOOJ 0 0 » 0.00?

Q

Fig 7,8. Magniication of part of the phase diagram in Fig. 7.3. Dark gray regions represent holes in the foliated structure. Here we observe a persistent chaotic transient leading at the end to a stable periodic orbit. The period-doubling curve PD^:l appears to pass directly through the cusp point ifl which the saddle-node curves SN1:1 aud SN2:1 meet.

Thus, there are two different tori involved in the process, and the transition between them is accomplished via a homoclinic bifurcation. In previous studies this type of transition was observed only for a limited region of relatively strong coupling [17, 20, 35], whereas we have found a tangent-like location of the SN and G curves. We suppose that the reason for this peculiarity rests with the specific features of the autonomous model (7.2)-(7,5), namely, the fact that the limit cycle trajectory passes very close to the stable and unstable manifolds of the saddle point at the origin (recall the phase space projection in Fig. 7.2 (b)).

Figure 7.8 shows a magnification of the upper part of the 1:1 and 2:1 resonance zones. Under variation of the forcing frequency we can follow the 1:1 cycle (and the modes developed from this cycle by period-doubling bifurcations) from the left hand edge of the diagram to the saddle-node bifurcation curve SNhl, Here, the 1:1 mode becomes unstable, and we can follow the saddle-solution backwards to the SN2:1 saddle-node bifurcation curve where the stable 2:1 solution is born. As we again increase the modulation frequency, the 2:1 solution


proceeds through a period-doubling cascade to chaos and back again. For the 1:1 solution, the first period-doubling bifurcation PD\:1 continues around the fold SN1:1 to become a period-doubling curve for the saddle cycle. The second period-doubling curve PD\l appears to pass right through the cusp point at which the SN1:1 and SN2:1 curves meet. This implies the existence of a codimension-3 bifurcation point.

Regions in Fig. 7.8 that are shaded dark gray are "holes" in the foliation. Here, the amplitude of the chaotic set becomes large enough to make contact with the inset of the saddle cycle situated between the stable layers, the chaotic set loses its stability in a boundary crisis [36, 37], and the coexisting cycle in the underlying layer becomes the only stable solution. As illustrated by Fig. 7.7, however, the chaotic set continues to manifest itself in the form of a persistent chaotic transient. The figure follows such a transient until the trajectory finally finds its way to the periodic orbit marked by small diamonds. If we were to interprete this behavior for our microbiological system, it would imply that the bacterial population could oscillate chaotically for months before it suddenly attained a stationary periodic solution. The parameter values for the chaotic transient in Fig. 7.7 correspond to point A in Fig. 7.8.

7.4 Cascaded Two-Pool System

The above analysis provides insight into the kinds of behavior that can arise in a forced population dynamic model. Although only part of these dynamics may be observed in real life, it is clear that the potential complexity is significant. Let us now try to extend the analysis to consider a cascaded system of two microbiological population pools. As bifurcation parameters we shall take the nutrient concentrations a\ and oi in the local supplies to the two pools.

Figure 7.9 shows the phase diagram obtained by means of a manually directed continuation program. Now, the flow rate is p = 0.003/mm. For small values of the nutrient supplies o\ and o<i (i.e., in the lower left corner of the diagram), the system exhibits a stable equilibrium point. As o\ is increased beyond a threshold value er# = 7.015 ng/ml, the upstream population pool becomes unstable and starts to perform self-sustained oscillations. This transition is represented by the Hopf bifurcation curve H?. Alternatively, if we again consider the situation a\ < GH, increasing a^ leads the second pool to become unstable at the Hopf bifurcation curve H\. For o\ = 0, this curve


passes through the point a2 = ffH- For finite values of <xi, overflow of resources from the upstream pool implies that smaller values of a-i are required to elicit self-sustained oscillations in the second pool.

0 2 4 6 8 10 12 14 14 IS 20

0

Fig 7.9. Phase diagram for two cascaded microbiological reaction systems, <7j and <r2 represent the local resource supplies to each of the two systems. The second habitat also receives resources from the overflow of the first habitat. Hn,Tn and PD„ denote Hopf, torus and period-doubling bifurcation curves, respectively. SN and G are curves of saddle-node arid global bifurcations. The insert shows details of part of the bifurcation structure.

In the region of parameter plane to the right of H\, the upstream habitat is stable, and the system performs a simple limit cycle oscillation embedded in the four-dimensional subspace of total phase space associated with the downstream pool. Increasing the nutrient supply to the first pool leads the system through the torus bifurcation T\. Above T\, both habitats perform self-sustained oscillations, and we observe the classic structure of resonance tongues on a background of quasiperiodic motion [33]. Only the main tongues are shown, and the intervening regions with quasiperiodic (high-periodic and chaotic) behaviors are indicated by a light gray shading. The period of oscillation for the individual pool tends to decrease with increasing nutrient supply. Hence, a particular synchronization can be followed under simultaneously increasing values of ai


and 02, and the tongues are tilted to the right. From left to right the winding numbers of the main tongues are 2:3, 3:5, and 1:2. It is interesting to note that along the torus bifurcation curve T\ the winding number exhibits a minimum at approximately O<L = 13.5 fig/ml. Beyond this value, the winding number starts to increase again.

Each of the main tongues displays an internal structure in the form of a cascade of period-doubling bifurcations leading to chaos. As shown in the insert, some of the minor tongues display a substructure resembling the characteristic swallow-tail structure that one can observe for bimodel maps such as, for instance, the sine-circle map [38, 39, 40]. These structures may be considered as relatively well-understood, and we shall not discuss them in more detail here.

The 1:1 resonance region, on the other hand, displays the same peculiar structure as we observed for the resonance regions of the single pool model with external forcing. The 1:1 resonance area develops from the Hopf bifurcation #2 where the upstream pool starts to perform self-sustained oscillations. By virtue of the overflow of nutrients, these oscillations act as a forcing signal for the bacterial population in the second pool. However, the forcing is so strong as to practically quench the tendency for oscillations in this pool. As a consequence, we observe a structure which is similar in certain respects to that of a forced system with a damped resonance.

The edge of the 1:1 resonance area is delineated by the saddle-node bifurcation curve SN (see Fig. 7.9). Along this curve a stable and an unstable 1:1 solution, existing to the left of the curve, are annihilated. Somewhat to the left of the saddle-node bifurcation curve we find the global bifurcation curve G that proceeds all the way down to the lowest forcing amplitudes, i.e., to the torus bifurcation curve T\. Along G, a quasiperiodic solution that exists to the right of G makes contact with the inset to the 1:1 saddle cycle and hereby abruptly looses its stability. In the region between G and SN we thus have a quasiperiodic solution (with resonance tongues) coexisting with the stable and unstable 1:1 cycle.

For o\ values lower than approximately 10 fig/ml, we observe an even more complex type of transition. This is illustrated in Fig. 7.10. Let us fix <j\ = 7.5 fig/ml and follow the stable synchronous solution C\ from the lowest value of o"2- Let us also indicate the number of unstable directions (number of Floquet multipliers numerically larger than 1) each by a superscript "+". While cj-i increases, C\ remains stable until the point 1 on the torus bifurcation curve T2,


Fig 7.10. Bifurcation curves associated with the transition from quasiperiodic to synchronous oscillations for two cascaded microbiological reaction systems at p = 0.003/min. The regions of torus existence are shaded gray.

where the periodic solution becomes unstable C\ -4- C*+. However, no stable torus appears to be born at this point. Instead, the trajectory follows the already existing quasiperiodic attractor (shaded gray). With further increase of 0-2, C*+ can be followed until the point 2 on the SSN2 line of saddle-saddle-node bifurcation. At this point, C*+ merges with the saddle cycle C}- Now, one can follow C j with decreasing cr2 until the point 3 on the SSNi line where it is annihilated together with the unstable cycle C^. When scanning again towards larger values of cr2, the C%+ solution can be found to exist inside the two-torus (shaded gray). Finally, the two-torus undergoes a crisis at the line G of homoclinic bifurcation, as described in the previous section. In this way, each of the four bifurcations at the points 1 - 4 plays an essential role in the transition between asynchronous and synchronous behaviors.

In the upper part of the main resonance area, i.e. for a\ > 12 jig/ml, the stable 1:1 solution undergoes a cascade of period-doubling bifurcations leading to chaos. The bifurcation curves for these transitions and the resulting chaotic


region are denoted by a superscript 1 in Fig. 7.9. Overlaying this cascade we find another set of period-doubling bifurcations (denoted by a superscript 2). These bifurcations originate from the 3:1 cycle produced in a resonance tongues to the left of the diagram (i.e., in the region of negative 02 values). Hence, in the top left corner of the phase diagram we have a chaotic attractor originating in the 1:1 resonant solution coexisting with a chaotic attractor originating in the 3:1 solution.

7.5 Homoclinic Synchronization Mechanism

In the previous section we have seen how two coupled microbiological reactors, besides various types of periodic dynamics, can produce quasiperiodic and chaotic dynamics. By virtue of the cascaded nature of our system, these dynamics will give rise to quasiperiodic and chaotic forcing of the third habitat. The purpose of the present section is to examine the response of this habitat to such complex forcing signals. Analogous with the case of periodic forcing, one can distinguish between a synchronous response in which the third habitat entrains with the external forcing signal, and an asynchronous response in which such entrainment does not occur. However, as discussed in Chapter 6 the distinction between the two types of response, particularly to a chaotic signal, is not so straight forward.

Obviously, one can imagine situations in which a minute change in the temporal variation of the forcing signal can make the difference between the two types of response. In general, however, it seems plausible that the ability of the third habitat to synchronize with the forcing signal rests with some general characteristics of this signal such as, for instance, the typical amplitudes and most dominant frequencies. Problems of a somewhat similar nature have previously been dealt with, for instance by Anishchenko et al. [41], by Pikovsky et al. [42], by Shulgin et al. [43], and by Postnov and Balanov [44] who have considered synchronization of a chaotic system to a periodic signal. Postnov et al. [45] have investigated the converse case of a periodic system being driven by a chaotic forcing.

From a general point of view, synchronization can be considered as a process by which two systems lock to one another in such a way that specific conditions are realized in each of the systems at certain characteristic time intervals. As discussed in Chapter 6, this phenomenon is denoted as chaotic phase synchro-


nization [42, 46, 47]. For our cascaded microbiological reactors, the characteristic time intervals may be defined as the mean return times < T; > to a Poincare secant of the subspaces for each of the population pools, i = 1, 2,3. As these return times correspond to the peaks of the Fourier spectrum we can introduce the rotation numbers

<T2> <n> n2 = Tf^> r23 = T7FS' (7'8)

and by monitoring the variation of r2s with some parameter we can determine whether the third habitat is synchronized with the behavior of the second population pool. In the numerical calculations to be presented here we have used Bi + 0.0IP, = 0 as the Poincare plane for subsystem i.

Let us first consider the response to a quasiperiodic forcing. By choosing <Ti = 7.5 fig/ml and a2 = 4.1 fig/ml for the nutrient supplies to the first two pools, one can obtain a quasiperiodic modulation of the concentration of primary resources in the overflow from the second habitat. Figure 7.11(a) shows the variation of the winding number r2$ with the additional local nutrient supply 03 to the third habitat. At relatively low nutrient supplies, r2^ = 1. Here, the third habitat synchronizes with the forcing signal to produce a similar quasiperiodic signal. Figure 7.11(c) shows a projection of the Poincare section for this dynamics for 0-3 = 6.3 fig/ml. The flow rate is p — 0.003/mm.

As the nutrient supply to the third habitat is increased, this 1:1 synchronization between the two quasiperiodic signals is maintained up to 03 = 6.4 fig/ml, at which point the synchronization abruptly breaks down and the winding number jumps to r23 = 1.38. Figure 7.11(b) shows an example of the asynchronous behavior observed in this interval. The motion here takes place on an ergodic three-torus. If, with an initial asynchronous response, 03 is adiabati-cally reduced, the asynchronous response can be followed all the way down to cr3 = 4.9 /ig/ml. Hence, in the interval from 4.9 to 6.4 fig/ml, a synchronous and an asynchronous response to the quasiperiodic forcing coexist. This is completely analogous to the coexisting periodic and quasiperiodic response observed in the two-pool model. Only, the two types of behavior now take place on a two- and a three-torus, respectively.

As a3 is increased beyond 7.2 fig/ml a new region of synchronization is reached, now with r2% = 1.5. This region extends from 0-3 = 7.2 fig/ml to 03 = 9.5 fig/ml. The existence of such finite intervals of synchronization between the two quasiperiodic motions results in a structure similar to the devil's staircase


Fig 7.11. Asynchronous (b) and synchronous (c) responses of the third population pool to a quasiperiodic forcing from the upstream pools. Variation of the rotation number r-a with the nutrient supply a3 (a). r23 = 1.0 indicates a synchronous response. Note the hysteresis phenomenon in the transition between the two types of response. There is also a region of synchronization (without hysteresis) for 7-23 = 1.5.

structure we know for the case of periodic forcing of a self-oscillating system [33].

Let us now consider the case in which 02 = 0 and G\ is varied from 12.0 till 14.5 fig/ml. Comparison with Fig. 7.9 shows that in this case the first two population pools demonstrate a transition to chaotic dynamics of relatively low dimension and developed through period-doubling of the 3:1 synchronous solution. The main regimes observed (and the corresponding bifurcation curves) are displayed in Fig. 7.12(a). For o\ < 12.48 fig/ml, the nutrient concentration in the outflow from the second chemostat follows a simple limit cycle oscillation C\, and variation of 03 towards the left-hand part of the diagram (not shown) realizes the homoclinic synchronization mechanism as described for the nonautonomous bacteria-virus population.


" ~i i i | i i i | i I T | i i r

: ( b ) :

4- J^ 2~_

n" — >

Nfl.

-

•

•

R i i i 1 i i i 1 i i i 1 i i i

10.0 11.0 12.0 13.0 11.5 12.0 12.5

Fig 7.12. (a) Bifurcation diagram for the system of three bacteria-virus populations; (b) hys-teretic transition between synchronous and asynchronous responses to a chaotic variation in the supply of nutrients to the third population pool as revealed by the variation in the winding number T23 with o$. r23 = 1 implies synchronization between the two chaotic motions. <7i = 14.25 fig I ml, a2 = 0.

The right border (Ti) is the torus bifurcation curve. However, no stable torus appears at this line, only the loss of stability for the periodic solution C\ near the saddle cycle. As it was proved for the two-dimensional system, this kind of transition is accompanied by a global bifurcation involving the homoclinic orbit of the saddle [48, 49]. Higher values of o\ allow us to observe qualitatively the same phenomena for the period-2 and period-4 limit cycles C2 and C4. Finally, when o\ exceeds 14.07 fxg/ml we observe a hysteresis behavior and a crisis-like transition between two chaotic attractors.

To characterize the involved attractors and the crisis itself let us recall some results on phase synchronization of chaos [24, 41, 46, 47]. For periodically forced chaotic systems it has been found that there are two main types of chaotic attractors. The first is qualitatively equivalent to the period-doubling chaos for the autonomous chaotic system with a fundamental frequency governed by the frequency of the external signal. Following Anishchenko et al. [41] let us term this chaotic regime as "synchronous chaos". The second type of chaotic attractor is torus-chaos with a power spectrum that exhibits two fun-

14.0

13.0

12.0

; CHAOS Gy-

- y^\ / 4

-^ y^T PD2 y

Y C2 A

-PD1 /

:C A7 ' 1 1 / 1 1 1 1

- / ( , ' W / BC

/ 4

, , ,


damental frequencies above the continuous background. By analogy with the quasiperiodic (asynchronous) non-chaotic behavior let us refer to this regime as " asynchronous chaos".

Besides the specific form of their phase projections, Poincare sections, and power spectra, the two types of behavior can be easily distinguished by following the phase of the chaotic attractors [46, 47, 50] or by calculating their mean return time to a Poincare secant [51].

Fig 7.13. Poincare sections of the two coexisting chaotic attractors for <r3 = 12.25 (a) and a3 = 11.73 ng/ml (b). Oi = 14.25 fig/ml, a2 = 0. The secant was defined as Bt = Pi/5.

The value of r^ was found to be 1.0000 ± 0.0001 at all points of the diagram in Fig. 7.12. This is consistent with the fact that the choice of 02 = 0 in the second chemostat produces synchronous (chaotic or regular) regimes only. Figure 7.12 (b) shows the variation of the winding number r<i% with the nutrient concentration 03 over the range from 03 = 11.5 to 12.5 fig/ml. In the lower end of this interval, T23 = 1, and the third population pool synchronizes with the chaotic forcing to produce a similar, low-dimensional chaotic response.

As 03 is increased, synchronization between the two chaotic motions is maintained all the way up to 03 = 12.3 fig /ml, where the winding number abruptly jumps to r23 = 1.34. For larger values of 03, the response is asynchronous. Loosely speaking, this response may be characterized as partly quasiperiodic and partly chaotic. If 03 is now reduced, we again observe the characteristic hysteretic transition between asynchronous and synchronous behavior in the model. Following the asynchronous response, the nutrient supply to the third habitat can be reduced to 03 = 11.75 fig /ml before synchronization sets in. Together with other observations the above results clearly show that the considered


chaotic attractors represent synchronous (branch r23 = 1.0) and asynchronous chaos in the sense of Anishchenko et al. [24, 41], Rosenblum and Pikovsky [46, 47], and Yalcinkaya and Lai [52]. However, the transition between the two chaotic attractors is of a different type. In the above works, the coexistence of synchronous and asynchronous chaotic regimes was not found, and the whole picture of transition between these regimes differed significantly from the one observed here. In our case the transition is consistent with the homoclinic synchronization mechanism that we have illustrated for the regular oscillations in the previous sections.

0.04

0.03

DroinO.02

0.01

0.00 11.7 11.8 11.9 12 12.1 12.2

O-3

Fig 7.14. The minimal distance Dm{n between the asynchronous chaotic attractor and the saddle limit cycle calculated at o~\ = 14.25 jig/ml, <72 = 0. The distance profile along the closed trajectory of saddle cycle is given in the insert.

To support this conclusion the Poincare section was calculated at the center of the coexistence area and at the homoclinic bifurcation curve H. The results are shown in Fig. 7.13. Inspection of the figure clearly shows how the asynchronous chaos touches the unstable cycle at the point of crisis 03 = 11.73 fig/ml.

Of course, for such a high-dimensional system like our coupled bacteria-virus model it is difficult to make precise statements about the mutual configuration of attractors based on Poincare sections alone. However, useful information can


be obtained by calculating the distance between the specified objects in phase space. In Fig. 7.14 the variation of the minimal distance between the asynchronous chaos and the saddle cycle is plotted as a function of the parameter 03, In the insert the distance profile along the saddle cycle at 0-3 = 11.73 fig/ml is shown. 5000 points were recorded along the saddle cycle. The minimal distance is shown for each point. The tangency of attractors is about to occur at the point number 4367.

7.6 One-Dimensional Array of Population Pools

Having observed how variations of the resource concentration in the overflow from one pool can entrain the population dynamics in the next pool and how, under different conditions, asynchronous responses of increasing complexity can arise, the natural questions to ask are: Can extended regions of synchronized behavior develop along the chain? What are the conditions, and how can they be characterized? In this section we shall attempt to give a preliminary answer to some of these questions by examining the dynamics of a chain of 20 population pools.

Equilibrium state , 1

0 5' 10 15 20 Chemostat number

Figure 7.15 provides an overview of the behavior of our one-dimensional array of microbiological reactors. Here, the pool (or chemostat) number i and the local nutrient supply a have been used as parameters. The dilution rate is p — 0.003/mm and the nutrient concentration is specified in fig/ml. Obviously, for a system as large as a 20-pool chain it is not realistic to consider all possible


parameter combinations. Hence, we have assumed that all habitats receive the same local resource supply cr, = a, i — 1,2...20.

For sufficiently low values of a, the system attains a stable equilibrium state that extends along the entire chain. As the nutrient supply is increased, starting in the downstream end of the chain, habitat after habitat begins to perform self-sustained oscillations. An interval then exists in which all habitats from a given number and up execute a synchronous periodic behavior. As a is further increased, the downstream habitats show quasiperiodic, chaotic and higher order chaotic behaviors of increasing complexity, and only the intermediate or first pools perform simple periodic oscillations. For a > 7 fig/ml, only the motion of the first habitat remains periodic.

Fig 7.16. Phase space projections for the cascaded population pools with a = 6.0 fig/ml. Each projection is plotted on the plane (-Bj_i,.Bj), i = 3 . . . 18. A region of synchronous chaotic behavior is seen to extend from pool number 10 to pool number 12.

Even in the region of chaotic dynamics one can observe domains of synchronized behavior. As an illustration of this phenomenon, Fig. 7.16 shows a sequence of phase space projections for 16 subsequent population pools starting with i = 3 in the top left corner and ending with i — 18 in the lower right corner. Here, a — 6.0 fig/ml. All the depicted subsystems exhibit chaotic dynamics. However, inspection of the figure clearly reveals the coherence in behavior between pools 10, 11 and 12. There are also certain similarities in the phase space projections for habitats 5 and 6.

To examine the formation of localized domains of chaotic synchronization from a somewhat different point of view we have calculated the inverse recurrence times (or basic frequencies) < F > for each of the population pools. Figure 7.17 shows the variation in < F > with the chemostat number for three


different values of the local resource supply: a = 7.785 fig/ml (a), 8.452 fig/ml (b), and 9.235 fig /ml (c). Synchronization to the periodic motion of the first population pool does not occur with these values of a. It is interesting to note, however, that synchronization to the more complicated dynamics of the second population pool is quite common. Figures 7.17(a) and (b), for instance, demonstrate how the third pool synchronizes with the second pool. In Fig. 7.17(a) we can also observe synchronization between pools 4, 5, and 6, between pools 7, 8, and 9, and between pools 10, 11 and 12.

0.40

0.38

A 0 3 6

LL V

0.34

0.32

(a)

10.0 15.0

Chemostat number 10.0 15.0

Chemostat number

Fig 7.17. Variation of the inverse recurrence time < F > with pool number for different values of the nutrient supply a. Note how synchronization with the second population pool can reappear at a considerable distance down the chain, a = 7.785 fig/ml (a), 8.452 ng/rrd (b), and 9.235 fig/ml (c).

5.0 10.0 15.0

Chemostat number

Intuitively, one would expect that once synchronization with a specific pool had failed, it could not be reestablished again further down the chain, i.e., information about the dynamics of a pool would be lost if the subsequent pool did not synchronize with it. By contrast, Fig. 7.17(b) shows how synchronization with the dynamics of pool number two can reappear far down the chain. In this figure the dynamics of pools 4-10 bear no obvious relation with the dynamics of the second pool. Nonetheless, pools 12-16 again synchronize with pool number 2. Figure 7.17(c) also shows locking of several chaotically oscillating pools, only


the synchronization domain has now moved all the way up along the chain.

Fig 7.18. Variation of the inverse recurrence time < F > with the nutrient supply for the different pools in the population dynamic chain. The figure illustrates the phenomenon of sliding synchronization regimes. Letters a, b, and c refer to the conditions in Fig. 7.17. The small square emphasizes a region with synchronization between pools 6 and 7.

This interesting phenomenon of sliding synchronization domains is illustrated in a more direct manner in Fig. 7.18. Here, we have plotted the inverse recurrence time < F > for the various population pools as a function of the nutrient supply a over the interval from 7.5 to 9.6 fig/ml. In this display, synchronization is witnessed by the fact that several habitats show the same value of < F >. As we follow the nearly straight line starting around a = 8.8 fig/ml and extending to the lower right corner of the figure we can see how pool after pool falls out of synchrony. In the beginning all pools between number 2 and number 13 are synchronized. As a is increased, however, starting with pool number 13, one pool after the other loses synchronization. In the interval between a = 8.3 and 8.6 fig /ml, increasing a causes pool number 20, pool number 19, etc., to lose synchronization. At the same time, however, pools with lower chain number gradually entrain with the behavior of the second and third population pools.


7.7 Conclusions

The aim of this chapter has been to examine synchronization phenomena in a chain of population dynamic systems, each capable of producing self-sustained oscillations in the presence of sufficient nutrients. As a specific example we considered a microbiological reaction system involving bacteria and phages. Coupling between the subsystems took place via the flow of primary resources from one population pool to the next. In this way, the bacterial growth rate of one habitat was modulated by variations of the nutrient concentration in the overflow from the previous habitat. Hence, in several cases we could observe how the complexity in behavior increased from pool to pool.

We also observed how localized domains of chaotic phase synchronization developed along the chain, and how the position of these domains shifted in dependence of the nutrient supply. Such domains of chaotic synchronization may play a role for many other types of biological and physiological systems. As discussed in Chapter 5, for instance, the insulin producing /?-cells in the pancreas are known to produce complex patterns of slow and fast oscillations in their membrane potential [53, 54]. The /3-cells interact with one another through gap junctions that allow ions and small molecules to pass from cell to cell. Most likely, however, they also interact via the flow of blood in the small vessels along which they are situated. Similar to the coupling discussed for the cascaded microbiological system, this would produce an interaction where a /3-cell reacts to the insulin release from the /3-cells that are upstream to it. It is clearly of interest to understand to what extent synchronization between the cells can arise, and whether such a synchronization has an effect on the overall pancreatic function.

It seems natural to assume that the influence of a modulation generated by a particular population pool would rapidly decrease along the chain to soon be "forgotten". Hence, once broken, synchronization with an upstream pool should not appear further down the chain. Contrary to this expectation, we have observed that synchronization (particularly to the dynamics of the second habitat) tends to reappear at a downstream position. If we think of a river passing through a system of lakes, the same type of phenomena might be expected to occur.

Even more interesting, however, is the observation that our model seems to exhibit a synchronization mechanism that differs in an essential manner from


the well established mechanism originally proposed by Arnol'd [33]. This new mechanism is characterized by the existence of global bifurcations right down to the smallest forcing amplitudes. Hence, it leads to the coexistence of a synchronous and an asynchronous response to a forcing signal. We have observed this phenomenon for the single-pool model with sinusoidal external forcing, for the model of two coupled population pools and for the case of quasiperiodic or chaotic forcing of the third pool. It is likely that this peculiar synchronization mechanism is associated with the specific form of the phase space structure for the individual pool as discussed in connection with Fig. 7.2(b). It is possible, however, that the form of the forcing term also plays a role. With a given resource supply to a pool there is a maximum to the nutrient concentration in the overflow to the next habitat. As the bacterial population in the first habitat grows larger, the nutrient concentration in the overflow will decrease. If the bacterial population in the first habitat becomes large enough to oscillate, not only will it produce oscillations in the nutrient concentration of the overflow, but the average concentration of nutrients in the overflow will also be reduced.

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Chapter 8

CLUSTERING OF GLOBALLY COUPLED MAPS

8.1 Ensembles of Coupled Chaotic Oscillators

In physics, biology, and other fields of science one often encounters systems in which a large assembly of oscillators through their mutual interaction produce different forms of collective behavior at the macroscopic level [1, 2, 3]. Examples of this type of system are most easily found in the living world, e.g., the phase-locking of chirps of neighboring males of certain species of grasshoppers [4], the widespread synchronous calcium oscillations associated with the bursting electrical activity of pancreatic /3-cells [5], or the recently reported sustained glycolytic oscillations in suspensions of yeast cells [6]. However, similar phenomena also arise in the study of Josephson junction arrays [7], in multimode laser systems [8], and in charge density waves [9].

In general, the individual oscillators of the ensemble will not be identical, but their parameters will be distributed over certain intervals. The interaction network may also have a so-called heterogeneous structure with combinations of local and global coupling mechanisms and with more or less random variations in the individual coupling strengths [10, 11]. In view of obtaining a better understanding of problems of this nature, a significant number of investigators have studied the properties of large populations of self-sustained oscillators [12, 13, 14, 15, 16, 17, 18]. In particular, it has been shown [19] that the onset

291


of mutual entrainment, which occurs when the interaction exceeds a certain threshold, bears certain analogies with a second-order phase transition.

However, under far-from-equilibrium conditions, besides self-sustained oscillations, each element in the ensemble may exhibit complicated bifurcation sequences leading to deterministic chaos and other forms of complex dynamics. An early study of coupled chaotic oscillators was performed by Bohr et al. [20] who simulated 256 interacting nephrons on a parallel computer. More recently, Pikovsky et al. [21] have performed a numerical study of synchronization phenomena in a population of 3000 — 5000 globally coupled Rossler oscillators. They demonstrated that the emergence of coherent behavior (and, hence, of a non-vanishing macroscopic mean field) is related to a synchronization of the phases of the individual oscillators whereas the amplitudes continue to behave quite differently and chaotic [22]. With further increase of the coupling parameter, amplitude synchronization of the interacting oscillators can occur, and a large number of coexisting cluster states may be observed. This was shown, for instance, by Zanette and Mikhailov [23] for a system of 1000 globally coupled Rossler oscillators. At a first sight, the assumed global (or all-to-all) coupling may appear a little unusual. However, it does represent a realistic coupling type in many biological systems where the cells (or functional units) are stimulated by signals that are controlled by the aggregate activity of the same units.

To fully account for the high-dimensional dynamics of a large population of interacting chaotic oscillators is outside the range of present understanding. Such systems will typically exhibit an extremely large number of coexisting limiting states each characterized by its own basin of attraction. Some of the states may be point cycles or quasiperiodic attractors on tori, and others may represent various forms of synchronous chaotic states. Among these there may be a dominance of Milnor attractors [24, 25] for which the basins of attraction are riddled with initial conditions leading to other limiting states [26].

As a simplified approach to the above problem, Kaneko [27, 28, 29] has proposed to study a system of A'' globally coupled, identical one-dimensional maps (oscillators)

N

Xi(n + l) = (l-e)f(xi(n)) + ^J£f(xi(n))> " = 0 , 1 , . . . , (8.1) j=i

where i = 1,..., JV is a space index for the ^-dimensional state vector x(n) —

Clustering of Globally Coupled Maps 293

{xi(n)}i=1. e £ R is the coupling parameter, and / : 1 -> 1 is a one-dimensional nonlinear map that can generate chaotic dynamics for the individual oscillator in the absence of coupling. The simplest form of asymptotic dynamics that can occur in system (8.1) is the fully synchronized (or coherent) behavior in which all elements display the same temporal variation. This type of the behavior occurs for a range of coupling parameters around the value e = 1. The motion is then restricted to a one-dimensional invariant manifold, the main diagonal in phase space.

For certain values of the coupling parameter, the state of full synchronization may attract all or almost all initial conditions. For other values of e, different types of clustering are observed [27, 28, 30, 31], i.e., the population of oscillators splits into groups with different dynamics, but such that all oscillators within a given group asymptotically move in synchrony. Two-cluster behavior, for instance, is characterized by the dominance of dynamics for which

•^i^+i — XiN1+2 — • • • — Xi*i —

def = X

def IN ~ V

(8.2)

with N\ and N% = N — N\ denoting the number of synchronized elements in each of the two clusters. This is usually the first type of clustering to occur as the coupling parameter e is reduced from values around e = 1, and the state of full synchronization breaks down. An interesting question therefore relates to the way in which this desynchronization occurs. On one hand, one could expect the coherent state to break up into a symmetric (or nearly symmetric) two-cluster state with an even (or nearly even) distribution of oscillators between the two clusters. On the other hand, it is also possible that the desynchronization proceeds through the splitting off of a single oscillator (or a few oscillators) while the majority of oscillators maintain synchrony. Other interesting questions relate to the types of dynamics that the two-cluster state can support and to the bifurcations through which the two-cluster state breaks up into multi-cluster states.

Under the conditions (8.2), the coupled map system (8.1) transforms precisely into a system of two coupled one-dimensional maps of the form

x(n+l) y(n + 1) = F

x(n) y{n)

f(x(n))+P£[f(y(n))-f(x(n))} f(y(n)) + (l-p)e[f(x(n))-f(y(n))}

(8.3)


with the parameter p describing the distribution of oscillators between the two clusters. (More precisely, p = N2/N denotes the fraction of the total population that synchronizes into state y). For N — 3, for example, with the clusters

def def

x\ — X2 = x and £3 = y, the dynamics of (8.1) is described by Eqs. (8.3) with p — 1/3. Clearly, for N = 3, two-cluster dynamics can be realized in 3!/(2!l!) = 3 different ways. Hence, we have three two-cluster states differing by the index of the subspaces. For larger values of N, the possible realizations of a given cluster distribution grows very rapidly.

The simplification of the problem that we have obtained through the reduction of the ./V-dimensional map (8.1) into the two-dimensional map (8.3) is enormous. The system (8.3) describes the dynamics of all the N ,,^iN y different two-cluster states defined by (8.2), and by studying the desynchronization of the coherent state x — y for (8.3) we can determine the precise bifurcation scenarios for all possible one- to two-cluster transitions in system (8.1). The map (8.3) says nothing about the stability of the two-cluster states in the full iV-dimensional phase space. However, by generalizing the approach to K-cluster dynamics with K = 3, 4 , . . . , we can examine both the stability of the low-dimensional states and the bifurcations through which they break up into states of higher dimension.

In the present chapter we shall consider the mechanisms involved in the transition from one- to two-cluster dynamics for the whole range p e [0,1] of possible distributions of oscillators between the two clusters. The case of linear and symmetric coupling [32, 33, 34]

x(n+l) = f(x(n))+e[y(n)-x(n)} . y(n + 1) = f(y(n)) + e[x(n) - y(n)} l ° ^ j

was considered in significant detail in Chapters 2 and 3. Here, we discussed the different processes by which chaotic synchronization is lost for a system of two coupled logistic maps f(x) = ax(l — x). We also presented different scenarios for the local and global bifurcations that take place after the initial transverse destabilization of a periodic orbit embedded in a synchronized chaotic state. In this connection we emphasized the role of the absorbing area [35, 36, 37] in restraining the dynamics of the coupled map system, once the synchronization breaks down [38].

The case of two symmetrically coupled logistic maps with nonlinear coupling (i.e.,Eq. (8.3) wi thp= 1) was considered by Astakhov et al. [39]. They followed


some of the bifurcations that take place after the first transverse destabilization of a low-periodic synchronous saddle cycle in the diagonal D = {(x, y) \ x = y}. A subsequent paper [40] examined the influence of a parameter mismatch on the desynchronization transitions in the same system.

The purpose of the present chapter is to study the transition from fully synchronized behavior to two-cluster dynamics in the system (8.1) for an arbitrary distribution of oscillators between the two clusters, i.e., for arbitrary values of the asymmetry parameter p £ [0; 1]. For symmetrically coupled, identical maps (p = 5), the first transverse destabilization of a saddle cycle embedded in the synchronized chaotic state (and, hence, the transition to riddling) occurs via a transverse period-doubling bifurcation or via a transverse pitchfork bifurcation. The presence of asymmetry in the coupling (p ^ 5) does not change the general form of the transverse period-doubling bifurcation. The transverse pitchfork bifurcation, on the other hand, is transformed into a saddle-node bifurcation leading to the formation of a couple of point cycles off the synchronization manifold. As shown in Sec. 8.2, the riddling bifurcation then becomes transcritical, i.e., it involves the exchange of stability between one of the off-diagonal point cycles and the saddle cycle on the diagonal [41, 42].

We continue our investigations by studying the sequence of bifurcations that the asynchronous point cycles undergo as the coupling strength e and the asymmetry parameter p are varied. We show how the unstable manifolds of one or both of these cycles control the global dynamics of the system after riddling has occurred and, hence, determine the character of the riddling bifurcation. The transcritical riddling bifurcation is found to always be hard.

In the case where there is a parameter mismatch so that two slightly different maps fai(x) or fa2{

x) a r e associated with the individual space points Xi, i = 1,...,N, the transcritical riddling scenario may be replaced by two saddle-node bifurcations involving a transversely unstable saddle and a repeller. Alternatively, if the sign of the parameter mismatch is different (in relation to the sizes of the two sub-populations), the saddle cycle involved in the transverse destabilization of the synchronized chaotic state smoothly shifts away from the synchronization manifold. In this way, the transcritical riddling bifurcation is replaced by a symmetry breaking bifurcation that destroys the thin invariant region existing around the nearly symmetric chaotic state. Problems of this type are considered in Sec. 8.5. Again we show that the stable and unstable manifolds of the asynchronous point cycles play an essential role for the


dynamics of the system. In Sec. 8.6 we establish general conditions of stability for K-cluster states,

i.e., for states of the globally coupled map system (8.1) in which the oscillators distribute themselves between K groups, such that the oscillators within each group operate in synchrony. The purpose of Sec. 8.7 is to amend the phase diagram originally proposed by Kaneko [27, 28, 29] for the distribution of different clustering states in parameter space. In this connection we also discuss different ways of desynchronization for the coherent state. Analysis of the two-dimensional map (8.3) allows us to add a significant amount of detail that one can hardly obtain through simulation of the full system (8.1) [43, 44]. Sections 8.8 and 8.10 are devoted to studying the formation of nearly symmetric (p ~ | ) and of strongly asymmetric (p « 0) two-cluster states. Our analyses show that the strongly asymmetric states usually are first to arise. Moreover, with increasing nonlinearity, new periodic states are found to emerge in the same order as the periodic windows arise in the logistic map. In Sec. 8.10 we also discuss an example of a cluster splitting bifurcation by which a period-3 two-cluster state breaks up into a period-6 three-cluster state. In Sec. 8.9 we examine the transverse stability of chaotic clusters. It is found that low-dimensional chaotic clusters in general are unstable in the full TV-dimensional phase space.

8.2 The Transcritical Riddling Bifurcation

Let us consider the two-dimensional coupled map system (8.3) with f(x) defined as the one-dimensional logistic map fa(x) = ax(l — x). Let the parameter a G (3; 4) be chosen such that the map / : x i—> fa(x) has a chaotic attractor / C [0,1]. Transverse to the diagonal D = {(x,y) | x = y} the eigenvalue of the map F (8.3) is found to be v± = f'a(x)(l — e). Hence, for e — 1 we have VS- = 0, and the diagonal D is superstable. Moreover, at e — 1 any initial point (x(0),y(0)) G R2 will be mapped into D in a single iteration under the action of F .

The superstability of the diagonal obviously ensures the existence of an interval for the coupling parameter e around s = 1 where the fully synchronous chaotic state A^ = {(x, y)\x = y G / } is asymptotically stable. When the coupling parameter e varies beyond this interval, the synchronous state A^ loses its asymptotic stability in a riddling bifurcation [45, 46, 47, 48]. As previously discussed this bifurcation takes place when some saddle cycle embedded in the


synchronous chaotic state loses its stability in the direction transverse to the diagonal. Often the saddle periodic cycle that first loses its transverse stability is of relatively low periodicity [49].

For values of a > ao — 3.678... where the map fa has a one-piece chaotic attractor, the riddling bifurcation takes place when the saddle fixed point PO(XO,XQ) embedded in A^ loses its stability in the direction transverse to D and becomes a repeller. As before XQ = 1 — 1/a denotes a fixed point of the logistic map fa. It is easy to show that the bifurcations occur at e = sfid = 1 ± l / ( a — 2) when the transverse eigenvalue u± = (2 — o)(l — e) of the fixed point PQ becomes greater than 1 in absolute value.

0

Brid p £rid

Fig 8.1. Transverse bifurcations of the fixed point P0 lying in the synchronous state A^ for the map (8.3) in the case of symmetrical coupling (p = | ) . To the left (at e = e~id), a supercritical period doubling bifurcation takes place giving rise to a saddle period-2 cycle 72 = {Si, S2}. To the right (at e = £*id), a supercritical pitchfork bifurcation occurs producing two saddle fixed points P and Q. The variable r\ is defined as r\ = (y — x)/2. Dashed lines at 77 = 0 denote a repelling fixed point P0.

Let us start by considering the system (8.3) with symmetrical coupling, i.e., for v = \- A schematic diagram of the transverse bifurcations of the fixed point Po is presented in Fig. 8.1. At e = e~id the riddling bifurcation involves a transverse period-doubling bifurcation (the transverse eigenvalue of Po leaves the unit circle through —1). This bifurcation gives rise to the birth of an antisymmetric period-2 saddle cycle P2 = {Si, S2} whose points gradually move away from the diagonal when e continues to decrease. At e = e^id the riddling bifurcation has the form of a transverse pitchfork bifurcation (the transverse

Po


eigenvalue of PQ becomes greater then +1). After the bifurcation, two saddle fixed points P and Q close to the diagonal appear. They again move away from the diagonal when e increases.

The transverse period-doubling and pitchfork bifurcations at e = eHd are both supercritical. It follows that the corresponding riddling bifurcations are soft [50, 51], i.e., immediately after the bifurcation there exists an invariant region of infinitesimal transverse size, the absorbing area [33, 34, 38], that envelops all trajectories starting close to the chaotic attractor A^ in the diagonal.

Fig 8.2. Absorbing area (hatched) for the map F in the symmetric case (p = |) after a supercritical pitchfork bifurcation of the fixed point P0. The absorbing area is bounded partly by arcs of the critical curves L\ and i 2 and partly by unstable manifolds of the asynchronous point cycles P and Q born in the pitchfork bifurcation. The parameters are a = 3.8, p = 0.5, and e = 1.57.

An example of the absorbing area that appears after the supercritical pitchfork riddling bifurcation is shown hatched in Fig. 8.2. As discussed in Chapters 2 and 3, the boundary of the absorbing area is partially composed by unstable manifolds of the antisymmetric saddle fixed points P and Q, and partially (near the corner points) by the critical curves L\ and Li which are the first and the second iterations by F of the locus of points where the Jacobian of the map F vanishes:

L0 = {{x, y)\x = l/2ory = 1/2}. (8.5)


The absorbing area that appears after the period-doubling bifurcation has a similar shape. The only difference is that the unstable manifolds of the antisymmetric period-2 saddle cycle P<i = {Si, £2} n o w take part in creating the boundary of the absorbing area. Immediately after the supercritical riddling bifurcation (whether this is caused by a transverse period-doubling or pitchfork bifurcation), the width of the absorbing area is infinitesimal. When moving away from the bifurcation point the width grows proport ional to y/\& — ^H</I'

and this width determines the amplitude of the maximal possible bursts away from the synchronized state.

As noted by Maistrenko et al. [52, 53], the transverse eigenvalue of any symmetrical point cycle depends only on the sum of coupling parameters, which for the system (8.3) is given by ep + e(l — p) = e. Since this sum is independent of p, the parameter points e = efid of the riddling bifurcations are the same for a l l p e [0,1].

0.15

0-

-0.15 0.35 o Srid 0.47

Fig 8.3. Diagram for the riddling bifurcation when the fixed point P0 € A^ loses its transverse stability through a supercritical period-doubling bifurcation at e = e~id giving rise to a saddle period-2 cycle P2 = {Si, S2}. The dashed line at 77 = 0 corresponds to a repelling fixed point Po- The parameter p = 0.33.

It is well-known [54, 55] that period-doubling is a generic, i.e. codimension-1, bifurcation. Hence, the riddling bifurcation at e — e~id persists under perturbations of the system, and it maintains the form of a period-doubling even if the symmetry is broken (p =£ | ) . As illustrated in Fig. 8.3, an asymmetric coupling only causes a small asymmetry in the position of the points 5i and 52 for the period-2 saddle cycle Pi = {S\, S2}. Let us therefore focus on how the


supercritical pitchfork bifurcation at e = e*id is affected by the introduction of an asymmetry in the coupling. With this purpose we use a method proposed by Maistrenko et al. [33, 34] and described in detail in Chapter 3 in connection with the unfolding of the riddling bifurcation for two linearly coupled maps.

Applying the variable transformation

Z=(y + x)/2, ri={y-x)/2 (8.6)

to (8.3), we can rewrite the map F in the new variables £, r\:

F H-> f(0 + f(0(2p~l)ev-a^-

where / (£) = a£(l — £) is the logistic map and / '(£) = a( l — 2£) its derivative. The variable change (8.6) represents a simple 7r/4 rotation of phase space such that the diagonal x = y for the original map F corresponds to the axis rj — 0 in the map F. Equation (8.6) also involves a scaling with the factor y/2.

For the map F the riddling bifurcation at s = e*id takes place when the transverse eigenvalue vL = /'(£o)(l — e) of the fixed point Po(£o)0) passes through + 1 . Here, £o = ^o = 1 — l / a - For the considered values of a the fixed point £o is unstable, and the absolute value of the longitudinal eigenvalues i/|l = f'(xo) is always greater than 1.

Let the coupling parameter be such that the fixed point PQ is a transversely attracting saddle, i.e., \v±\ < 1 and the synchronous chaotic state is still before the riddling bifurcation. Then, there exists an invariant one-dimensional transverse manifold Wj_ = {(£, rj) \ £ = <p(r))} passing through PQ. The Taylor series expansion of </?(•) is

<p{n) = 6 + (1 - 2p)rj + ^ " / V + ... (8.8) J/|| - V±

where dots denote terms of higher order. From this expansion it follows that the manifold Wj_ is smooth up to C2 in the vicinity of the fixed point Po, provided that the non-resonant conditions i/|| ^ v±_ and i/y ^ v\ are fulfilled.

Having calculated asymptotically the invariant transverse manifold (8.8) we can study the action of the two-dimensional map (8.7) along it. In this way we obtain a one-dimensional map ft : 77 (->• h(rj) which is a restriction of F onto W± in the vicinity of zero. To third order in 77, the expansion of the map ft takes


the form

h(v) = viT) + 2a(2p - 1)(1 - e)r]2 +

8qMl 2-p)( l - £V + ^ (8-9)

The non-resonant conditions v\\ v±_ and v\\ v\ guarantee C3-smoothness of the map h(-) near zero.

Bifurcation of the fixed point 770 = 0 for the one-dimensional map h now corresponds to the transverse destabilization of the fixed point Po for the original two-dimensional map F given by (8.3). Assume first that p = \ so that the coupling is symmetrical. Then the quadratic term in (8.9) vanishes. It immediately follows that before the bifurcation, at e < e*id, the map h has a single stable fixed point 770 = 0- After the bifurcation, at e > e*id, the fixed point 770 is unstable and two stable fixed points appear in its neighborhood. Therefore, with symmetrical coupling, the riddling bifurcation has the form of a supercritical pitchfork bifurcation as shown in Fig. 8.1.

When the symmetry of the map (8.3) is broken, the bifurcation diagram undergoes an essential change. Indeed, it can be shown from (8.9) that for p ^ \, and for values of the coupling parameter below the riddling bifurcation (esn < e < e*id), the one-dimensional map h now acquires two additional non-trivial fixed points. Both of these are positive for p < ^ and negative for p > | . One of the non-trivial fixed points, being an unstable node, approaches 770 = 0 as e —> e*id and passes through the fixed point 770 at e = e^id in a transcritical bifurcation, in which the crossing fixed points exchange their stability.

For the original two-dimensional map (8.3) the riddling bifurcation proceeds according to the following scenario, see Fig. 8.4. When increasing the coupling parameter e beyond 1, a saddle-node bifurcation occurs at some value of the parameter e = esn, producing two fixed points P and Q. Both of these fixed points are situated above the diagonal D if p < | , and below the diagonal if p> \. With further increase of e, the repelling fixed point Q approaches the diagonal D. At e — e*id it passes through the saddle fixed point Po £ D. The fixed points Po a n d Q exchange their stability. After the transcritical riddling bifurcation, the symmetrical fixed point Po on the diagonal becomes a repeller, and the fixed point Q becomes a saddle lying below the diagonal for p <\ and above D for p > \. The bifurcation diagram for the transcritical riddling is shown in Fig. 8.4.


Ssn Srid

Fig 8.4. Diagram of the transcritical riddling bifurcation when the fixed point PQ(XO, XQ) loses its transverse stability through the exchange of stability with another solution. The parameter p < | is fixed. For p > 5 the bifurcation diagram changes symmetrically with respect to the line 77 = 0. Dashed curves correspond to repelling fixed points, and solid curves to saddle points.

In terms of the variables (£,rj) (defined in Eq. (8.6)), the coordinates of the fixed points P(£+,r)+) and Q{£~,r}~) can be expressed as:

1

r = H 2 2 o ( e - l ) '

and ± _ (1 - 2p)e ± y/(a - l)2(e - l ) 2 - 4e2p(l - p)

2a(e - 1)

The fixed points P and Q exist for e > esn, where

(8.10)

(8.11)

(a - l ) 2 + 2(o - l ) V p ( l - p )

( a - l ) 2 - 4 p ( l - p )

is the value of the coupling parameter at the saddle-node bifurcation of F

(8.12)

8.3 Global Dynamics after a Transcritical Riddling

In this section we shall show that the riddling bifurcation in the system (8.3), if it is caused by a transverse transcritical bifurcation of Po, is always hard. This means that immediately after the bifurcation there will be a path for the


trajectories to go far away from the synchronized attractor A^ C D, even if they start in a very thin neighborhood Us of A^s\ In other words, any slight coupling asymmetry transforms the soft riddling bifurcation caused by the supercritical pitchfork bifurcation (for p=\) into a hard riddling transition associated with the transcritical bifurcation (for p ^ i ) . We presume that this type of hard bifurcation will be typical for coupled identical oscillators as soon as the coupling is non-symmetrical.

To throw light on the properties of the transcritical riddling bifurcation, the global dynamics of the non-invertible map F given by Eq. (8.3) has to be examined. As discussed in Chapters 2 and 3, absorbing areas play a fundamental role in the theory of two-dimensional non-invertible maps [35, 36, 37, 38]: They absorb all or almost all trajectories from their neighborhoods and retain these trajectories forever.

When N logistic (or, more general, non-invertible) maps are coupled it is easy to see that the resulting ./V-dimensional map Fff will also be non-invertible, i.e., there will exist critical hyper-surfaces such that \DFN\ = 0. Hence, when analyzing the global dynamics of Fff one may expect the existence of " absorbing volumes", i.e., of ./V-dimensional invariant regions in phase space whose boundaries are formed (completely or partially) by images of the critical hyper-surfaces. To the best of our knowledge, however, only two-dimensional non-invertible maps have been analysed so far with respect to this property. Generalization of the concept of an absorbing area to the case of ./V-dimensional non-invertible maps remains an unsolved and challenging problem.

Consider the synchronized chaotic state A^ after the transcritical riddling but before the blowout bifurcation. The chaotic attractor A^ is still attracting on average, i.e., the Lyapunov exponent Aj_ responsible for the growth of transverse perturbations is negative. For example, if a = 3.8 the riddling bifurcation takes place at e = 1.55, and the synchronous state A^ is attracting on average up to e = 1.65, where Aj_ changes sign and the blowout bifurcation takes place.

In the parameter interval 1.55 < e < 1.65 (a = 3.8), the state A^ is a measure-theoretic Milnor attractor: It attracts a positive Lebesgue measure set of points from its neighborhood [26, 46, 47, 48]. The topological properties of the basin of attraction are still far from being understood in detail. Certain progress has been made in the case of piecewise-linear maps. In particular, Pikovsky and Grassberger [56] have conjectured that for two coupled tent maps, in the regime of weak synchronization, periodic points are dense in the vicinity


of the synchronous chaotic state. A proof of this conjecture was recently given by Glendinning [57, 58].

1.0

y

0.1 0.1 X 1.0

Fig 8.5. Absorbing areas A (hatched) and A\ after a transcritical riddling bifurcation. The smaller area A is bounded by segments of unstable manifolds of saddle fixed points P and Q, and by the segments L\ and L2 of critical curves. The larger absorbing area A\ is bounded only by the critical curves L\ and L2. The parameters are a = 3.8, e = 1.57, and p = 0.45.

A state portrait of system (8.3) after the transcritical riddling bifurcation is presented in Fig. 8.5. There exists an invariant region A (hatched) around the chaotic attractor A^ C D. As before, its boundary is composed partially by segments (arcs) of unstable manifolds of the saddle fixed points P and Q and partially, by segments of two consecutive F-images of the critical curves LQ denoted by L\ and L2 respectively. Inspection of Fig. 8.5 also shows how the absorbing area A is embedded in a larger absorbing area Ai whose boundary is delineated entirely by the critical curves L\ and Li- In certain parts of the phase space, the boundaries of the two absorbing areas coincide.

Our calculations provide evidence that this type of phase portrait containing two nested absorbing areas is typical for system (8.3) in the parameter region after the transcritical riddling bifurcation and before stabilization of the fixed point P (see Fig. 8.7). When a trajectory starts near the chaotic set A^s\ after a number of iterations it will typically fall in a small neighborhood of


the fixed point PQ € A^ and here it attains a finite probability of going away from the diagonal along the separatrix connecting PQ with P (or, alternatively, connecting PQ with Q). The deviation of P from the diagonal is larger than that of Q. Hence, the distance to P provides an upper bound to the maximum amplitude in the burst of the trajectories away from the synchronous state. Moreover, at the point of transition the distance PPo is finite, and right after the transition, bursts of finite amplitude can therefore be observed. This is in contrast to the symmetrical case, where the amplitude of the bursts grows gradually with the parameter.

Let us follow the evolution of the global dynamics of the map F, when the control parameter e passes through the bifurcation value e*id of the transcritical riddling (Fig. 8.4). As we shall see the sequence of bifurcations observed differs from that of the symmetrical case p — \, as outlined in Sec. 8.2 above.

1.0

y

0.1 0.1 x 1.0

Fig 8.6. Absorbing area A (hatched) at the moment of the transcritical riddling bifurcation. Parameters a = 3.8, e = 1.555, and p = 0.4.

Before the riddling bifurcation, the attractor A^ in the diagonal is asymptotically stable for system (8.3); it attracts all trajectories from a sufficiently small neighborhood [46, 47, 48]. At the bifurcation moment e — e*id, the attractor A^ loses its transverse stability (Fig. 8.6). The smallest invariant region enveloping A^ is now an absorbing area A created by the unstable manifolds


of the saddle fixed point P and, partially, by the critical curves L\ and Li. The maximal distance of its boundary from the diagonal is approximately given by the coordinates of the saddle fixed point P. Its deviation from the diagonal is

• p p . A , (1 ~ 2p)e + V(a ~ l)2(e - l ) 2 - ieM^P)

The distance from PQ to the lower boundary of A is clearly smaller. Nevertheless, as it can be seen in Fig. 8.6, it is also finite beginning from the bifurcation moment e = e^id of the transcritical riddling bifurcation considered. This result is associated with the fact that the lower boundary of A is created by the unstable manifolds of the saddle P .

Based on the phase portrait in Fig. 8.6 at the moment of riddling bifurcation, we claim that under the action of F any small neighborhood of the point PQ = Q eventually will spread throughout the whole region A. Indeed, there is a separatrix (Po, P) connecting Po with P. So, a positive measure set of points from the neighborhood will move towards P along (Po, P)- Having approached P at some distance, these trajectories will follow the unstable manifolds of P and then fold on the critical curve L\ (those moving to the right) or on Li (those moving to the left). Then they cross the diagonal and pass near the right boundary of A (which is created by the continuation of the unstable manifolds of P). After this, as computer simulations show, the trajectories begin to move between the upper-right and the lower-left cones of A.

While restrained to the absorbing area A, the trajectories (more precisely, the invariant curves created by the trajectories) cross the separatrix (Po, P)-This implies that some points of the trajectories (those near the intersections) can again be involved in the motion along (Po, P) in the direction towards P , and the situation described above will be repeated, giving rise to new intersections of stable and unstable manifolds of P . Hence, intersections of stable and unstable manifolds of P are evident from our numerical simulations and lead to a complicated dynamics of the map F inside A. We have not looked in detail for possible homoclinic tangencies or considered their persistence. It would be interesting, however, to examine this question to see whether Newhouse regions [59, 60] exist for systems of two coupled logistic maps.

From the above discussion we conclude that the boundary of the absorbing area A can be approached, with any given precision, by trajectories originating in any small neighborhood of the fixed point Po belonging to the attractor A^ in the diagonal. The location of the boundary gives exact limits for the deviations


of the trajectories from the diagonal when they enter into asynchronous bursts. Hence, beginning right from the moment of the transcritical riddling bifurcation at e = e*id, the amplitude of the asynchronous bursts is of order 0(1). We conclude that the transcritical riddling bifurcation is always hard [41, 42].

This property differs from the analogous riddling bifurcation in the symmetrical case p — \ as well as from the riddling bifurcation caused by the transverse period-doubling bifurcation, which is soft or hard depending on the super- or sub-critical nature of the bifurcation causing it [33, 34].

Let us continue to vary the coupling parameter e beyond the bifurcation value e+id to observe further changes in the structure of the absorbing area A. After the bifurcation, the fixed point Q becomes a saddle and moves down away from the diagonal. First, it lies inside A so that its unstable manifolds do not participate in the boundary of A. Then, Q emerges from the interior of A, and its unstable manifolds begin to contribute to the boundary (Fig. 8.5). The parts of the boundary of A that are created by the manifolds of Q move down and away. However, due to asymmetry (p < | ) , the distance to the lower boundary of A from the diagonal remains smaller than the distance to the upper boundary given by the manifolds of P.

8.4 Riddling and Blowout Scenarios

In this section we examine the riddling and blowout transitions for the chaotic synchronous state A^ in more detail, emphasizing particularly the structure of its basin of attraction. Due to the obvious symmetry of the map (8.3) with respect to p = | , we only consider the interval p £ [0, ^]. To be more concrete, let us fix the nonlinearity parameter to a = 3.8 at which the logistic map / is considered to possess a one-piece chaotic attractor I.

Both the riddling and the blowout transition may be significantly modified by the presence of another attractor whose basin can reach close to the diagonal. In particular, this concerns an attractor Aup (above the diagonal) which may coexist with the stable synchronous state A^ and thus affect the riddling and blowout transitions.

The two-parameter (p, e) bifurcation diagram for the attractor Aup is presented in Fig. 8.7. In order to understand the main aspects of this diagram let p be fixed and consider what happens as e increases starting from 1. The lowest curve in Fig. 8.7 (denoted "saddle-node") represents the saddle-node bifurca-


S4 , S3, S2 ,Si

2.0

1.8

1.6 8

1.4

1.2

""0 0.1 0.2 0.3 0.4 0.5 P

Fig 8.7. Bifurcation diagram for the attractor Aup situated above the diagonal. The hatched region corresponds to the parameter values where the attractor Aup exists. Riddling (e^id « 1.555 ...) and blowout (ew ss 1.65 ...) bifurcation lines of the one-piece chaotic attractor A<s> on the diagonal as well as the curve of saddle-node bifurcation of P and Q (Fig. 8.4) are also shown. Letters Hi, H2, H3, IZ4 (Si, S2, S3, <S,j) denote p-intervals (separated by dashed lines) with different types of riddling (blowout) transitions (see text for details). Parameter a = 3.8.

tion in which the asymmetric fixed points P and Q are born, P as a saddle and Q as a repelling node. For p=\, the saddle-node bifurcation takes the form of a pitchfork bifurcation and coincides with the riddling bifurcation. Otherwise, the saddle-node bifurcation occurs before (i.e., at lower values of e than) the riddling bifurcation. As the parameter point (p, e) enters the hatched region, at e = sst, P becomes a stable node, and with further increase of e, the asymmetric fixed point transforms into a stable focus. Hereafter (not shown), the fixed point undergoes a supercritical Hopf bifurcation, and a stable invariant closed curve appears with quasiperiodic or periodic dynamics. This curve is destroyed with further increase of e, and an attracting chaotic set is born. Finally, the upper boundary of the hatched region corresponds to the disappearance of the chaotic attractor Aup in a boundary crisis at e = £CT .

The upmost bifurcation curve in Fig. 8.7 denoted as "absorbing area crisis" represents the contact bifurcation, at e — econ, of the absorbing area A with the basin of infinity (see [35, 36, 37, 38]). By destroying the invariant region around the synchronous chaotic state A^, this absorbing area crisis results in the appearance of holes belonging to the basin of infinity within the former basin of A^s\ Note that econ > £„. Hence, as e is increased, the boundary crisis

absorbing area crisis

blowout \

V^ffl riddling /is'fQ^-^^1'

7 #4 |/?3| A?2

!'

•

saddle-node


(at which Aup disappears) takes place before the contact bifurcation (at which global riddling with the basin of infinity arises).

As discussed in Sec. 8.2, the riddling bifurcation of the chaotic synchronous state A^ is independent of p and takes place at e = e+id = 1 + l / (a - 2). It is caused by the transverse destabilization of the symmetric fixed point PQ(XO,XQ) G A^s\ xo = 1 — 1/a. For p — \ this occurs via a supercritical pitchfork bifurcation (Fig. 8.1), whereas for p G [0, | ) the transition is trans-critical (Fig. 8.4). The riddling bifurcation at e = e*id may result in a locally or globally riddled basin of A^. This depends on the existence of an absorbing area A around A^s\ and on the presence of another attractor within A [38].

Let us first consider the case when the transcritical riddling bifurcation leads to a locally riddled basin for the chaotic synchronous state A^s'. Such a transition takes place when the absorbing area A still exists (that is, before the contact bifurcation) and there are no other attractors within A. As it can be seen in Fig. 8.7, for the considered value a = 3.8, these conditions are satisfied if p 6 Tli U Tl3 = (0.377...; 0.5] U (0.245...; 0.273...).

The difference between the first and the second p-interval of local riddling is that for p G (0.377...; 0.5] the fixed point P has not yet stabilized at the moment of the riddling bifurcation, i.e., e^id < est (Fig. 8.5). In the second case p G (0.245...; 0.273...), the attractor Aup has already been destroyed through a boundary crisis bifurcation, but the absorbing area A still exists, i.e., £„. < 6 rid < £c<™- I n p l a c e °f Aup a chaotic saddle Aup embedded in the basin of the synchronous chaotic state exists. Figure 8.8 shows some of the repelling periodic orbits associated with the chaotic saddle Aup.

According to the definitions proposed by Ashwin et al. [46, 47, 48], the basin of the synchronous state A^ is locally riddled when arbitrarily close to any point in A^ there exists a positive measure set of points which move away from the diagonal for a finite distance. If there is no other attractor within the invariant absorbing area A, almost all of the trajectories will return to the synchronous chaotic state. Some of them will be attracted by A^s\ whereas others will repeat the asynchronous bursts. Such a behavior is produced by the transversely repelling fixed point PQ G A^ which has an everywhere dense set of preimages in A^s>. Trajectories starting close to such a set will be mapped, first, into a sufficiently small neighborhood of Po. Hereafter, they can move away from the diagonal following the separatrix which connects the repelling fixed point P0 with the saddle fixed point P (for p G (0.377...; 0.5], see Fig. 8.5)


1.25

-0.23 -0.1

Fig 8.8. Synchronous attractor A^ and its basin boundary (outer closed curve) after the transcritical riddling bifurcation. Black crosses indicate some of the periodic orbits belonging to the chaotic saddle Aup. Also shown is the boundary of the semi-invariant absorbing area A that is created by arcs Lk, k = 1 , . . . , 5 of critical lines. Parameters a = 3.8, p — 0.27, and e = 1.58.

or with the chaotic saddle Aup (for p G (0.245...; 0.273...), see Fig. 8.8). If p G K2 U TZ4 = (0.273...; 0.377) U [0; 0.245...) the riddling bifurcation leads

to a globally riddled b&sin of the synchronous state A^s\ For p G (0.273...; 0.377) one has est < e^id < £CT, and there exists an attractor Aup above the diagonal in the moment of the bifurcation. The repelling fixed point Q, causing the riddling in the transcritical bifurcation with P0 G A^s\ brings with it the basin boundary of the attractor Aup to the diagonal. Therefore, there exists a tongue of points, with vortex in PQ, which belong to the basin of attraction of Aup. This is illustrated in Fig. 8.9 where Aup is an invariant closed curve. The fixed point PQ has an everywhere dense set of preimages in A^ (the map F in is non-invertible and mixing). The tongue with vortex in PQ has preimages in a neighborhood of any point of A^8\ It follows that arbitrary close to any point of A^ there is a positive measure set of points attracted by Aup. This is the case when the basin of the synchronous chaotic state A^ is globally riddled with the basin of Aup.

For p G [0; 0.245...) an absorbing area contact bifurcation with the basin of infinity takes place before the riddling bifurcation, i.e., econ < e*id. After


Fig 8.9. The transciitical riddling bifurcation may lead to a globally riddled basin of the synchronous state A^ with the basin of the attractor Aup above the diagonal. For the chosen parameters, Aup is a closed invariant attracting curve born in a supercritical Hopf bifurcation of the asynchronous Ixed point P. The basin of Aup is shown in gray. The basin of A*-*'1 is densely llled by "dots" which actually are small diameter regions belonging to the basin of Aup. Parameters a — 3,8, p = 0.33, and e = 1.6.

the crisis one can observe "holes to infinity" of a small diameter within the former basin of the synchronous chaotic state A^. These holes accumulate in the neighborhood of the fixed point Q. Hence, in the moment of the riddling bifurcation, Q brings with it part of the basin of infinity to the fixed point Pa € AW. This riddling scenario is similar to what we have discussed for the case when the basin of A^ is globally riddled with the basin of Aup (Fig. 8.9). Therefore, immediately after the riddling bifurcation, the basin of attraction of the synchronous state A^ becomes globally riddled with the basin of infinity (Fig. 8.10).

The blowout bifurcation of the attracting chaotic state A^ corresponds to its transformation into a chaotic saddle [46, 47, 48]. This bifurcation takes place when the transverse Lyapunov exponent

1 j r - i

Ax = Hm — Y In |/'(at(n))(l - e)|, (8.13) n=0


1.25

y

- 0 . 2 5 - 0 . 1 X 1.1

Fig 8.10. Globally riddled basin of the synchronous state A^ with the basin of ininity after a transcritical riddling bifurcation. Basin of infinity is shown in gray. Parameters o = 3.8, p = 0.245, and e = 1.6.

that is responsible for the average growth of transverse perturbations, changes sign from negative to positive. The calculation is performed on a typical trajectory {(x(n),x(n))}™=0 C A(*\ x(n + 1) = fa(x(n)). For the considered parameter value o = 3.8, the blowout bifurcation of A^ is found to occur at e = £« ~ 1-65.

The blowout bifurcation for the map (8.3) may follow different scenarios depending on whether it takes place from a locally or from a globally riddled basin of A®. In Fig. 8.7, for p e Si U <S3 = (0.488. . . ; 0.5] U ( 0 . 3 . . . ; 0.336...), the basin of A^ is locally riddled in the bifurcation moment. In this case, the blowout bifurcation is supercritical [47], which means that the chaotic attractor A^ gradually spreads into the two-dimensional phase space. Indeed, as illustrated in Fig. 8.11, trajectories of the new "swelled" attractor A spend most of the time close to the former attracting synchronous state A^s\ At the same time we can see how the attractor is bounded by the critical curves L*.

If p € (0.488. . . ; 0.5], e« < est and the blowout bifurcation takes place before the stabilization of the fixed point P above the diagonal (Fig. 8.11(a)). In the second interval p G (0 .3 . . . ;0 .336. . . ) , e« > So- and the blowout bifurcation takes place when the chaotic attractor Aup developed from P has already been


Fig 8.11. Attractors for the map (8.3) and their basin boundaries after a supercritical blowout bifurcations in the presence of a locally riddled basin. Parameters a = 3.8, (a) p = 0.498, E = 1.658, and (b) p = 0.33, e = 1.67.

destroyed (Fig. 8.11(b)). Hence, there is a chaotic saddle Aup in place of it (see also Fig. 8.8). In this case, the chaotic saddle AUp may be included in A.

The blowout bifurcation is subcritical [46, 47, 48] when it occurs from a globally riddled basin of the synchronous chaotic state A^s\ This is the case for p £ S2 U <S4 = (0.336.. . ; 0.488 .. .) U [0; 0.3.. .) . For the first p-interval, in the moment of blowout bifurcation, there exists an attractor Aup over the diagonal and the basin of A^ is globally riddled with the basin of Aup (Fig. 8.9). Hence, above the bifurcation, trajectories are typically attracted by Aup. In the second case, p 6 [0; 0.3 . . . ) , the blowout bifurcation takes place when the basin of A^s' is globally riddled with the basin of infinity (Fig. 8.10). Above the bifurcation, typical trajectories escape to infinity.

8.5 Influence of a Parameter Mismatch

In the previous section we described the riddling and blowout transitions when the coupling is asymmetrical (p ^ | ) , but the coupled one-dimensional maps are the same. It follows that for any p 6 [0; 1] the diagonal D remains invariant under the action of the two-dimensional map (8.3).

To remove the remaining symmetry we now add a small parameter mismatch


between the coupled maps. For this purpose, we consider the system

x(n + 1) = fai(x(n)) + pe[fa2(y(n)) ~ fai(x(n))] (R 1 4 \

y(n + 1) = fa2(y(n)) + (1 - pM/aM")) ~ f«Mn))} [ '

where, as before, fa denotes the logistic map with parameter a: fa{x) — ax(\ — x). The cluster distribution and coupling parameters p and e are similar to those for Eq. (8.3), but maps that synchronize in one cluster now have slightly different parameters from maps that synchronize in the other cluster. The mismatch between the one-dimensional maps fai and fa2 can be introduced as follows: ai = a and a^ — a8. For small values of the mismatch, the factor 8 will be close to 1.

It is evident that for the two-dimensional map F given by Eqs. (8.14) the diagonal D is no longer invariant with 8 ^ 1 . In place of A^ C D a two-dimensional invariant absorbing area arises for some range of the coupling parameter e. Its transverse diameter is small as long as 8 is close to 1, and vanishes with 8 —> 1. We claim that the transcritical riddling bifurcation in this case is replaced by an interior or an exterior crisis of the absorbing area A^. The moment when this happens is shifted with respect to the case without mismatch. Moreover, this crisis bifurcation is always hard.

The above phenomena can be explained by the following changes in the global dynamics of the system (8.14). There are two distinctive cases depending on the sign of the mismatch, i.e., if 8 is smaller or greater than 1. For p belonging to the interval [0; \), the majority of maps are of type /Ql, and 8 < 1 implies that most of the maps have the larger nonlinearity parameter a\. Conversely, for 6 > 1, the larger sub-population of maps have the smaller nonlinearity parameter.

First we fix 8 = 0.998. When the original system (8.3) is far enough below the riddling bifurcation, the mismatch produces a shift of the fixed point PQ by a small distance above the diagonal. Po is still a saddle, and its unstable manifolds define an invariant absorbing area A^ of a fairly small transverse diameter (hatched region in Fig. 8.12). The boundary of A^ is of the same type as that of A described in Sec. 8.3, i.e., it consists partially of the unstable manifold of Po and partially of arcs of critical curves L\ and Li (see the zoom in Fig. 8.12).

With increasing e, a saddle-node bifurcation occurs which gives birth to two fixed points P and Q over the region A^ in a similar way as described in


1.0

y

0.1 0.1 x 1-0

Fig 8.12. Absorbing area A^ (hatched) that replaces the chaotic synchronous state A^ (dashed) when the parameter mismatch <5 = 0.998 is introduced. Parameters o = 3.8, e = 1.538, and p = 0.4.

Sec. 8.2 above. The repelling node Q moves towards A^ and, at some moment e = e'sn, two fixed points Q and PQ meet and are annihilated with one another in an inverse saddle-node bifurcation. At some larger value e = e"sn > e'sn a new saddle-node bifurcation takes place giving rise to the birth of two new fixed points (a repelling node and a saddle) below the diagonal. Subsequently, one of them (the saddle) moves away from the diagonal and the other (the repeller) approaches it. An example of the bifurcation diagram is shown in Fig. 8.13.

The crisis bifurcation of the absorbing area A^s' at e = e'sn can be considered the analogue of the hard riddling bifurcation in the presence of a parameter mismatch. Indeed, before the bifurcation, the trajectories cannot escape from the thin absorbing area A^s\ Asynchronous bursts remain small and grow smoothly with the magnitude of the mismatch and coupling: the maximal amplitude of the bursts is determined (approximately) by the distance of the point PQ from the diagonal. After the bifurcation the invariance of A^ is destroyed. Trajectories starting from A^"' get access to a neighborhood of the saddle point P located far away from the diagonal. The subsequent behavior of the trajectories is similar to that described in Sec. 8.3: Following the unstable


Fig 8.13. Diagram for the bifurcations that replace the transcritical riddling bifurcation shown in the Fig. 8.4 in the case of a small parameter mismatch S = 0.998.

manifolds of P they fold at L\ or L<i and come close to the diagonal. Then trajectories spend some (usually long) time within the former region A^s' before entering a new burst to P; and this type of behavior will be repeated. This crisis of A^s' can be considered as interior. It replaces the transcritical riddling bifurcation of the symmetrical system in the case when riddling results in a locally riddled basin.

If, at e = £'sn) the fixed point Q brings with it part of the basin of an attractor Aup developed from P, or part of the basin of infinity, the crisis of A^s> is exterior. In this case, after the crisis, most of the trajectories from A^ will move to the attractor Aup over the diagonal, or escape to infinity, so that they never come back to A^s>. The exterior crisis replaces the transcritical riddling bifurcation of the symmetrical system in the case when riddling results in a globally riddled basin.

In this way, we have exposed the moment when the smoothly growing small desynchronous bursts in the system with mismatch (8.14) abruptly change into excursions that move far away from the diagonal. This occurs when an inverse saddle-node bifurcation of the repeller Q and the saddle fixed point PQ destroys the thin invariant absorbing area

Now we consider the case with S = 1.002. At such a mismatch the saddle fixed point Po is placed slightly below D. Unstable manifolds of Po again bound an invariant absorbing area A^s\ as shown in Fig. 8.14.

With increase of e, two fixed points P and Q arise above the diagonal in a


Fig 8.14. Absorbing area A(s) (hatched) replacing the synchronous state AW (dashed) in the case of a parameter mismatch 5 = 1.002. Parameters a = 3.8, e = 1.55, and p = 0.4.

Ssn

Fig 8.15. Diagram of the bifurcations that replace the transcritical riddling bifurcation shown in Fig. 8.4 in the case of a parameter mismatch 6 = 1.002.

saddle-node bifurcation. The repelling node Q moves towards the diagonal. At the same time, PQ slowly moves down and away from the diagonal, leading to an increasing size of the absorbing area A^. The bifurcation diagram is shown in Fig. 8.15.


With further increase of e, the repelling node Q enters into the absorbing area A^ and destroys it. Again, the associated crisis bifurcation of A'") is the form that the hard transcritical riddling bifurcation assumes in the case of a parameter mismatch. Asynchronous bursts of the trajectories abruptly grow so that the trajectories can go far away from the diagonal. In analogy with the above description, the further behavior depends on whether there is an attractor Aup above the diagonal, and whether an absorbing area A still exists. After the crisis, trajectories from the former absorbing area A^ may go to the attractor Aup (if it exists), or escape to infinity (if the absorbing area A is destroyed), or (otherwise) fill the whole area A.

Therefore, in the both cases 8 < 1 and 8 > 1, the crisis bifurcation of A^ appears as a hard symmetry breaking bifurcation of the system (8.14). The hard transcritical riddling bifurcation and the subsequent blowout bifurcation observed 8—1 are replaced by an interior or an exterior crisis of the thin invariant absorbing region i « for 8 ± 1.

8.6 Stability of if-Cluster States

Let us now suppose that system (8.1) falls into a i\-cluster state, i.e. that the coordinates of the state vector x = {xi}i=l split into K groups such that in each group the coordinates are identically the same

_ _ _ def xii — x%i — • • • — xiNi — 2/1

_ _ _ def X'N1+I — xiNx+2 — • • • — xiNl+N2 — 2/2 (8 .15)

'J_ _ ' _ ' def XiN1+N2+...+NK_l+l ~ XiN1+N2 + ...+NK_l+2 ~ • • • — XiN — ]JK.

The positive integer Nj represents the number of variables X{ belonging to the j'th cluster, j = 1,2, . . . , K, so that Nx +N2 +... + NK = N. We note that, by virtue of the complete symmetry of the system (i.e., the fact that all the individual maps are the same), for any set {Nj} the .^-dimensional subspace defined by Eqs. (8.15) remains invariant for the dynamics in the corresponding if-cluster state.

Introducing the set of parameters pj = Nj/N,j = 1,2, . . . , K, the dynamics in the .ftT-cluster state can be described by the system of K coupled one-dimensional maps


K

W(n + l) = ( l - e ) / ( w ( n ) ) + e 5 3 p i / ( y i ( n ) ) > i = \,...,K. (8.16)

This system is also a globally coupled map system, but with different weights Pj associated with the contribution of the jth. cluster to the global coupling. Varying the parameters pj in (8.16) we can obtain the governing map for the in-cluster dynamics of any possible K-cluster state of our original system (8.1).

A necessary condition for the presence of stable if-cluster behavior in system (8.1) is that the map (8.16) with the assumed values of the parameters pj has a stable invariant set A^K\ but that there is no stable invariant sets A^L\ A^ I> A^K\ with L < K. For example, system (8.1) with even number of oscillators N may demonstrate symmetric two-cluster dynamics (8.2) if the two-dimensional map (8.3) with p = \ has a stable invariant set A^ % D = {(x, y) \ x = y}.

Provided that it is stable in the cluster subspace, the conditions for an at-tractor A^ of system (8.16) to be stable in the whole iV-dimensional phase space are that it is also stable in the transverse directions. The transverse stability of A™ may be asymptotic, when it attracts all trajectories from its neighborhood, or weak, when A^ is stable in the Milnor sense only, i.e., it attracts trajectories from a positive Lebesgue measure set of initial conditions from any iV-dimensional neighborhood of A^ [24].

In order to examine the conditions for transverse stability of the two-cluster state (8.2) we consider the Jacobian matrix D<$> of the iV-dimensional map $ denned by Eq. (8.1). Reduced on the subspace defined by Eqs. (8.2), the matrix Z?$ can be represented as

D $ = M(x) L(y) LT(x) N(y)

where M{x) and N(y) are symmetric matrices of dimensions iVi x N\ and JV2 x JV2, respectively, and T denotes the operation of transposition. It is easy to show that the matrix D<& has two distinct eigenvalues i/x,i = f'(x)(l — e) and I/J.,2 = f'(y)(l — e) that occur with the multiplicities JVi — 1 and ./V2 — 1, respectively.

Let now the two-dimensional map (8.2) have an attractor A^ which does not belong to the diagonal D = {(x,y) | x — y}. By virtue of the form of the transverse eigenvalues u±i and u±$ and the fact that the corresponding


eigenvectors do not depend on the phase coordinates, the transverse Lyapunov exponents of the two-cluster state are given by

Afi = lim \ X > | f(x(n))(l - e) |= lim \ £ l n | f(x(n)) | + l n | l - e\ fc->oo K

n = 0 fc->oo n = 0

(8.17)

Af2 = lim i £ In | / ' (y(n))( l - e) \= lim ± £ In | f'(y(n)) | + In |1 - e | , * - * ° ° n = 0 & - > ° ° n = 0

evaluated for a typical trajectory {(a;(^),y(n))}^=0 C A^2'.

As discussed above, the attractor A^ for the system (8.3) of two coupled maps is at least a Milnor attractor for the TV-dimensional system (8.1) when it attracts a positive Lebesgue measure set of points from M.N. For this to occur, both the above Lyapunov exponents must be negative [46, 47, 48]. Hence, a procedure for finding stable two-cluster states in system (8.1) can be the following. First, we find an attractor A^ % D = {(x,y) | x = y} for the system of two coupled maps (8.3). Then two Lyapunov exponents Aj^, i = 1,2 of the form (8.17) are calculated for typical trajectories on A^2\ For the parameter region where both of these Lyapunov exponents are negative, the system of globally coupled maps (8.1) has a stable (at least on average) two-cluster state with a dynamics given by the two-cluster attractor A^2\ Note that this procedure does not depend on the number N of coupled oscillators in Eq. (8.1). The only restriction is that this number should allow the assumed distribution of variables between the clusters. For example, if the two-dimensional system (8.3) with p = 1/3 has an attractor A^ (not belonging to the diagonal D), and the two transverse Lyapunov exponents are negative, then the iV-dimensional system (8.1) will have stable two-cluster states for N = 3 (JVi = 2,N2 — 1),N — 6(Ni = 4,N2 = 2),N = 9 (Ni = 6, N2 = 3), N = 12 (Ni = 8, N2 = 4), etc.

In the case of periodic dynamics, it can be shown that if the two-dimensional map F of the form (8.3) with 0 < e < 1 has a stable period-2™, m = 1, 2 , . . . , cycle out of the diagonal with symmetric distributions of its points with respect to the diagonal (e.g., for p = \) then system (8.1) exhibits stable period-2m two-cluster dynamics.

By analogy with the two-cluster state, for a /ST-cluster state (8.15) with the attractor A^K\ one has to iterate the map (8.16) on A^ and calculate K


transverse Lyapunov exponents as given by

1 fc-i

Ag = lim v 5Z l n I /'(2/i(^)) I + In |1 - e | , j = l,2,...,K. (8.18) n=0

When all the Lyapunov exponents are negative, A^ is also an attractor in N dimensions in the Milnor sense [24]. This provides the conditions for the existence of stable if-cluster states for system (8.1).

8.7 Desynchronization of the Coherent Chaotic State

The purpose of this section is to discuss the phase diagram proposed by Kaneko [27, 28, 29] for the occurrence of the various clustering states and to identify the different types of bifurcations that occur as the coupling constant e and the nonlinearity parameter a are varied.

Coherent motion of the coupled map system (8.1) takes place on the main diagonal D = {(xi, X2,.--, £JV) \ x\ = X2 = ... =• XM} of the ^-dimensional phase space and is governed by the logistic map / = fa. Depending on the value of a, the coherent dynamics may be either periodic or chaotic, as characterized by the sign of the Lyapunov exponent

n=0

calculated for a typical trajectory {z(n)}^L0 of fa. For an ensemble of coupled logistic maps this implies that, for any particular value of a, only a single one-dimensional attractor, periodic or chaotic, can exist on D. Let us again denote it by AW, where the superscript stands for "symmetric".

The average transverse stability of the attractor A^ is determined by the transverse Lyapunov exponent A^ = Aa + In |1 — e|. Actually, there are N — 1 transverse Lyapunov exponents but, due to the symmetry of system (8.1), they are all equal to A_ (see Sec. 8.6) . Hence, the coherent motion loses its transverse stability simultaneously in all N — 1 independent transverse directions.

In the phase diagram of Fig. 8.16, the uppermost (dotted) curve denotes the transverse destabilization of the fixed point PQ = (xo, xo,..., XQ), X0 = 1 - \. In the parameter regime a > ao = 3.678573 where the attractor A^ is one-piece chaotic, PQ is the first trajectory on A^ to lose its transverse stability, and,


0.50

0.40

0.30

8

0.20

0.10

0.00

3.50 3.60 3.70 3.80 3.90 4.00 a

Fig 8.16. Phase diagram for cluster formation in a system of globally coupled logistic maps. a is the nonlinearity parameter for the individual map, and e is the coupling parameter. The uppermost (dotted) curve represents the riddling bifurcation of the one-piece coherent chaotic state A^ in which the fixed point PQ G A^ loses its transverse stability, and the fully drawn fractal curve delineates the blowout bifurcation. The smooth fully drawn and dashed bold curves represent stabilization of the asynchronous period-2 and period-4 cycles in the symmetric two-cluster states, respectively. The lowermost (dashed-dotted) curve represents the stabilization of (another) period-4 cycle in the symmetric three-cluster state. Regions denoted by SH correspond to parameter values where the system has stable clusters, and subscripts indicate the cluster numbers. fHc denotes the region where the dynamics is high-dimensional chaotic.

hence, the dotted curve represents the riddling bifurcation curve. This curve can be easily determined analytically [32, 33, 34]. Below the riddling bifurcation curve the coherent chaotic state is weakly stable only. Destabilization of PQ takes place via a transverse period-doubling bifurcation and produces an asynchronous period-2 saddle around the fixed point. For slightly lower values of the coupling parameter, the synchronous period-2 cycle embedded in the coherent chaotic state also undergoes a transverse period-doubling, producing an asynchronous period-4 cycle.

The fractal curve in Fig. 8.16 denotes the blowout bifurcation of A^SK The blowout occurs at e = ey — 1 — e~^a when the transverse Lyapunov exponent


X± of the synchronous chaotic set changes sign from minus to plus. After the blowout bifurcation, A^ is no longer an attractor but has turned into a chaotic saddle. Almost all trajectories now go away from the coherent state described by the chaotic set A^, and in general only a zero measure set of trajectories will approach A& [46, 47, 48]. One of the main questions of the present chapter is to determine the fate of the diverging trajectories. We find that, depending sensitively on a, there are two different possibilities associated with the mutual disposition of the blowout and two-cluster stabilization curves.

Let a be fixed and let us consider what happens as the coupling parameter e is reduced. If the blowout bifurcation occurs before the appearance of a stable two-cluster state, the coherent phase turns into a high-dimensional chaotic state. With further reduction of parameter e, this may be captured into one of the periodic two-cluster states. In the opposite situation, i.e. when the asynchronous periodic cycles stabilize before the blowout bifurcation, two-cluster states appear before the blowout of the coherent state. As a consequence, both types of dynamics (fully synchronized chaotic and two-cluster periodic) coexist in some region of the (a, e)-parameter plane [41, 42].

In Fig. 8.16, the solid and dashed bold curves represent the stabilization of the asynchronous cycles P2 (period-2) and P4 (period-4) forming the possible symmetric, i.e. equally distributed (p = 5) two-cluster states. These cycles remain stable in some regions under the curves to destabilize with further reduction of e in a Hopf bifurcation. The symmetric two-cluster state p j , which arises as the asynchronous saddle cycle produced through a transverse period-doubling bifurcation of the symmetric fixed point P{*\ stabilizes in a subcritical, inverse pitchfork bifurcation along the fully drawn bold curve. P4, which arises from a transverse period-doubling of the symmetric period-2 orbit, stabilizes along the dashed bold curve. As inspection of Fig. 8.16 shows, for a > 3.93, Pi stabilizes before (i.e., for higher values of e than) P%. In Section 8.8 we shall perform a detailed analysis of the influence of cluster asymmetry on the stabilization of the cycles Pi and P\ (and the dynamics developed from these cycles). The idea is to illustrate the important role played by the exactly symmetric two-cluster states for the desynchronization phenomena in system (8.1).

The last (dotted-dashed) bifurcation curve shown in Fig. 8.16 represents the stabilization of the symmetric three-cluster state. In the moment of this bifurcation, stable period-4 cycles appear in each of the subspaces for the sym-


metric three-cluster states with a dynamics governed by (8.16) with K = 3 and Pj = 1/3, j = 1, 2,3. In the region of interest this three-cluster curve lies below the two-cluster curve given by the stabilization of Pi and P4. Hence, we assume that the two-cluster bifurcation curve delineates the first moment of formation of symmetric clusters in the globally coupled map system (8.1).

In the two-cluster states, the dynamics is governed by the two-dimensional map (8.3). Figure 8.17(a) shows a characteristic phase portrait after the riddling bifurcation. The fixed point P0 = {XQ = yo = 1 — £} belonging A^ has become transversely unstable in a period-doubling bifurcation giving rise to a saddle period-2 cycle Pi. The thin curves connecting PQ with the points of Pi represent a separatrix. Close to this separatrix the trajectories will first approach Pi and then proceed along one of the unstable manifolds of the saddle cycle. Hence, there exists a positive measure set of the trajectories which, when starting near P0 , can move away from A^ to a distance given approximately by the deviation of Pi from PQ . As the preimages of the fixed point PQ are dense in A^, we conclude that in the neighborhood of any point of A^, there exists a positive measure set of points which give rise to trajectories that go away from A^ in the direction towards Pi, i.e., the basin of attraction of A^ is locally riddled. Trajectories that burst away from A^ are restricted to an absorbing area denoted in Fig. 8.17(a) by A.

As we can see from Fig. 8.16, this type of locally riddled dynamics occurs for a relatively wide region (denoted 9ti) of the (a, e)-parameter plane. The lower boundary of this region consists of two very different parts: a fractal boundary defined by the blowout bifurcation curve, and a smooth boundary corresponding to the symmetric two-cluster formation curve. The corresponding transformations of the dynamics of the system clearly involve very different processes.

If the parameter point (a, e) leaves the region !?ti through the fractal (blowout) curve, the absorbing area A defines a new attractor in the plane of the two-cluster state. This is illustrated in Fig. 8.17(b). As we shall see in Sec. 8.9, however, this type of two-dimensional attractor arising from the coherent state in a blowout bifurcation is not stable in the whole A^-dimensional phase space. Transverse to the two-cluster state, the maximal Lyapunov exponent A^ is positive although small, growing according to the power law \e - su\a , 1 < a. < 2, where eu is the blowout bifurcation value and the exponent a depends on the asymmetry parameter p.


1,1'

<• . ' ( I

1.1 C'-

-O.i i l i

l » ' l

'•) I

-0.1

/

/

-0.1 1.1

Fig 8.17. Typical phase portraits for the globally coupled map system (8.1) reduced to the symmetric two-cluster subspace (p = | in (8.3)): (a) locally riddled basin of attraction for the coherent state A^ after the riddling bifurcation (o = 3.8, e = 0.42), (b) ori-off intermittency after the blowout bifurcation of A^ (a — 3.8, e = 0.34), and (c) globally riddled basin of attraction for A^ after stabilization of the asynchronous period-2 cycle P% (o = 3.75, e = 0.315). The light gray region in (a) and (c) represents the basin of attraction for A^\ and the basin of attraction for on-off attractor in (b). The dark gray regions in (c) represent the basin of attraction for the cycle P2 whose points are plotted by crossed circles. The curves L\ and L2 delineate the absorbing area A, and PQ is the fixed point embedded in A^'\ Note that the on-off intermittency state iu (b) is unstable in jV-dimensional phase space.

Consider now the second possibility where the (a, e ) -parameter point leaves

9ti th rough the smooth two-cluster formation curve. T h e characterist ic phase


portrait after this transition is presented in Fig. 8.17(c). Two different asymptotic states coexist: a coherent state given by the synchronous chaotic set A^ and a periodic two-cluster state given by the cycle P?,- The basin of attraction for the coherent attractor becomes globally riddled with the basin of P2. In the phase diagram of Fig. 8.16, the parameter region where this kind of globally riddled dynamics occur, is denoted by 9 1,2- When (a, e) belongs to £Hi,2, both coherent and two-cluster regimes can be realized in system (8.1) when calculations are performed with randomly chosen initial conditions. This follows from the stability of the of the symmetric period-2m cycles in N dimensions as soon as they are stable in the two-cluster state (see Sec. 8.6).

The lower boundary of 9^2 in Fig. 8.16 is given by the blowout bifurcation curve for the coherent attractor A^. Under this curve, is no longer stable on average. Hence, only stable two-cluster regimes can be observed in system (8.1), provided that the parameter point (a, e) lies above the three-cluster dotted-dashed curve (9^ region). Below the latter curve two- and three-cluster states coexist (9^2,3 region). Moreover, in the lower left corner of Fig. 8.16, one can observe a parameter region where the blowout curve falls even below the three-cluster curve (5Hi,2,3 region). Here, the coherent chaotic state coexists with two- and three-cluster states.

The last region in Fig. 8.16, denoted by d\c, is bounded by the blowout curve from above and by the symmetric two-cluster formation curves from below. Here, the dynamics of system (8.1) can be high-dimensional chaotic, provided that strongly asymmetric clusters do not arise. We justify this statement in Sec. 8.9 by showing that symmetric two- and three-cluster chaotic states are unstable in the whole iV-dimensional phase space of system (8.1). The role of strongly asymmetric clusters, that can be observed for relatively large numbers N, will be considered in Sec. 8.10.

8.8 Formation of Nearly Symmetric Clusters

As shown above, the appearance of the symmetric (or nearly symmetric) two-cluster dynamics in the globally coupled map system (8.1) is caused by stabilization of the period-2 or period-4 asynchronous cycles Pi and P4. In this subsection we shall consider how the moment of stabilization depends on the presence of a small cluster asymmetry, as realized when the parameter p in system (8.3) starts to differ from | . A main conclusion is that the symmetric


clusters, i.e. with p = | , stabilize at higher values of the coupling parameter e than do slightly asymmetric clusters. Moreover, the stabilization occurs the later the larger the asymmetry is. For the symmetric two-cluster state (p = | ) , the cycles P2 and P4 are born in transverse period-doubling bifurcations of the coherent fixed point P^ and P2 , respectively. After the bifurcations they are first unstable (saddles) to later stabilize in inverse subcritical pitchfork bifurcations. A characteristic phase portrait for the situation when both P% and P4 have stabilized is presented in Fig. 8.18. This figure corresponds to a parameter point in the region *K2 of Fig. 8.16 where the synchronized state is already a chaotic saddle.

Fig 8.18. Phase portrait of the globally coupled map system (8.1) reduced to the symmetric two-cluster subspace (p = | in (8.3)) after stabilization of both the asynchronous period-2 (denoted by P2 and plotted by crossed circles) and the asynchronous period-4 (denoted by P4

and plotted by stars) cycles. After the blowout bifurcation, the coherent state A^ (dashed line segment) is a chaotic saddle. After the transverse period-doubling bifurcations giving birth to Pi and Pt, the symmetric fixed point (denoted by p j s ' and plotted by crossed square) and symmetric period-2 cycle (denoted by P$'' and plotted by triangles) are repellers. Basins of attraction for the cycles P2 and F4 are shown in dark and light gray, respectively. Parameters a = 3.9 and e — 0.345. With further reduction of e each of the cycles P2 and F4 undergo a sequence of additional bifurcations leading to various forms of quasiperiodic and chaotic two-cluster dynamics.


For the case of slightly asymmetric clusters, the cycles Pi and P4 can be obtained by continuation of those in the symmetric case with the parameter p (starting with p — | ) . When p 7 \, the cycles stabilize in saddle-node bifurcations off the main diagonal rather than via inverse subcritical pitchfork bifurcations as in the symmetric case (p = | ) .

0.30-

0.25

0.10

:(a)

~

7

Crisis^

Crisis

j? \ \pa N \ \ >

\ XptH V X N N

s -C"

•

\ p o p2 :

\ _ H _ _ _

\SN

^

0.20 0.30 0.40 0.50

0.40

0.35 -

£ 0.30

0.25-

0.20

(b)

Crisis ~^**^

1 . p

VPV>

1

1 1 '

s^-

P4<2)

\PD

— r

\PD

^

P2

SN

i

<^

0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 P

Fig 8.19. Stability regions in the (p, e)-parameter plane for the various types of dynamics in system (8.3) that develop from the asynchronous period-2 (P2) and period-4 (P4) cycles and represent two-cluster states in (8.1). Bifurcation curves denoted by SN, PD, and H correspond to saddle-node, period-doubling, and Hopf bifurcations, respectively. With decreasing values of p we can follow P2 through a cascade of period-doubling bifurcations into a chaotic off-diagonal attractor that finally destroys in a boundary crisis. The bold dashed curve bounds the region where the largest Lyapunov exponent transverse to the two-cluster state is negative. Here, system (8.1) displays stable two-cluster states with a distribution between clusters as defined by p and a dynamics that is given by the attractors developed from P2. Parameters a = 3.8 in (a) and o = 4 in (b).

Figure 8.19 shows the regions of stability for the various types of dynamics that evolve from P<i and P4 under variation of p and e for two different values of the nonlinearity parameter a. In Fig. 8.19(a) (a = 3.8), the upper boundary of the stability region (solid curve denoted SN) defines the moment of stabilization of the asynchronous period-2 cycles Pi in the above mentioned saddle-node bifurcation. This curve is clearly seen to assume its maximal value for p = | , representing the fact that symmetric clusters will stabilize before slightly asymmetric clusters as e is reduced.


For a = 3.8, stabilization of Pj occurs at lower values of the coupling parameter than stabilization of P2, and we find the stability region for P4 (and for solutions developed from P4) in the upper right corner of the stability region for P2. For a = 4.0 (Fig. 8.19(b)), on the other hand, P4 stabilizes before Pi (see Fig. 8.16), and the stability region for P4 falls above that of Pi-

As stated in Sec. 8.6, the stability of a periodic cycle in the two-cluster phase plane implies its stability in TV-dimensional phase space. Hence, the uppermost curves in Figs. 8.19 (a) and (b) are the bifurcation curves in the (p, e) -parameter plane for the appearance of symmetric (or nearly symmetric) two-cluster states. The overlapping stability regions for Pi and P4 implies that the system has two coexisting types of two-cluster dynamics (see, e.g., Fig. 8.18). With further variation of the parameters p and e, the cycles P2 and P4 undergo a variety of different bifurcations in which more complicated two-cluster dynamics arises. Besides periodic cycles of higher periodicity, quasiperiodic and chaotic dynamics occur. Some of the bifurcation curves are indicated in Figs. 8.19 (a) and (b) where period-doubling, saddle-node, and Hopf bifurcation curves are denoted by PD, SN, and H, respectively. A more detailed examination of some these transitions was presented in the beginning of this chapter [41, 42]. The bold dashed curves in Figs. 8.19 (a) and (b) denote the transverse destabilization of the two-cluster attractors developed from P2, and the lower right curves represent their final boundary crises. As we can see, there is a fairly large parameter region where the attractor in the two-cluster state is transversely unstable.

8.9 Transverse Stability of Chaotic Clusters

In this section we show that chaotic motions in two- and three-cluster states that arise after the blowout bifurcation of the coherent attractor, in general, are transversely unstable. To verify this conjecture, we show that the largest transverse Lyapunov exponents \]_ (for the two-cluster state) and X± (for the three-cluster state) are positive. Moreover, immediately after the blowout bifurcation e = eu they grow in accordance with a power law.

Figures 8.20 (a) and (b) display scans of \ L over the range from e = 0 (uncoupled system) to right above the blowout bifurcation (eu = 0.5) for a = 4 and for two different values of the asymmetry parameter : p = \ (symmetric two-clusters) and p = 1/3 (1:2 cluster distribution). The scans of \\_' are shown as


| o .

*-> 0.0

— 1 - - , 1 1 1 1 1 1 1 1

(b)

\ XX ^ 1

- v | \ 1

t "

s

s

1 1

N

0.2 0.3 0.4 0.5

Fig 8.20. Variation of the largest transverse Lyapunov exponent (solid bold curve) with the coupling parameter e for the two-cluster states being (a) symmetric (p = | ) or (b) with 2 : 1 variable distribution (p = 1/3). Parameter a = 4. Dashed curves represent Lyapunov exponents within the two-cluster state. Note that, when e decreases, the state stabilizes in TV dimensions if it becomes an attracting cycle. Our interest is focused on the behavior immediately after the blowout bifurcation of the coherent state, that occurs at e = 0.5.

bold curves. The dashed curves show the variation of the two other Lyapunov exponents that control the two-dimensional cluster dynamics (8.3). In both cases the blowout bifurcation at eu = 0.5 gives rise to a hyperchaotic attractor (all three Lyapunov exponents are positive) bounded in two-dimensional phase space by the absorbing area A (see Fig. 8.17 where the characteristic form of A is illustrated). In Fig. 8.20(a) there is an interval around e = 0.23 where

(2)

A± is negative while the Lyapunov exponents in the two-cluster plane are positive. Here, we have a transversely stable chaotic two-cluster state. However, through most of the scan the transverse Lyapunov exponent is positive when the longitudinal exponents are positive. In particular this is true immediately after the blowout bifurcation.

Figure 8.21 shows an enlargement of the rightmost parts of the graphs from Fig. 8.20 in order to illustrate the power law of growth for A^ after the blowout bifurcation for different values of the asymmetry parameter p. a = 4. As we can see

Here, again


0.010

la'

Fig 8.21. Variation of the largest transverse Lyapunov exponent Aj_ for the chaotic two-cluster state A'2' arising immediately after the blowout bifurcation of the coherent state A'"'. Here, A^ has a form similar to those shown in Fig. 8.17(b), and the blowout bifurcation occurs at e = 0.5. Parameter a = 4. The three curves in (a) represent different values of the asymmetry parameter p. In all cases, the transverse Lyapunov exponent is positive (although small). The dashed curve gives a variation of the transverse Lyapunov exponent X]_' of the coherent state A's'. In (b), the same graphs in logarithmic scale illustrate the power law dependence (8.19). Here, the transverse Lyapunov exponents for p = 0.5, p = 0.4, and p = 0.3 are plotted by circles, squares, and triangles, respectively. As one can see, straight lines within the marks fit the values of the exponents and have slopes a = 2 (p = 0.5), a = 1.8 (p = 0.4), and a = 1.7 (p = 0.3). We conclude that the chaotic two-cluster state formed in this process cannot be stable in TV-dimensional phase space.

A? ~ |ew - e\a , e-^eu, (8.19)

where a = 2 for the symmetric case p = \ and decreases with decreasing p. This

result is supported by plotting the graphs on a logarithmic scale (Fig. 8.21(b)).

Here, Ae = ew - e, and the slopes of the linear part of the graphs determine

the exponent a in the power law (8.19).

We conclude that chaotic two-cluster motion, when it appears after desta-

bilization of the coherent phase, is transversely unstable. It follows that the

chaotic motion must be at least three-dimensional. We now show that the

dimension must also be larger than three. To this end, we give numerical evi

dence that chaotic motion in the symmetric three-cluster states is transversely

unstable.


0.004

§ 0.002

j j 0.000

-0.002

'

^J^X^^*"" •

0.48 0.485 0.49 e

0.495 0.5

Fig 8.22. Transverse Lyapunov exponent X± (shown by circles) for a symmetric three-cluster (2) state as a function of the coupling parameter e. The largest \\\ and the second \\_'2 transverse

Lyapunov exponents for the two-cluster state with 2 : 1 (p = 1/3) variable distribution between clusters are also shown. The value (2A^\ + A^2 J /3 is represented by the bold dashed curve

and reproduces the values of A^'. We conclude that A^ becomes positive immediately after the blowout bifurcation (e = 0.5) and grows in accordance with a power law. Here, a = 4.

Figure 8.22 shows a plot of the transverse Lyapunov exponent A^' versus pa-rameter e for a symmetric three-cluster state. A^ becomes positive immediately after the blowout bifurcation (e = 0.5) and appears to grow in accordance with a power law similar to (8.19). This can be justified as follows. As illustrated in Fig. 8.23, the typical trajectory in the chaotic three-cluster state behaves in such a way that it spends most of the time near the diagonal two-dimensional planes az = {x = y, z} , ay = {x = z, y} , and ax = {x, y = z}. Moreover, it switches between these planes in an apparently random manner. From this observation we conclude that an approximate value for the transverse (to the three-cluster state) Lyapunov exponent A^ can be obtained as calculated on the planes <rx,cry, and az, with the additional assumption that the average time spent near each of these planes is the same. This gives

A i s >s (2A? i + A?i)/3, (8.20)


0.08

0 x

-0.08 0 20000 40000 60000 80000 100000 120000

0.08

N 0

w

-0.08 0 20000 40000 60000 80000 100000 120000

0.08

N 0 >*

-0.08 0 20000 40000 60000 80000 100000 120000

Iterations

Fig 8.23. Synchronization errors calculated on a typical trajectory for the chaotic three-cluster state (considering system (8.16) with K = 3 and pj = 1/3, j = 1,2,3). We have added a small noise of maximal amplitude 10~22. The first 104 iterations are skipped, and the next 1.2 • 105

iterations are plotted. The trajectory spends most of its time near the two-dimensional planes <Jz = {x ~ y, z}, ay = {x = z, y} and ax = {x, y = z}, and it switches between these planes in an apparently random manner. Parameters a = 4 and e = 0.495.

where A± 1 and A^ 2 a r e the largest and the second transverse Lyapunov expo

nents for the chaotic motions in the two-cluster planes crx,ay, and az. Using the expression (8.17) for the transverse Lyapunov exponents for two-cluster states and the formula (8.18) for three-cluster states, we arrive at the approximate formula (8.20).

To conclude our considerations, we note that the numerical calculation of K^ has required the introduction of small noise of the order of 10~22. Without this noise, trajectories are captured by the two-cluster dynamics because of finite precision in the calculations. We suppose that this capturing phenomenon can explain why high-dimensional chaotic motions arising after desynchroniza-tion of the coherent phase have not previously been reported. Indeed, in the case considered, any regular calculation (by standard double or triple precision but without noise) gives evidence of two-cluster dynamics even though this is actually transversely unstable as soon as it is chaotic [44, 61].


8.10 Strongly Asymmetric Two-Cluster Dynamics

System (8.1) can demonstrate two-cluster behavior (for proper initial conditions) if the two-dimensional map F has an attractor A^ that does not belong to the diagonal Di = {(x,y) \ x = y}. Such an attractor usually originates in the stabilization of a periodic cycle out of the diagonal [43, 44]. Under variation of the parameters this stable cycle undergoes different types of bifurcations that may lead to other stable periodic cycles (e.g., via a period-doubling bifurcation) or to a stable closed invariant curve (Hopf bifurcation). Further developments of the attractor A^ may turn it into an attracting chaotic set which finally disappears in a boundary crisis. Therefore, to investigate the emergence of clusters in the system of coupled maps (8.1) we have to examine the appearance of stable point cycles within the corresponding subspace and then to examine the transverse stability of these cycles with respect to the cluster subspace. More specifically, for two-cluster behavior, stable periodic cycles of the two-dimensional map (8.3) should be found by varying the parameters a, e, and p.

Figure 8.24 presents regions in (p, e)-parameter plane where the map (8.3) has an attracting cycle (cycle periods are indicated by numbers) away from the diagonal £>2- If the (p, e)-parameter point falls in one of the black regions of Fig. 8.24, with appropriate initial conditions, system (8.1) will exhibit periodic two-cluster behavior for the corresponding values of parameter a. The distribution of maps between the clusters is given by the parameter p. To obtain the bifurcation diagrams in Fig. 8.24, we fixed parameter a, took a fine grid in the (p, e)-parameter plane and, with 20 randomly chosen initial conditions for each grid point, iterated the map F to look for an asymmetric stable cycle of a period less than 50. When such a cycle was found (at least for one initial condition), the corresponding (p, e)-parameter point has been plotted black.

Inspection of Fig. 8.24 suggests that with decreasing coupling strength s, the first two-cluster states to appear are highly asymmetrical with respect to the distribution of oscillators between the clusters, i.e., states for which the parameter p is small.

Figure 8.24 also displays a surprising organization of the periodic regions to the right of the p = 0 value: They follow the well-known sequence of windows of the logistic map. Indeed, as one can see in Figs. 8.24 (a) and (b) the widest window is of period 3. The next, relatively large window is of period 5 followed


Fig 8.24. Regions of parameter plane where the map (8.3) displays an attracting periodic cycle outside of the diagonal, (a) a = 4,(b) a = 3.84, and (c) a = a0 = 3.6785735104... For a = 3.84 regions of period-3 two-cluster dynamics are being formed both along the e = 0 axis and along the p = 0 axix (b).

by period 7 and 9. In between the period-3 and -5 windows there is a window of period 8.

To the left of the period-3 window we find a period-adding sequence of windows of periods 4, 5, 6 and so on (Fig. 8.24(a), o = 4). These stability regions correspond to stable cycles jk = {xi}i=i, A; = 4 , 5 , . . . of so-called maximal type


(coordinate X{ increases: X{ < Xi+{) which arise in the bifurcation diagram for the logistic map fa beyond the period-3 window. For a = 3.84 (which is inside the period-3 window) such cycles have not yet appeared for the logistic map and, as it can be seen in Fig. 8.24(b), the corresponding windows are not present in the (p, s:)-parameter plane of the map (8.3).

Figure 8.24(c) shows the phase diagram for a = CLQ — 3.6785735104..., i.e., at the moment of the first homoclinic bifurcation of the nontrivial fixed point XQ = 1 - ^ for the logistic map fa. It is known that for a < ao, the logistic map fa does not display odd-period windows. Instead, the period-6 and period-10 windows are the widest and, as a sequence, one can clearly see the windows corresponding to these periods in Fig. 8.24(c). Broadly speaking, Fig. 8.24(c) is similar to Fig. 8.24(a), only the periods of the asymmetric windows are multiplied by 2. Indeed, instead of the aforementioned windows of periods 3, 4, 5, 6, etc. in Fig. 8.24(a) one can see windows of periods 6, 8, 10, 12, etc., which are ordered in the same way.

Figures 8.25 (a) and (b) show the most prominent periodic windows for the two-dimensional map F at small values of the asymmetry parameter p. Here, a = 4 and a = 3.83, respectively. It is interesting to note that the windows reach the higher with respect to e the smaller p is. This implies that the first two-cluster states to synchronize when decreasing the coupling strength e are those with very strong asymmetry.

In Figs. 8.24 (a) - (c) one can also observe a similar system of periodic windows for small e and large p. These windows arise in the (p, e)-parameter plane through the e = 0 axis and lead to a manifestation of periodicity in the so-called turbulent regime where the globally coupled map system displays many, relatively small clusters. This phenomenon was recently investigated by Shimada and Kikuchi [62]. They showed how the maximally symmetric three-cluster attractor with period-3 motion is related to the period-3 window of the individual map [63].

Using the period-3 window as an example, let us now discuss the mechanism underlying the emergence of the highly asymmetric periodic windows. Consider system (8.3) at a parameter-a value for which the logistic map fa has a stable period-3 cycle. Numerical evidence suggests that as soon as the logistic map fa enters the period-3 window at a = a^ = 1 + 2\/2 = 3.8284..., the two-dimensional map F of the form (8.3) acquires a stable period-3 cycle _P3 out of the diagonal. The stability region for the cycle P3 emerges from the p = 0


0.65 0.44

0.25, 0.01 0.02 0.03 0.04 0.05 0.06 0.07 P

0.28, 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 P

Fig 8.25. Windows of two-cluster dynamics for small values of p. (a) a = 4, and (b) a = 3.83. With decreasing p the windows reach higher and higher with respect to e.

axis and expands to the right as a increases. This is demonstrated in Fig. 8.26. Here, we have plotted the stability regions in the (p, e)-parameter plane for four fixed values of the parameter a immediately above a^.

Figures 8.27 (a)-(d) present phase portraits of system (8.3) where the asymmetric period-3 cycle P3 bifurcates as it leaves the region of stability. Inside the period-3 window, the logistic map fa has a pair of stable and unstable period-3 cycles 73 and 73 born in a saddle-node bifurcation at a = a3. Correspondingly, the two-dimensional map F has a symmetric pair of attracting and

3W 5M saddle period-3 cycles P3 ; and P3 , respectively, placed on the diagonal D^.

In Figs. 8.27 one of the points of P3 (marked by a cross) and one of the points of P3 (marked by a cross with circle) can be seen on the dashed line which is part of the diagonal Z?2- Besides the symmetric period-3 cycles there are four asymmetric period-3 cycles out of the diagonal marked as follows: A cross is for an asymmetric attracting period-3 cycle P3 , a triangle is for an asymmetric repelling period-3 cycle, and a cross with circle is for each of two asymmetric saddle period-3 cycles.

When increasing the coupling strength, the parameter point (p, e) in Fig. 8.26 leaves the region of stability through its upper boundary. At this bifurcation the asymmetric stable period-3 cycle P^ annihilates with one of the asymmet-


0.45

0.40

e

0.35

0.30;

0.25,

_— ^ 5 » _ -

- 1

' — =3.83' ' - ' • — a = 3.8315 ••••

'• = "r" jf^r.: r.: r.: r.;3< • • •

( ( 1 \

1 1 •

- a = 3.833 •• a = 3.8345 -

Cd

i

\

0.003 0.006 0.009

Fig 8.26. Birth and growth of the period-3 stability region of the two-dimensional map F. Note the broadening of the window with increasing values of a.

ric saddle period-3 cycles in an inverse saddle-node bifurcation (Fig. 8.27(b)). If the parameter point leaves the region of stability through its right border, P3 collides with a different asymmetric saddle period-3 cycle to disappear in another inverse saddle-node bifurcation (Fig. 8.27(c)).

The third possibility is realized when the coupling strength s decreases: the parameter point (p, e) then leaves the region of stability of P3 through the lower boundary. A supercritical period-doubling bifurcation of P3 occurs giving rise to an asymmetric stable period-6 cycle PQ • Figure 8.27(d) illustrates the phase portrait just after the bifurcation, where two points of P6 are plotted by stars.

In Fig. 8.26 the letter B denotes a codimension-two bifurcation point of 1:1 strong resonance where both eigenvalues of the cycle P^ are equal to 1. By virtue of the particular shape of the stability region, this singular bifurcation point B determines the very first moment for the emergence of asymmetric period-3 two-cluster states in system (8.1) when the coupling parameter e decreases.

With increasing a, the period-3 window starts to move into the (p, ^-parameter plane to the right from the axis p = 0. The first stages of this movement


.DU

y

13

(a) P&

4

n»

E = 0.4 • P = o

A

£ ( s ) * = X

0.50

0.14 0.50

y

0.13 0.

X 0.18

(c) e = 0.4155 p = 0.0019

^ 4

4 A

5(8) ,, p(s) *3 * 5 J -

y

0.13

(b)

p(a)

p(s) ^r~~

e = 0.4155 p = 0

4

A P ( s )

0.14

0.70

y

0.10

X 0.18

(d) * * e = 0.28 j>% A P = °

J ? ^ ^ .14 0.18 0.14 X 0.18

Fig 8.27. Different bifurcations occurring along the edges of the period-3 window (a)-(d). By stars with circles we have denoted two points of an asymmetric saddle period-6 cycle that is born in a period-doubling bifurcation from the asymmetric saddle period-3 cycle, a = 3.83

are illustrated in Fig. 8.26 for four different values of a. Figure 8.28(a) (here a = 3.86) shows a typical shape of the window when it has already separated from the p = 0 axis. In particular, as one can see, decreasing parameter p causes a cascade of period-doubling bifurcations of P^'. This is again in agreement with the bifurcation diagram for the logistic map. We also note the period-6 region in the lower right corner of the window. H denotes a Hopf bifurcation curve below which the coupled map system displays quasiperiodic dynamics (with resonances). Immediately to the right of the window we find a two-dimensional form of type-I intermittency.

In Fig. 8.28(b) the structure of the period-3 window is presented for a — A. The boldly dashed curve bounds the region where the cluster attractor A^


028.0\ 0.02 0.03 0.04 0.05 °'3§.12 0.14 0.16 0.18 0.2 0.22

P P

Fig 8.28. Detailed structure of the period-3 window, (a) a = 3.86 and (b) a = 4.

originating from the asymmetric period-3 cycle P3 is stable in the whole N-dimensional phase space of system (8.1). This curve is obtained by calculation of transverse Lyapunov exponents. Indeed, for any two-cluster state A^ there are two distinct transverse Lyapunov exponents of the form (8.17) [44, 42] evaluated for a typical trajectory {(x(n),y(n))}°°_0 C A^2\ The first exponent

(2) \]_i is responsible for the breakdown of the cluster symmetry of the state x in (8.2), whereas A^'2 is responsible for the breakdown of the cluster symmetry of the state y.

When the parameter point crosses the curve TPD (Fig. 8.28(b)), the asym

metric stable period-3 cycle P3 undergoes a transverse period-doubling bifur-(3)

cation giving rise to a stable period-6 cycle Pg ' that does not belong to the cluster subspace (x,y). This bifurcation occurs when the transverse Lyapunov exponent X(

±2 of the period-3 cycle P3 becomes positive. As a result, cluster state y is no longer stable but splits into two subclusters. A stable 3-cluster state is born with period-6 temporal in-cluster behavior.

In this way, a transverse period-doubling bifurcation can lead to a "cluster-splitting" phenomenon where the number of synchronized clusters grows by 1. The stability region for the 3-cluster period-6 cycle P6 in the whole N-dimensional phase space of system (8.1) is bounded by the dotted curve in Fig. 8.28(b).

To provide an overview of the above results, Fig. 8.29 presents a general


bifurcation diagram in the (a, e)-parameter plane for desynchronization of the coherent motion and for the emergence of two-cluster states in the system of globally coupled maps (8.1). The fully drawn noisy (fractal) curve represents the blowout bifurcation of the fully synchronized coherent state. Below this curve the coherent motion is repelling on average. The dotted curve denotes the transverse destabilization of the symmetric nontrivial fixed point (XQ, XQ) causing the riddling bifurcation (provided that the logistic map fa has a one-piece chaotic attractor).

0.60

0.50

0.40

8

0.30

0.20

01§.60 3.70 3.80 3.90 4.00

a

Fig 8.29. Bifurcation diagram for desynchronization of the coherent motion and emergence of symmetric and asymmetric two-cluster states. Compare with Fig. 8.16 where the formation of strongly asymmetric two-cluster states is left out.

The solid stepped curve represents the emergence of highly asymmetric two-cluster states in system (8.1) by the mechanism described above. The numbers associated with this curve denote the periods of stable in-cluster cycles that cause the emergence of asymmetric two-cluster states.

Two additional bifurcation curves in Fig. 8.29 relate to the stabilization of the symmetric two-cluster states with period-2 (thin solid curve) and period-4 (thin dashed curve) in-cluster dynamics. As one can see from Fig. 8.29, for


a > 3.7, the highly asymmetric clusters control the cluster formation in the considered system of globally coupled maps (8.1). The asymmetric clusters clearly appear before the symmetric ones.

In conclusion, we have demonstrated how the transition from coherent to two-cluster dynamics in a system of N globally coupled logistic maps (8.1) can be studied by means of two asymmetrically coupled maps in the form (8.3). Our investigations have shown that the formation of strongly asymmetric clusters can play a significant role in the desynchronization process. Such clusters are typically first to appear as the coupling between the maps is reduced. In the (p, e)-parameter plane, stability regions for the strongly asymmetric two-cluster states emerge through the p = 0 axis as a increases. This happens at the same bifurcation moments as the corresponding periodic windows arise in the individual logistic map fa. With further increase of a, these stability regions separate from the p = 0 axis and move to the right. This scenario implies that the cluster formation process depends in an essential manner on the system size, i.e., the number N of interacting oscillators. Highly asymmetric two-cluster states can only be realized at large enough numbers N of interacting elements.

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Chapter 9

INTERACTING NEPHRONS

9.1 Kidney Pressure and Flow Regulation

The kidneys play an essential role in regulating the blood pressure and maintaining a proper environment for the cells of the body. This control depends partly on regulatory mechanisms associated with the individual functional units, the nephrons. However, a variety of cooperative phenomena that arise through interactions among the nephrons may also be important. The purpose of this chapter is to present experimental evidence for a coupling between nephrons that share a common piece of afferent arteriole, to develop a mathematical model that can account for the observed cooperative phenomena, and to discuss the possible physiological significance of these phenomena. We shall be particularly interested in the synchronization effects that can occur among neighboring nephrons that individually display oscillatory or chaotic dynamics in their pressure and flow regulation. As discussed in Chapter 1, in-phase synchronization, for instance, in which the nephrons simultaneously perform the same regulatory adjustments of the incoming blood flow, is likely to produce synergetic effects in the overall response of the system to external disturbances. Out-of-phase synchronization, on the other hand, will produce a slower and less pronounced response of the nephron system in the aggregate.

It has long been recognized that the ability of the nephrons to compensate for variations in the arterial blood pressure primarily rests with the so-called

349


tubuloglomerular feedback (TGF) by which the nephron can regulate the incoming blood flow in dependence of the ionic composition of the fluid leaving the loop of Henle [1]. Early experiments by Leyssac and Baumbach [2] and by Leyssac and Holstein-Rathlou [3, 4] demonstrated that this feedback regulation can become unstable and generate self-sustained oscillations in the proximal in-tratubular pressure with a typical period of 30-40 s. With different amplitudes and phases the same oscillations are manifest in the distal intratubular pressure and in the chloride concentration near the terminal part of the loop of Henle [5]. While for normal rats the oscillations have the appearance of a limit cycle with a sharply peaked power spectrum reflecting the period of the cycle, highly irregular oscillations, displaying a broadband spectral distribution with significant subharmonic components, were observed for spontaneously hypertensive rats (SHR) [3].

It has subsequently been found [6, 7] that irregular oscillations can be observed for normal rats as well, provided that the arterial blood pressure is increased by ligating the blood flow to the other kidney (so-called 2 kidney-1 clip Goldblatt hypertension). In a particular experiment, where the function of the nephron was accidentally disturbed, evidence of a period-doubling transition was observed [8]. Together with the above mentioned subharmonic components in the spectral distribution for the hypertensive rats, this type of qualitative change in behavior provides evidence in support of the hypothesis that the pressure and flow regulation in the rat nephron operates close to a transition to deterministic chaos [9, 10]. For non-oscillatory nephrons, self-sustained oscillations can often be elicited through microperfusion of artificial tubular fluid into the proximal tubule, demonstrating that the source of oscillations lies within the regulatory mechanisms of the individual nephron [4, 6].

As illustrated in the schematic drawing of Fig. 9.1, the TGF regulation is made possible by the interesting anatomical feature that the terminal part of the ascending limb of the loop of Henle passes within cellular distances of the afferent arteriole. At the point of contact, specialized cells (the macula densa cells) monitor the NaCl concentration of the tubular fluid and produce a signal that activates the smooth muscle cells in the arteriolar wall. The higher the glomerular filtration is, the faster the fluid will flow through the loop of Henle, and the higher the NaCl concentration will be at the macula densa cells. A high NaCl concentration causes the macula densa cells to activate the vascular smooth muscle cells in the arteriolar wall and thus to reduce the diameter of the

Interacting Nephrons 351

vessel. Hence, the blood flow and thereby the glomerular filtration are lowered.

Proximal tubule

Fig 9.1. Sketch of the main structural components of the nephron. Note particularly how the terminal part of the loop of Henle passes within cellular distances of the afferent arteriole, allowing the TGF mechanism to control the incoming blood flow in response to the ionic composition of the fluid leaving the loop of Henle.

The steady state response of the arteriolar flow regulation can be obtained from open-loop experiments [11] in which a paraffin block is inserted into the middle of the proximal tubule and the rate of filtration is measured as a function of an externally forced flow of artificial tubular fluid into the loop of Henle. Reflecting physiological constraints on the diameter of the arteriole, this response follows an S-shaped characteristic with a maximum at low Henle flows and a lower saturation level at externally forced flows beyond 20 — 25 nl/min. The steepness of the response is significantly higher for spontaneously hypertensive rats than for normotensive rats [12]. Together with the delay in the TGF regulation, this steepness plays an essential role for the stability of the feedback system [5, 13], and the experimentally observed higher stepness for spontan-iously hypertensive rats may therefore explain the more complicated pressure variations observed in these rats.

A main component in the regulatory delay is associated with the finite transit time of the fluid through the tubular system. The length of this delay can be


estimated from the phase shift between the pressure oscillations in the proximal tubule and the oscillations of the NaCl concentration in the distal tubule. A typical value is 10-15 s [14]. In addition there is a transmission time of 3-5 s for the signal from the macula densa cells to reach the smooth muscle cells in the arteriolar wall [5, 14]. In total this delay is sufficient for the nephrons in normotensive rats to operate close to or slightly beyond a Hopf bifurcation [13, 15]. This is the point of transition where the equilibrium state becomes unstable, and self-sustained oscillations arise in the pressure and flow regulation. It is worth mentioning that there is evidence to show that similar oscillations occur in man [16].

Besides reacting to the TGF signal, the afferent arteriole also responds to variations in the pressure difference across the arteriolar wall. This response consists of a passive elastic component in parallel with an active muscular (or myogenic) component. A similar response appears to be involved in the au-toregulation of the blood flow to many other organs, and the significance of this element in the nephron pressure and flow regulation is clearly revealed in experiments where the spectral response to a noise input is determined [17]. Here, one observes a peak at frequencies considerably higher than the frequencies of the TGF regulation and corresponding to a typical arteriolar dynamics. Based on in vitro experiments on the strain-stress relationship for muscle strips, Feldberg et al. [18] have proposed a mathematical model for the reaction of the arteriolar wall in the individual nephron. This model will play an essential role in our description of the pressure and flow regulation for the nephron.

However, as previously noted, the functional units do not operate independently of one another. The nephrons are typically arranged in couples or triplets with their afferent arterioles branching off from a common interlobular artery (or cortical radial artery) [19], and this proximity allows them to interact in various ways. Experimental results by Holstein-Rathlou [20] show how neighboring nephrons tend to adjust their TGF-mediated pressure oscillations so as to attain a state of in-phase synchronization. Holstein-Rathlou has also demonstrated how microperfusion with artificial tubular fluid in one nephron affects the amplitude of the pressure variations in a neighboring nephron, and how reactivation of the oscillations in the first nephron may be followed by reactivation in the non-perfused neighbor [20].

As an illustration of these results, Fig. 9.2 shows how microperfusion into the proximal tubule of one nephron can influence the pressure oscillations in a neigh-


Fig 9.2. Results of a microperfusion experiment for a pair of neighboring nephrons. Arrows indicate the start and stop of the perfusion phase. In the mi-croperfused nephron (top trace), the tubular pressure oscillations are blocked during the perfusion. During the same period, the amplitude of the oscillations is reduced in the nonperfused nephron (lower trace).

boring nephron. In the microperfused nephron (top trace) the proximal tubular pressure oscillations are blocked during the microperfusion. Arrows indicate the start and stop of the perfusion phase. During the same period, the amplitude of the oscillations is decreased in the non-perfused nephron (lower trace). Note how the oscillations are reactivated simultaneously in both nephrons, and how they are in phase both before and after the microperfusion.

This type of cross-talk among the nephrons is assumed to be produced by signals transmitted along the afferent arterioles [20]. The mechanisms underlying such a coupling are not known in detail. However, two different types of interaction seem plausible,

(i) A direct coupling between the TGF mechanisms of neighboring nephrons. The presence of such an interaction is well-established experimentally, but the underlying cellular mechanisms remain less understood. It is likely that the coupling is associated with a vascular propagated response where electrical signals, initiated by the TGF of one nephron, travel across the smooth muscle cells in the arteriolar wall from the region close to the macula densa and upstream along the arteriole to the branching point with the arteriole from the neighboring nephron. Because of the relatively high speed at which such signals propagate as compared with the length of the vessels and the period of the TGF-mediated oscillations, this type of coupling tends to produce in-phase synchronization. If the afferent arteriole of one nephron is stimulated by the TGF-mechanism to contract, the vascular signals almost immediately reach the neighboring nephron and cause it to contract as well.

(ii) A much simpler type of coupling that we shall refer to as hemodynamic

25nl/min, mm Hg

I I 1 | l ! i

2 U 6 8 10 12 14


coupling. This coupling arises from the fact that if one nephron is stimulated by its TGF-mechanism to contract its afferent arteriole, then the hydrostatic pressure rises over the neighboring nephron, and the blood flow to this nephron increases. Half a period later when the increased blood flow activates the TGF-mechanism in the neighboring nephron and causes it to contract its afferent arteriole, the blood flow to this nephron is again reduced, and the blood flow to the first nephron increases. This type of coupling tends to produce out-of-phase or anti-phase synchronization between the pressure oscillations of the two nephrons. In reality, we expect both mechanisms to be present simultaneously and to compete for dominance. Depending on the precise structure of the arteriolar network this may cause one mechanism to be the stronger in certain parts of the kidney and the other mechanism to dominate in other parts. Let us end this introductory discussion by noting that simulation results for systems of interacting nephrons were published already by Jensen et al. [8] and by Bohr et al. [21]. These studies describe a variety of different synchronization patterns including a chess-board pattern of anti-phase synchronization for nephrons arranged in a square lattice. However, at the time when these studies were performed the physiological mechanisms underlying the nephron-nephron interaction were not yet understood. The present discussion of interacting nephrons is based on our recent publications [22, 23, 24, 25]. A more detailed description of the physiological processes that take place in the individual nephron may be found in Topics in Nonlinear Dynamics [10].

9.2 Single-Nephron Model

In our model of the individual nephron [22], the proximal tubule is considered an elastic structure with little or no flow resistance. The pressure Pt in the proximal tubule changes in response to differences between the in- and outgoing flows,

dP 1 -JT - -?T~ [Ffilt - Freab ~ -Fffen] , (9 .1) "£ ^tub

where, Ffm is the glomerular filtration rate and Ctub the elastic compliance of the tubule. The Henle flow,

F - Pt~ P<i r Hen — D

ttHen

(9.2)


is determined by the difference between the proximal (Pt) and the distal {P4) tubular pressures and by the flow resistance i?#en in the loop of Henle. This description is clearly a simplification, since a significant reabsorption of water and salts occurs during passage of the loop of Henle. However, within the physiologically relevant flow range it provides a good approximation to the experimentally determined pressure-flow relation for the loop of Henle [9].

As the filtrate flows into the descending limb of this loop, the NaCl concentration in the fluid surrounding the tubule increases significantly, and osmotic processes cause water to be reabsorbed. At the same time, salts and metabolic biproducts are secreted into the tubular fluid. In the ascending limb, on the other hand, the tubular wall is nearly impermeable to water. Here, the epithelial cells contain molecular pumps that transport sodium and chloride from the tubular fluid into the space between the nephrons (the interstitium). These processes are accounted for in considerable detail in the spatially extended model developed by Holstein-Rathlou et al. [13]. In the present model, the reabsorption Freab in the proximal tubule and the flow resistance R,Hen are treated as constants. Without affecting the composition much, the proximal tubule reabsorbs close to 60% of the ultrafiltrate produced by the glomerulus.

The glomerular filtration rate is expressed as [26]

FfM = (1 - Ha) ( l - g ) £ _ j _ £ (9.3)

where Ha is the hematocrit of the afferent arteriolar blood (i.e., the fraction that the blood cells constitute of the total blood volume at the entrance to the glomerular capillaries). Ca and Ce are the protein concentrations in the afferent and efferent plasma, respectively, and Ra is the flow resistance of the afferent arteriole. (Pa — Pg) /Ra determines the incoming blood flow. Multiplied by (1 — Ha) this gives the plasma flow. Finally, the factor (1 — Ca/Ce) relates the filtration rate to the change in protein concentration for the plasma remaining in the vessel.

The glomerular pressure Pg is determined by distributing the arterial to venous pressure drop between the afferent and the efferent arteriolar resistances, i.e., as the solution to the linear equation

Pg = Pv + Re (^jf^ - FfUt) , (9-4)

where the venous (Pv) and arterial (Pa) pressures and the efferent arteriolar


resistance Re are considered as constants. The protein concentration Ce in the efferent blood is determined from the

assumption that filtration equilibrium is established before the blood leaves the glomerular capillaries, i.e., that the glomerular hydrostatic pressure minus the efferent colloid osmotic pressure Posm equals the tubular pressure. The experimentally determined relation between the colloid osmotic pressure and the protein concentration C can be described as [27]

Posm = aC + bC2. (9.5)

This leads to an expression of the form

e 2b ^a2-4b(Pt-Pg)-a (9.6)

In the computer model the simultaneous equations (9.3), (9.4), and (9.6) are combined into a single third-order equation for Ce. For relevant values of the various parameters, this equation has a single positive solution.

The glomerular feedback is described by a sigmoidal relation between the muscular activation t/j of the afferent arteriole and the delayed Henle flow £3

min /r. « \

1 + exp [a {3x3/TFHeno - S)\

This expression is based on empirical results for the relation between the glomerular filtration and the flow into the loop of Henle as obtained in the aforementioned open-loop experiments [12]. In Eq. (9.7), V'max and tpmin denote, respectively, the maximum and the minimum values of the muscular activation, a determines the slope of the feedback curve. We have already discussed how this slope plays an important role in controlling the stability of the pressure and flow regulation. In the next section we shall use a as one of the main bifurcation parameters. S is the displacement of the curve along the flow axis, and Fneno is a normalization value for the Henle flow.

The delay in the tubuloglomerular feedback is represented by means of three first-order coupled differential equations,

dx\ „ 3 .„ „N

-£ = FHen ~ TfXl, (9.8)

^ = | ( « i - » > ) . (9-9)


^ = f ( * 2 - * 3 ) , (9-10)

with T being the total delay time. This representation implies that the delay is represented as a smoothed process, with xi,x% and £3 being intermediate variables in the delay chain. By using a delay of finite order we can implicitly account for dissipative phenomena in the form, for instance, of a damping of the pressure oscillations from the proximal to the distal tubule.

The afferent arteriole is divided into two serially coupled sections of which the first (representing a fraction ft of the total length) is assumed to have a constant flow (or hemodynamic) resistance, while the second (closer to the glomerulus) is capable of varying its diameter and hence the flow resistance in dependence of the tubuloglomerular feedback activation,

Ra = Rao[P+(l-(3)r-4]. (9.11)

Here, Rao denotes a normal value of the arteriolar resistance and r is the radius of the active part of the vessel, normalized relatively to its resting value. In accordance with Poiseuille's law for laminar flows, the hemodynamic resistance of the active part is assumed to vary inversely proportional to r4.

Experiments have shown that arterioles tend to perform damped, oscillatory contractions in response to external stimuli [28]. This behavior may be captured by a second-order differential equation of the form

^ + k%-Pav~ Peq = 0. (9.12) at1 at OJ

Here, A; is a characteristic time constant describing the damping of the arteriolar dynamics, and w is a parameter that controls the natural frequency of the oscillations.

Pa, = \{Pa~ (Pa ~ Pg) + P f l) (9-13)

is the average pressure in the active part of the arteriole, and Peq is the value of this pressure for which the arteriole is in equilibrium with its present radius at the existing muscular activation.

As previously noted, the reaction of the arteriolar wall to changes in the blood pressure is considered to consist of a passive, elastic component in parallel with an active, muscular response. The elastic component is determined by the properties of the connective tissue, which consists mostly of collagen and elastin.


The relation between strain e and elastic stress ae for homogeneous soft tissue may be described as [29]

ae = C0 (e^ - 1), (9.14)

where Co and 7 are constants characterizing the tissue. For very small values of e (76 <C 1), we have an approximately linear strain-stress relation. However, for larger e values, the stress rises exponentially with the strain.

The active component in the strain-stress relation appears to be surprisingly simple. For some emax the active stress aa is maximum, and on both sides the stress decreases almost linearly with |e —emax | . Moreover, the stress is proportional to the muscle tone tp. By numerically integrating the passive and active contributions across the arteriolar wall, one can establish a relation among the equilibrium pressure Peq, the normalized radius r, and the activation level ip [18]. This relation is based solely on the physical characteristics of the vessel wall. However, computation of this relation for every time step of the simulation model is time consuming. To speed up the process we use an approximation in the form of the analytic expression [22]

Peg = 2.4 x e10^14) + 1.6(r - 1) + V ( 1 + e t L - 0 + 7"2(r + 0 9 ) ) ' (9"15)

where Peq is expressed in kPa (1 kPa = 103 N/m2 = 7.5 mrriHg). The first two terms in (9.15) represent the pressure vs. radius relation for the nonactivated arteriole. It consists of an exponential and a linear term arising from the two terms in the expression (9.14) for ae. The terms proportional to ip represent the active response. This is approximately given by a sigmoidal term superimposed onto a linear term. The activation from the TGF mechanism is assumed to be determined by (9.7). The expression in (9.15) closely reproduces the output of the more complex, experimentally based relation [22].

The above equations complete our description of the single-nephron model. In total we have six coupled ordinary differential equations, each representing an essential physiological relation. Because of the need to numerically evaluate Ce in each integration step, the model cannot be brought onto an explicit form. The applied parameters in the single-nephron model are specified in Table 1. They have all been adopted from the experimental literature, and their specific origine is discussed in Jensen et al. [9].


Arterial blood pressure Venous blood pressure Distal tubular pressure Flow resistance of afferent arteriole Flow resistance of efferent arteriole Flow resistance in loop of Henle Elastic compliance of tubule Equilibrium flow in loop of Henle Proximal tubule reabsorption Arterial hematocrit Arterial plasma protein concentration Linear colloid osmotic coefficient Nonlinear colloid osmotic coefficient Lower activation limit Upper activation limit Equilibrium activation Damping of arteriolar dynamics Stiffness parameter Fraction of afferent arteriole Feedback delay (standard value) Feedback amplification (normal rats) Vascular coupling parameter Hemodynamic coupling parameter Elastic compliance of glomerulus

Pa = 13.3 kPa P„ = 1.3 kPa Pv = 1.3 kPa Ra = 2.3 kPa/(nl/s) Re = 1.9 kPa/(nl/s) P-Hen = 5.3 kPa/(nl/s) Ctub — 3.0 nl/kPa •FffenO = 0.2 nl/s

Freab = 0.3 nl/s

Ha = 0.5

Ca = 54 g/l

a = 22 • 10~3 kPa/(g/l)

b = 0.39 • 1 0 - 3 kPa/(g/l)2

V w = 0.20 V w = 0.44 i>eq = 0.38 jfc = 0.04 Is u) = 20 kPa • s2

P = 0.67 T = 16s a = 12 7 = 0.2 (base case) £ = 0.2 (base case) Cgi0 = 0.11 nl/kPa

Table 1. Parameters used in the nephron model. The experimental basis for most of these

parameters is discussed in [9].

9.3 Bifurcation Structure of the Single-Nephron Model

Figure 9.3 shows an example of a one-dimensional bifurcation diagram for the single-nephron model obtained by varying the slope a of the open-loop response characteristics (9.7) while keeping the other parameters constant. In particular, the delay in the feedback regulation is assumed to be T — 16 s, which agrees


with physiological expectations [14]. The diagram was constructed by combining a so-called brute force bifurcation diagram with a bifurcation diagram obtained by means of continuation methods [30, 31]. As previously noted, the continuation methods allow us follow stable as well as unstable periodic orbits under variation of a parameter and to find and identify the various bifurcations that the orbits undergo. Hence, in Fig. 9.3 fully drawn curves represent stable solutions and dotted curves represent unstable periodic solutions. (With two control parameters, the continuation technique can be used to follow curves of local bifurcations in parameter space as illustrated, for instance, in Fig. 9.6.)

0.9

0.8

0.7

0.6

0.5

0.4

18 20 22 24 26 28 30 32 34 a

Fig 9.3. One-dimensional bifurcation diagram for the single-nephron model obtained by varying the slope of the open-loop response characteristics. T = 16 s. Dotted curves represent unstable solutions determined by means of continuation techniques. Two saddle-node bifurcations of the period-1 cycle fold an uncompleted period-doubling structure over a complete period-doubling transition to chaos.

For a given value of a, the brute force bifurcation diagram displays all the values of the relative arteriolar radius r that the model attains when the steady state trajectory intersects a specified cross section (the Poincare section) in phase space. Due to the coexistence of several stable solutions, the brute force diagram must be obtained by scanning a in both directions. It may also be necessary to use several different initial conditions.

T T I I


For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the Figure). In this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PD\~2 in Fig. 9.3, this solution undergoes a period-doubling bifurcation, and in a certain interval of a-values the period-2 cycle is the only stable solution. This cycle is characterized by the fact that the stationary behavior performs two oscillations before it precisely repeats itself. As we continue to increase a, the period-2 solution undergoes a new period-doubling bifurcation at a = 22 (i.e., at the point denoted PDl~4). The presence of a stable period-4 cycle is revealed in Fig. 9.3 by the fact that r assumes four different values for the same value of a.

With further increase of a, the stable period-4 orbit undergoes two consecutive backwards period-doublings, so that the original period-1 cycle again becomes stable around a — 26. The stable period-1 cycle can hereafter be followed up to a = 31 where it is destabilized in a saddle-node bifurcation. The saddle cycle can be followed backwards in the bifurcation diagram (dotted curve) to a point near a = 22.5 where it undergoes a second saddle-node bifurcation, and a new stable period-1 orbit is born. This cycle has a considerably larger amplitude than the original period-1 cycle. As the parameter a is again increased, the new period-1 cycle undergoes a period-doubling cascade starting with the first period-doubling bifurcation at a = 25 and accumulating with the development of deterministic chaos near a = 27. At even higher values of a we notice the presence of a period-3 window near a = 28.5 and the appearance of a stable period-4 cycle around a = 33.5.

The above scenario is typical of nonlinear dynamical systems when the amplitude of the internally generated oscillations becomes sufficiently large. In the bifurcation diagram of Fig. 9.3 this occurs when the slope of the feedback characteristics exceeds a critical value. However, similar scenarios can be produced through variation of other parameters such as, for instance, the arterial pressure. This could explain the observation of chaos in normotensive rats made hypertensive by clipping one of the renal arteries. On a qualitative level, the bifurcation diagram also agrees with the aforementioned observation of a period-doubling in the response of a nephron to an external disturbance.

For normotensive rats, the typical operation point around a = 10 — 12 and T = 16 s falls near the Hopf bifurcation point. This agrees with the experimental finding that about 70% of the nephrons perform self-sustained


oscillations while the remaining show stable equilibrium behavior [5]. We can also imagine how the system is shifted back and forth across the Hopf bifurcation by variations in the arterial pressure. This explains the characteristic temporal behavior of the nephrons with periods of self-sustained oscillations interrupted by periods of stable equilibrium dynamics.

200 1.4 1.5 1.6 1.7 1.8

P/kPa

1.9 2.0

Fig 9.4. (a) Temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. (b) Corresponding phase plot. With the assumed parameters the model displays self-sustained oscillations in good agreement with the behavior observed for normotensive rats. The tubular pressure is given in kPa (lkPa = 7.5mmHg).

Figure 9.4(a) shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T — 16 s. All other parameters attain their standard values as listed in Table I. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out. With physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 9.4(b) shows the phase plot. Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion along the limit cycle proceeds in the clockwise direction.

As previously noted, spontaneously hypertensive rats (SHR) have significantly larger a-values than normal rats (a — 16.8 ± 12.0 vs. a = 11.4 ± 2.2


for normotensitive rats) [12]. On the other hand, it appears that the feedback delay is approximately the same for the two strains. Figure 9.5(a) shows an example of the chaotic pressure variations obtained for higher values of the TGF response. Here, a = 32 and T = 16 s. Under these conditions, the oscillations never repeat themselves, and calculations show that the largest Lyapunov exponent is positive [32]. The corresponding phase plot in Fig. 9.5(b) displays the characteristic picture of a chaotic attractor. One can interpret the behavior as resulting from an interplay between the rapid modulations associated with the arteriolar dynamics and the slower TGF-mediated oscillations. The two modes never get into step with one-another, however. We shall return to a discussion of the mode interaction and its significance for the synchronization phenomena in Sec. 9.5.

400

Fig 9.5. (a) Pressure variations obtained from the single-nephron model for a = 32 and T = 16 s. (b) Corresponding phase plot. With these parameters the model displays chaotic oscillations resembling the behavior observed for spontaneously hypertensive rats [9].

A more complete picture of the bifurcation structure of the single-nephron model is provided by the two-dimensional bifurcation diagram in Fig. 9.6. Here both the delay in the tubuloglomerular feedback and the slope a of the feedback characteristics are used as bifurcation parameters. Each time we pass one of the bifurcation curves in parameter plane, the steady state solution of the nephron model undergoes a qualitative change. Starting around a — 1.3 for T — 0, the lowest curve in the bifurcation diagram is the Hopf bifurcation curve. Below this curve, the model displays a stable equilibrium point, and above the curve the equilibrium point is unstable. The other curves represent either saddle-


node bifurcations (dotted curves) or period-doubling bifurcations (fully drawn curves). To the left we observe a very complicated structure of overlying period-doubling and saddle-node bifurcations. This structure is associated with 1 : 1 , 1 : 2, and 1 : 3 resonances between the arteriolar dynamics and the TGF-mechanism. However, with feedback delays of the order of 4 s, the structure falls outside the physiologically relevant parameter region.

35

30

25

20 b

15

10

5

0 0 4 8 12 16 20

77 s

Fig 9.6. Two-dimensional bifurcation diagram for the single-nephron model. The diagram illustrates the complicated bifurcation structure in the region of 1 : 1, 1 : 2, and 1 : 3 resonances between the arteriolar dynamics and the TGF-mediated oscillations. In the physiologically interesting regime around T = 16 s, another set of complicated period-doubling and saddle-node bifurcations occur. Here, we are operating close to a 1 : 4 (or 1 : 5) resonance.

The region of physiological interest is the region around T — 16 s. In this region we recover the bifurcation curves associated with the scenario described in connection with Fig. 9.3. As we scan vertically through the diagram for T = 16 s we first cross the Hopf bifurcation curve at a = 11. In the interval from a = 19 to a = 26 we pass the period-doubling curves at which the period-2 and the period-4 solutions first emerge and subsequently disappear again. These solutions are not affected by the lower branch of the saddle-node bifurcation curve. However, the reestablished period-1 solution that exists after we have passed out through the period-doubling curves is destabilized at the upper branch of the saddle-node curve, and the unstable (saddle) solution can


be followed down to the lower saddle-node bifurcation curve where a high-amplitude stable period-1 solution is born. This solution then proceeds to chaos through the uppermost period-doubling curve.

9.4 Coupled Nephrons

As illustrated in Fig. 9.7 the nephrons are often arranged in pairs or triplets that share a common interlobular artery [19]. Besides possible other mechanisms of interaction, this anatomical feature allows neighboring nephrons to influence each others blood supply either through electrical signals that activate the vascular smooth muscle cells of the neighboring nephron or through a simple hemodynamic coupling. The two mechanisms depend very differently on the precise structure of the arteriolar network. Hence, variations of this structure may determine which of the mechanisms that dominates. This is of considerable biological interest, because the effects produced by the two mechanisms tend to be opposite in phase, and their influence on the overall behavior of the nephron system may be very different.

Let us start by considering the vascular coupling. The muscular activation ip arises in the so-called juxtaglomerular apparatus and travels backwards along the afferent arteriole in a damped fashion. When it reaches the branching point with the arteriole from the neighboring nephron, it may propagate in the forward direction along that arteriole and start to contribute to its vascular response. In our model this type of cross-talk is represented by adding a contribution of the activation of one nephron to the activation of the other, i.e.,

Fig 9.7. Typical arrangement of a group of glomeruli with their afferent arterioles branching off from the same interlobular artery.

1pl,2tot - ^1,2 + 7^2,1 (9.16)

where 7 is the vascular coupling parameter, and V>i and fa are the uncoupled activation levels of the two nephrons as determined by their respective Henle flows in accordance with Eq. (9.7).


As previously mentioned, the vascular signals propagate very fast as compared with the length of the vessels relative to the period of the TGF-oscillations. As a first approach, the vascular coupling can therefore be considered as instantaneous. Experimentally one observes [33] that the magnitude of the activation decreases exponentially as the signal travels along a vessel. Hence, only a fraction of the activation from one nephron can contribute to the activation of the neighboring nephron, and 7 = e~'/'° < 1. Here, I is the propagation length for the coupling signal, and IQ is the characteristic length scale of the exponential decay. As a base case value, we shall use 7 = 0.2.

To implement the hemodynamic coupling, a piece of common interlobular artery is included in the system, and the total length of the incoming blood vessel is hereafter divided into a fraction e < f3 that is common to the two interacting nephrons, a fraction 1 — j3 that is affected by the TGF signal, and a remaining fraction /3 — e for which the flow resistance is considered to remain constant. As compared with the equilibrium resistance of the separate arterioles, the piece of shared artery is assumed to have half the flow resistance per unit length.

Defining Pe as the pressure at the branching point of the two arterioles, the equation of continuity for the blood flow reads

Pa ~ Pe _ Pe ~ Pgl Pe ~ Pgl /g ^ eRaQ/2 Ra\ Ra2

with

Ral = (/3 - e) Ra0 + (1 - p) Ra0r^ (9.18)

and

Ra2 = (P~e) RaO + (1 - £ ) RaO^4. (9.19)

Here, Rao denotes the total flow resistance for each of the two nephrons in equilibrium. r\ and r<i are the normalized radii of the active part of the afferent arterioles for nephron 1 and nephron 2, respectively, and Pg\ and Pgi are the corresponding glomerular pressures. As a base value of the hemodynamic coupling parameter we shall use e = 0.2.

Because of the implicit manner in which the glomerular pressure is related to the efferent colloid osmotic pressure and the filtration rate, direct solution of the set of coupled algebraic equations for the two-nephron system becomes


rather inefficient. Hence, for each nephron we have introduced the glomerular pressure Pg as a new state variable determined by

dP, 9J 1

Cgl0

P — P P • — P 1 e 1 g,i 1 g,i £ v — Ffilt,i (9.20)

dt (-sglo \ Ra,i Re

with i = 1,2. This implies that we consider the glomerulus as an elastic structure with a compliance Cg\0 and with a pressure variation determined by the imbalance between the incoming blood flow, the outgoing blood flow, and the glomerular filtration rate.

200

Fig 9.8. Phase plot for the steady-state behavior of one of the nephrons in the coupled-nephron model, a = 12, T — 16 s, e = 7 = 0.2. The two nephrons synchronize in phase and with a 1 : 4 synchronization between the two modes of the individual nephron.

Fig 9.9. Example of anti-phase synchronization in the temporal variation of the tubular pressures for two coupled periodically oscillating nephrons, a = 12, T = 16 s, e = 0.3, and 7 = 0.05. With these parameters, the hemodynamic coupling dominates.

From a physiological point of view, this formulation is well justified. Compared with the compliance of the proximal tubule, Cgi0 is likely to be quite small, so that the model becomes numerically stiff. In the limit Cgi0 —> 0, the set of differential equations reduces to the formulation with algebraic equations. Finite values of Cg\0 will change the damping of the system, and therefore also the details of the bifurcation structure. In practice, however, the model will not be affected significantly as long as the time constant Cgi0Reff is small compared with the periods of interest. Here, Reff denotes the effective flow resistance faced by Cgi0. For a simple estimate this resistance is determined by -j£— = •£- + ^-. In the present analysis we shall take Cgi0 = 0.11 nl/kPa. This gives Cgi0Reff = 0.12 s.


Figure 9.8 shows a phase plot for the steady-state behavior of one of the nephrons in the coupled nephron model. Here, we have displayed the normalized radius of the active part of the afferent arteriole vs. the proximal tubular pressure for 7 = e = 0.2. The two nephrons are assumed to have identical parameters, and with T = 16 s and a = 12 the uncoupled nephrons perform identical periodic motions with an arbitrary relation between their phases. The introduction of a coupling forces the nephrons to synchronize their phases. Depending on the initial conditions and on the relative strength of the two coupling mechanisms this synchronization may be either in phase or in anti-phase. The in-phase synchronization, which produces a symmetric motion for the coupled system, is favored if the vascular coupling is relatively strong. Anti-phase synchronization on the other hand, is more likely to occur in the presence of a strong hemodynamic coupling.

A typical example of anti-phase synchronization is demonstrated by the temporal variations of the tubular pressures of the two periodically oscillating nephrons in Fig. 9.9. Here, T = 16 s, a = 12, e = 0.3 and 7 = 0.05. With these parameters, the hemodynamic coupling dominates, and the nephrons operate precisely 180° out of phase.

The ability to synchronize is obviously not restricted to the case where the two nephrons are identical. In the presence of a small parameter mismatch between the nephrons, a sufficiently strong coupling will again force the nephrons to synchronize their pressure variations so that the periods become the same. In the nonlinear system each nephron will adjust its pressure regulation relative to the other so as to attain a precise 1:1 relation between the periods. This explains the experimental observation that many pairs of adjacent nephrons are found to exhibit precisely the same period, even though they cannot be expected to have identical parameters [20]. As long as the mismatch is small, the coupling strength required to synchronize the nephrons tend to scale in proportion with the size of the mismatch.

In the presence of a more significant parameter mismatch, the coupled nephrons will still tend to synchronize their motions. However, in this case 1:1 synchronization may not be attainable, and instead the dynamics will be attracted to a state where there is a rational relation (n : m with n and m being integers) between the periods. For different degrees of mismatch and different coupling strengths we expect to observe the full complexity of an Arnol'd tongue diagram with its associated devil's staircase of frequency-locked regimes


200

Fig 9.10. (a) Phase plot for one of the nephrons, and (b) temporal variation of the tubular pressures for a pair of coupled chaotically oscillating nephrons, a = 32, T = 16 s, and e = 7 = 0.2.

[34]. As discussed in Sec. 9.5, however, the problem is further complicated in the present case by the fact that the individual nephron involves two different modes. This allows for the phenomenon of multistability.

Let us hereafter examine the situation for larger values of a where the individual nephron exhibits chaotic dynamics. Figure 9.10(a) shows a phase plot for one of the nephrons in our two-nephron model for a = 32, T = 16 s, e = 0.0, and 7 = 0.2. Here we have introduced slight mismatch AT = 0.2 s in the delay times between the two nephrons, and as illustrated by the tubular pressure variations of Fig. 9.10(b), the nephrons follow different trajectories. However, the average period is precisely the same. This is a typical example of phase synchronization of two chaotic oscillators as discussed in Chapter 6.

Fig 9.11. Example of anti-phase synchronization of two chaotically oscillating nephrons. a = 32, T = 16 s, e = 0.3, and 7 = 0.05.


Let us finally consider a case where the hemodynamic coupling dominates the chaotic phase synchronization. Figure 9.11 shows an example of the type of dynamics that one can observe in this situation. Here, a = 32, T = 16 s, e = 0.3, and 7 = 0.05. Each nephron is found to produce a chaotic variation in its tubular pressure. The nephrons have synchronized their pressure variations with one another so that the average period is precisely the same, but the two nephrons clearly operate in anti-phase with one another.

9.5 Experimental Results

In order to study the interaction between the nephrons, experiments were performed with normotensive as well as with spontaneously hypertensive rats at the Department of Medical Physiology, University of Copenhagen and the Department of Physiology, Brown University [35].

During the experiments the rats were anesthetized, placed on a heated operating table to maintain the body temperature, and connected to a small animal respirator to ensure a proper oxygen supply to the blood. The frequency of the respirator was close to 1 Hz. This component is clearly visible in the frequency spectra of the observed tubular pressure variations. Also observable is the frequency of the freely beating heart, which typically gives a contribution in the 4-6 Hz regime. The frequencies involved in the nephron pressure and flow regulation are significantly lower and, presumably, not influenced much by the respiratory and cardiac forcing signals [9]

When exposing the surface of a kidney, small glass pipettes, allowing simultaneous pressure measurements, could be inserted into the proximal tubuli of a pair of adjacent, superficial nephrons. After the experiment, a vascular casting technique was applied to determine if the considered nephron pair shared a common piece of afferent arteriole. Only nephrons for which such a shared arteriolar segment was found showed clear evidence of synchronization, supporting the hypothesis that the nephron-nephron interaction is mediated by the network of incoming blood vessels [19, 36].

Figure 9.12 shows an example of the tubular pressure variations that one can observe for adjacent nephrons for a normotensive rat. For one of the nephrons, the pressure variations are drawn in black, and for the other nephron in grey. Both curves show fairly regular variations in the tubular pressures with a period of approximately 31 s. The amplitude is about 1.5 mmHg and the mean


imh "l ! ' /I I I I III I ' i l l ' l 1 I ,'i '• I:

Fig 9.12. Tubular pressure variations for a pair of coupled nephrons in a normotensive rat. The pressure variations remain nearly in phase for the entire observation time (or 25 periods of oscillation).

Fig 9.13. Anti-phase synchronization in the pressure variations for two neighboring nephrons in a normotensive rat. This type of synchronization is considered to be associated with a strong hemodynamic component in the coupling.

pressure is close to 13 mmHg. Inspection of the figure clearly reveals that the oscillations are synchronized and remain nearly in phase for the entire observation period (corresponding to 25 periods of oscillation).

Figure 9.13 shows an example of the opposite type of synchronization where the nephrons operate nearly 180° out of phase. These results are also from a normotensive rat. As previously mentioned, we consider anti-phase synchronization to be the signature of a strong hemodynamic component in the coupling, i.e., contraction of the afferent arteriole for one nephron causes the blood flow to the adjacent nephron to increase. In line with this interpretation, inspection of the vascular tree has shown that the nephrons in this case, while sharing an interlobular artery, are too far apart for the vascularly propagated coupling to be active.

Figures 9.14(a) and (b) show examples of the tubular pressure variations in pairs of neighboring nephrons for hypertensive rats. These oscillations are significantly more irregular than the oscillations displayed in Figs. 9.12 and 9.13 and, as previously discussed, it is likely that they can be ascribed to a chaotic dynamics. In spite of this irregularity, however, one can visually observe a certain degree of synchronization between the interacting nephrons. Figure 9.15


200 400 6O0 0 500 1000 1500 t/s t/s

Fig 9.14. Two examples (a and b) of the tubular pressure variations that one can observe in adjacent nephrons for hypertensive rats.

reproduces the results of a frequency analysis of the two pressure signals in Fig. 9.14(b). One can immediately identify the respiratory forcing signal at 1 Hz. The TGF-mediated oscillations produce the peak around 0.03 Hz, and the arteriolar oscillations show up as a relatively broad peak around 0.2 Hz. One can see how the spectral lines coincide for both the arteriolar oscillations and the TGF mediated oscillations. This implies that these oscillations are synchronized in frequency between the two interacting nephrons.

In order to investigate the problem of phase synchronization for the irregular pressure variations in hypertensive rats we have applied the method introduced by Rosenblum et al. [37, 38]. With this approach one can follow the temporal variation of the difference A$(i ) — <&2{t) — $i(*) between the instantaneous phases $i(t) and $2(i) for a pair of coupled chaotic oscillators. As discussed in Chapter 6, the instantaneous phase $(£) and amplitude A(t) for a signal s(t) with complicated (chaotic) dynamics may be defined from

A{t)emt) = s(t) + j s (t) (9.21)

where

00

(0 = W / ^ * (9.22)

Intel acting Nephrons 373

denotes the Hilbert transform of s(i), j being the imaginary unit. The notation PV implies that the integral should be evaluated in the sense of Cauchy principal value.

m : n phase synchronization between two oscillators is said to occur if

|n$2(<) - m$i(£) - C\ < (x (9.23)

where \i is a small parameter (LI < 2TT) that controls the allowed play in the phase locking. In particular, 1:1 phase synchronization is realized if the phase difference $2(*) — $i( t) remains bound to a small interval fj, around a mean value C. For systems subjected to external disturbances or noise one can only expect the condition for phase synchronization to be satisfied over finite periods of time, interrupted by characteristic jumps in A $ . Under these circumstances one can speak about a certain degree of phase synchronization if the periods of phase locking become significant compared to the characteristic periods of the interacting oscillators [39]. Alternatively, one can use the concept of frequency synchronization if the weaker condition

Fig 9.15. Spectral distribution of the irregular pressure variations iu Fig. 9.14(b). The peak at 1 Hz is the respiratory forcing signal.

A f l = ( n $ 2 ( * ) - m $ i ( t ) ) = 0 (9.24)

is satisfied. Here, () denotes time average, and AO is the difference in (mean) angular frequencies. As noted above, 1:1 frequency synchronization is already distinguishable from the spectral distribution of the experimental data.

Figure 9.16(a) shows the variation of the normalized phase difference A# /2TT

for the irregular pressure oscillations in Fig. 9.14(a). One can clearly see the locking intervals with intermediate phase slips. In particular, there is relatively long interval from t = 160 s to t = 460 s (corresponding approximately to six oscillations of the individual nephrons) where the phase difference remains practically constant. Figure 9.16(b) reproduces similar results for the irregular


pressure variations in Fig. 9.14(b). Here, we note in particular the interval from t = 400 s to t = 600 s (corresponding to eight oscillations of the individual nephrons) where the phase difference remains nearly constant. We also note that the phase slips typically assume a value of 2ir (or an integer number of 27r-jumps).

I . . . 1 . . . 1 . . . 1 _ , 5 l , 1 , , , , , 1 , 1 , 1

0 200 400 600 0 200 400 600 800 1000 1200 t / s t / s

Fig 9.16. Variation of the normalized phase difference A$/27r for the irregular pressure variations in Fig. 9.14(a) (a) and 9.14(b) (b).

We have analyzed the tubular pressure variations for about 10 pairs of chaotically oscillating nephrons. In most cases we have found indication of frequency synchronization and in some cases of phase synchronization. However, the above two examples (Figs. 9.16(a) and (b)) remain among the best. When judging this result, one has to consider that each nephron is surrounded by, and with varying strengths coupled to, several other nephrons. It should also be noted that, because of the interacting TGF-mediated and arteriolar oscillations, the chaotic dynamics in the nephrons is fairly complex and, hence, difficult to synchronize.

For comparison with the results obtained for the chaotically oscillating nephrons, Figs. 9.17(a) and (b) display the calculated variations in the normalized phase difference for the regularly oscillating nephron pairs in Figs. 9.12 and 9.13, respectively. For the interacting nephrons in Fig. 9.12, the phase difference is found to move in a narrow interval around A$/27r = 0, although with a tendency for the phase locking to destabilize towards the end of the trace. For the nephrons in Fig. 9.13, the phase difference moves around A<3? = n,


Fig 9.17. Normalized phase difference for the regular pressure variations in Figs. 9.12 and 9.13. Here one can clearly observe both in-phase ( A $ = 0) and anti-phase (A<3> = 7r) synchronization.

indicating the occurrence of anti-phase synchronization. In the next section we shall study the transitions to and between different regimes of synchronization in the two-nephron model. Particularly interesting in this connection is the role of multistability in the chaotic phase synchronization [40, 41].

9.6 Phase Multistability

Both the above mentioned experimental results [17] and our simulations reveal one of the most important features of the single-nephron model, namely the presence of two different time scales in the pressure and flow variations. Considering the model equations we can identify the two time scales in terms of (i) a low-frequency TGF-mediated oscillation with a period T/j = 2.2T arising from the delay in the tubuloglomerular feedback, and (ii) somewhat faster oscillations with a period Tv ss T/j/5 associated with the adjustment of the arteriolar radius.

To determine T/, and Tv in our numerical simulations we have used the mean return times of the trajectory to appropriately chosen Poincare sections

•Lv — < •'•ret ,: _n ^ < Tret . >, ur=0 " ' h

with < > denoting the average over many oscillations

(9.25)


0,21 ' i ^ -25 27 29 31 33 35

a

Fig 9.18. Two-mode oscillatory behavior in the single nephron model, (a) Rotation number diagram superimposed onto a bifurcation diagram obtained by means of 2D continuation (compare Fig. 9.6). (b) The rotation number r„/, as a function of the parameter a along the route A in (a); inserts in (b) show the phase projections on the (Pa, Pa) plane.


From these return times we can define the intra-nephron rotation number (i.e., the rotation number associated with the two-mode behavior of the individual nephron)

rvh = Tv/Th. (9.26)

rvh will be used to characterize the various forms of frequency locking between the two modes.

Superimposed onto the two-dimensional bifurcation diagram for the single-nephron model, Fig. 9.18 shows the existence of regions with different ratios between the two return times. In the bifurcation diagram (and in the following discussion) PD denotes a period-doubling bifurcation with the superscript indicating the period of the solution undergoing the transition, SN denotes a saddle-node bifurcation, and TB a torus birth bifurcation (also known as a secondary Hopf bifurcation).

With varying feedback delay, Fig. 9.18(a) shows how the two oscillatory modes can adjust their dynamics so as to attain different states with rational relations (n : m) between the periods. The main (1:1) synchronization regime is located near T = 2s (i.e., outside the figure), but regions of higher resonances (1:4, 1:5, and 1:6) are seen to exist in the physiologically interesting range of the delay time T € [12s, 20s].

While the transitions between the different locking regimes always involve bifurcations, bifurcations may also occur within the individual regime. A period-doubling transition, for instance, does not necessarily change r„/,, and the intra-nephron rotation number may remain constant through a complete period-doubling cascade and into the chaotic regime. This is illustrated in Fig. 9.18(b) where we have plotted rvh as a function of the feedback gain a along the route A as indicated in Fig. 9.18a. Phase projections from the various regimes are shown as inserts. Inspection of the figure clearly shows that rvh remains constant under the transition from regular 1:4 oscillations (for a — 25.0) to chaos (for a = 28.0), see inserts 1 and 2. With further evolution of the chaotic at-tractor (insert 3), the 1:4 mode locking is destroyed. In the interval around a = 31.5 we observe 2:9 mode locking.

We conclude that besides being regular or chaotic, the self-sustained pressure variations in the individual nephron can be classified as being synchronous or asynchronous with respect to the ratio between the two time scales that characterize the fast (vascular) mode and the slow (TGF mediated) mode, re-


spectively. As we shall see, this complexity in behavior may play an essential role in the synchronization between a pair of interacting nephrons. In these investigations we shall restrict ourselves to consider a parameter range around route A in Fig. 9.18a, i.e., a 6 [25,28] and T € [12s, 14s]. Moreover, we shall neglect the hemodynamic coupling (e = 0) and we shall adopt a more symmetrical representation of the vascular coupling, i.e.

^1,2 = ^1,2 + 7 (^2,1 ^ ^1,2) (3-27)

instead of the original Eq. (9.16). Symmetr ica l case T\ = Ti-

0.0-

0.0;

< fU d/ U t£& U

a

Fig 9.19. Simplified two-parameter bifurcation diagram for the coupled nephron model with Ti = T2 = 13.5s. 7 is coupling parameter, and a is the feedback stiength of the individual nephron.

Let us start by examining the bifurcations that occur in a system of two coupled identical nephrons as illustrated in Fig. 9.19. For T = 13.5s, the individual nephron exhibits stable period-1 dynamics in the entire interval between the Hopf bifurcation at a = 10.5 and the first period-doubling bifurcation at a = 25.52. However, for low coupling strengths and increasing a the corresponding in-phase solution loses its stability via a pitchfork bifurcation with the formation of two stable symmetrical solutions that each undergoes a cascade of period-doubling bifurcations. The corresponding bifurcational curves


are not depicted in Fig. 9.19. The synchronous anti-phase solution, on the other hand, is stable at low coupling strengths. Time and phase plots for this solution are illustrated in Fig. 9.20(a) for a = 26.0 and 7 = 0.01. The region of anti-phase solution is bounded by the slowly rising bifurcation curves TB. These curves extend all the way down to the first period doubling bifurcation for the individual nephron (lower curve) and to the point where the coupled system undergoes a pitchfork bifurcation (upper curve).

After the first period-doubling at a = 25.52, the individual nephron undergoes a series of additional period-doublings at a = 26.5, a = 26.7, etc. For the in-phase solution I\ of the coupled-nephron model, the corresponding transitions are delineated by the vertically running lines L+,PD2, and PDA. At L+ a stable in-phase period-2 solution arises (in a saddle-node bifurcation for the period-2 solution), and this solution hereafter undergoes a series of period-doubling bifurcations ending in a chaotic regime for a > 26.7. Figure 9.20(b) shows a couple of time and phase plots for the in-phase period-2 solution observed at a = 26.0 and 7 = 0.01. The other (nearly) in-phase period-2 solution I2 arises in a torus birth bifurcation. In its further development, this solution undergoes a torus bifurcation at the dotted line TB, and I2 hereafter produces resonant or nonresonant torus dynamics until it undergoes a transition to chaos via torus breakdown along the curve la- for large values of a. To the right of this curve we have two stable coexisting chaotic solutions. In one of these solutions, the two nephrons operate in complete synchrony. In the other solution they are synchronized with a small phase lag.

So far the picture is somewhat similar to prior findings for coupled Rossler oscillators [10, 42]. According to these results, the initial Hopf bifurcation for the individual oscillator with the introduction of coupling splits into a Hopf bifurcation producing a stable anti-phase solution and a Hopf bifurcation producing an unstable in-phase solution. In its further development, the first period-doubling bifurcation for the individual oscillator is replaced by a torus-birth bifurcation for the anti-phase solution in the coupled system. The in-phase solution produces a stable period-2 dynamics, which subsequently undergoes a period-doubling cascade to chaos.

However, due to the complicated dynamics of the single-nephron model, the coupled system allows for the appearance of additional solutions via so-called phase multistability [40, 41]. The nephrons may synchronize their slow TGF mediated dynamics with a phase difference corresponding (approximately) to an


2.2

2

1.8

1.4

1.2

1

(a)

0 20 40 60 80 100

t/s

0 20 40 60 80 100 D

t/S * \

Fig 9.20. Coexisting periodic solutions at a = 26.0, 7 = 0.01, and Ti = T2 = 13.5s. (a) represents the period-1 anti-phase solution A; (b) and (c) show the two period-2 solutions I\ and I2 with different phase shifts.


integer number of periods for the fast dynamics. And after a period-doubling, the nephrons may synchronize either in phase or in anti-phase with respect to the period-2 solution. Figure 9.20(c) illustrates the time and phase plots for such a solution. This is the solution that we have previously denoted J2. Thus for a = 26.0,7 = 0.01 and Tx = T2 = 13.5 s, the coupled nephron model displays three coexisting periodic solutions, an anti-phase period-1 solution (Fig. 9.20(a)), an in-phase period-2 solution (Fig. 9.20(b)), and another period-2 solution (Fig. 9.20(c)) that has a phase shift in the synchronization.

Nonidentical case T\ j^T%.

0.010

0.00!^

0.00I-

0.004:

0.002

;i , i i i , - ! " " 7 ' ' ' W

<3N /

Antiphase / sv / /

7 //

/ . ' j

PD 4 inphiise 1,2 '

' / \ V V /

\

i i i n i i i . n 11

13.4 13.5 13.6 TVs

Fig 9.21. Synchronization regions for three coexisting types of dynamics at a = 26.0 and 21 = 13.5 s. Note the nested character of the synchronization regimes for the two in-phase period-2 solutions.

Let us hereafter consider how the various dynamical regimes are effected by the introduction of a parameter mismatch between the functional units. To be concise we shall assume that the feedback delay 2\ for nephron 2 can differ from the delay 7\ = 13.5 s in nephron 1. Figure 9.21 shows the regions of stability in the (22,7) parameter plane for each of the three coexisting periodic solutions in Fig. 9.20. Inspection of the figure clearly shows that the two period-2 solutions /1 and h have close, but different stability regions (Arnol'd tongues). Moreover,


while the stability region for I\ is bounded by lines of saddle-node bifurcations, for 72 the region is bounded by torus-birth bifurcations. It has previously been shown by Postnov et al. [40] that this type of nested bifurcation structures is characteristic for systems exhibiting multistabihty because of the formation of subharmonics. Here, we observe a similar phenomenon for systems with two-mode dynamics.

The stability region for the anti-phase limit cycle appears not to be the classical resonance horn (Arnol'd tongue) and does not extend to small values of the coupling strength. It is bounded by period doubling lines PD where a Floquet multiplier becomes equal to —1, and the period-1 limit cycle losses its stability. However, no period-2 cycle appears on these bifurcation lines. We have found that each of the coupled subsystems is responsible for one PD line. For the symmetrical case T\ = Ti = 13.5 s, both bifurcations take place simultaneously, and the transition is diagnosed as a torus-birth bifurcation.

In the considered parameter range three coexisting synchronous solutions are detected. Two of them belong to different families arising from the period-doubling cascade in the individual system, and additional solutions may originate from these for larger a. The third mode is an anti-phase solution which only arises at finite coupling strengths.

9.7 Transition to Synchronous Chaotic Behavior

For weakly developed chaos the features of chaotic phase synchronization is quite well investigated using three dimensional models of chaotic oscillations [37, 38]. Our problem differs from previously studied cases because the individual oscillatory system has two modes that can be locked with each other. However, as we shall see, an interaction between the subsystems can break their mutual adjustment. It is also possible that a coupling can act in a different manner on the fast and slow oscillations. It is of interest to know to what extent the oscillatory modes adjust their motions in accordance to one another when the coupling is introduced. For the interacting systems we introduce two rotation numbers as follows:

r„ = T„i/T„2, rh = Thl/Th2 (9.28)

To provide more information, the variation of the phase difference is calculated separately for the slow and for the fast oscillations.


Let us consider the case of a = 27.30 that corresponds to a weakly developed chaotic attractor in the individual nephron. The coupling strength 7 and delay time Ti in the second nephron are varied. The obtained results are summarized in Fig. 9.22. Two distinct chaotic states can be detected and classified, respectively, as asynchronous and synchronous chaos. Outside the synchronization region, the phase projection has a square shape (Fig. 9.22 (a)). Both r-/, and rv

change continuously with T2 (Fig. 9.22 (b)), and the phase difference appears to be nearly uniformly distributed over the interval [0, 2-iv] for both time scales (Fig. 9.22 (c)).

Inside the synchronization region the projection of the phase trajectory while remaining chaotic changes its shape to become more alined with the main diagonal (Fig. 9.22 (a)). The rotation numbers Th and rv in this case are both equal to unity in every point of the synchronization area (Fig. 9.22 (b)). For both time scales there is a finite interval (located to around TT) of phase differences whose numerically calculated probability is equal to zero (Fig. 9.22 (c)). Thus, the phase difference for the synchronous chaotic oscillation is concentrated within a certain interval. This defines chaotic phase synchronization in the sense of Pikovsky and Rosenblum [37, 38].

The crosshatched trianglar zone on the (T2,7) parameter plane represents the region of stability of the synchronous chaotic attractor. Like the synchronization region for the periodic oscillations, it becomes wider with increasing coupling strength. Note that there are no qualitative differences in the dynamics of the slow and fast time scales. They both become synchronized at the same values of the control parameters. In this range of parameters the coupled two-mode chaotic oscillators operate as one-mode chaotic oscillators like the coupled Rossler systems discussed in Chapter 4. However, the two-mode oscillations demonstrate more complex behavior for other values of the control parameters.

With increasing a (moving to the right part of Fig. 9.19) the synchronized chaotic regime described in the previous section is destroyed. However, other transitions to chaotic synchronization can be found. The most interesting one, occurring at a RJ 28.0, is related to the bifurcations of the anti-phase family which originates from the limit cycle in Fig. 9.20 (a).

Figure 9.23 shows a diagram of the dynamical regimes. For the symmetric case Ti = T2 = 13.5 s (dashed line in Fig. 9.23), the period-one anti-phase limit cycle losses its stability when the coupling is decreased. However, for T2 < 13.5 s there is a range of a where the anti-phase solutions undergo a number of


0010

136

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JlO.OS o .

m-rpTrp

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A *J: L I"

Fig 9.22. Chaotic phase synchronization at a = 27.30. (a) Diagram of synchronous and asynchronous chaotic behavior with the corresponding (Pa, Pa) phase projections (in inserts); (b) The rotation numbers r/, (circles) and rv (squares) versus T2; (c) Distribution of the phase difference for 7 •=• 0.006 for asynchronous chaos (left) at T2 = 13.4 s and for synchronous chaos (right) at T2 = 13.5 s.


0.02C

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OJ

0.00?

0.004

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J- i-x- i_U-i_i- i - ._Li- i - . - i . - l , . i 12.0 12.4 12.8 13.2

1

_i= °

0.

0

Fig 9.23. Chaotic phase synchronization at a = 28.0: Transition from anti-phase to in-phase solutions, (a) Diagram of the main dynamical regimes; (b) rotation numbers rj, (circles) and r„ (squares) as functions of T2 at 7 = 0.01.

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bifurcations while maintaining synchrony. A reduction of the coupling strength 7 for Ti G [12.6s; 12.8s] leads to a period-doubling cascade and to the formation of an anti-phase chaotic regime (" A-chaos" zone in Fig. 9.23 (a)).

The anti-phase solution can lose its stability in two different ways:

(i) Increasing T<i produces a transition from synchronous to asynchronous chaos (panels (1) and (4) in Fig. 9.24) similar to the transition for the in-phase solution described in the previous section;

(ii) Decreasing T<L leads to a transition between the regimes of anti-phase and in-phase chaotic synchronization.

Let us consider the latter transition in more detail. Figure 9.23 (b) displays the rotation numbers r^ and rv plotted against T^ for 7 = 0.01. Besides the synchronous regime where both r^ and rv are equal to unity, we note that two different types of asynchronous behavior can be detected. For one of these rh = rv as it was demonstrated in the previous section. For Ti < 12.2, however, another type of asynchronous behavior arises which is characterized by the condition r/j 7 rv. This means that the rotation numbers rvh\ and rvh2 take different values in the interacting nephron models, i.e., the resonance relation between slow and fast oscillations is broken. Thus, the vascularly propagated coupling can lead to desynchronization of oscillations in coupled nephrons, and it can also break the entrainment of the time scales in each subsystem.

The plateau in Fig. 9.23(b) for Ti G [12.25s; 13.0s] indicates the presence of synchronous oscillations in the two nephrons. However, the different parts of this interval correspond to different oscillatory regimes. Namely, with varying T2 the anti-phase chaotic attractor (panel (1) in Fig. 9.24) transforms to an in-phase attractor at T2 « 12.5 s (panel (3) in Fig. 9.24). In panel (2) of Fig. 9.24 one clearly sees the intermediate state of desynchronization when the chaotic regime continuously drifts over different phase relations, sharing the signs of both in-phase and anti-phase regimes. Thus the appearance of anti-phase synchronization and transition to in-phase regime is related to vascularly propagated coupling that influences the fast and slow oscillatory modes in different ways.

Since we investigate a two-mode self-sustained oscillator, the following questions arise: Is it possible to reach a state of partial synchronization where only one of the mode is synchronized? How does it manifest itself in the regime of chaotic phase synchronization? It is known that desynchronization can take place either through varying nonlinearity or via increased mismatch parameter.


Fig 9.24. Chaotic phase synchronization at a = 28.0: The phase projections (Pa, Pa) and the distribution of the phase difference at the points marked in Fig. 9.23 (a)


Let us consider the first of these cases.

0.6 0.15

0 Jt/2 JI 3n/2 2JI

0.05

0 itfl Ji 3it/2 2%

Fig 9.25. Partial phase desynchronization when a changes from 27.2 (a) to 28.0 (b) at 7 = 0.06. Both h and v frequencies remain locked, but the phase synchronization of the fast mode is lost.

Figure 9.25 presents the phase distribution of the fast and slow oscillations. The distributions in Fig. 9.25(a) are clearly in accordance with the definition of phase synchronization for chaotic oscillations. The distribution functions are bounded and localized in the vicinity of some average value around 0 (and 2w). With increasing nonlinearity parameter a, the phase difference for the slow oscillations A(f>k maintains the the same distribution while the phase distribution of the fast oscillations A<f>v indicates the destruction of phase synchronization. Thus, with increasing a, the two internal time scales demonstrate different phase coherent properties. Note, however, that the nephrons remain frequency locked (insert 2 in Fig. 9.18(b)) because of the homogeneity of the interacting functional units. We conclude that a regime classified as partial phase synchronization of chaotic two-mode oscillations can be observed.

In all of this chapter we have considered a fully deterministic description both of the function of individual nephron and the nephron-nephron interaction. With physiologically realistic mechanisms and with independently determined parameters this has allowed us to explain how the pressure and flow regulation in the nephron becomes unstable in a Hopf bifurcation and how more complecated dynamics can arise as the feedback gain is increased. For coupled nephrons we have been able to explain both the observation of in-phase and anti-phase synchronization in the pressure variations for neighboring nephrons in normotensive rats and of chaotic phase synchronization in hypertensive rats.

In practice the nephrons exist and operate in a very noisy environment. The influence of noise is partly illustrated in Fig. 9.16 where the chaotic phase


synchronization is interrupted by phase jumps where synchronization is momentarily lost. Noise is also expected to wash out many of details in our bifurcation diagrams, and further investigations obviously have to consider this phenomenon in detail. A preliminary discussion of the effect of noise in nonlinear dynamic systems is given in Chapter 10. Here, we shall also consider the constructive effects that noise can have in biological systems.

Another problem of considerable interest concerns the range of the synchronization between the nephrons. Since the arteriolar network can be mapped out and the length and diameters of various vessels determined, it is possible to obtain an independent estimate of the typical strength of the hemodynamic coupling and its variation across the kidney. Similarly, determination of the decay length for the vascularly propagated signal will allow us to estimate the parameters of that coupling. The typical length of the vascular segments separating neighboring glomeruli is of the order of 250 — 300 um. This is only about 30% of the distance that the vascular signal is expected to propagate, suggesting that larger groups of nephrons might act in synchrony.

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[8] K.S. Jensen, N.-H. Holstein-Rathlou, P.P. Leyssac, E. Mosekilde, and D.R. Rasmussen, Chaos in a System of Interacting Nephrons, in Chaos in Biological Systems, edited by H. Degn, A.V. Holden, and L.F. Olsen (Plenum, New York, 1987), pp. 23-32.

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[12] P.P. Leyssac and N.-H. Holstein-Rathlou, Tubulo-Glomerular Feedback Response: Enhancement in Adult Spontaneously Hypertensive Rats and Effects of Anaesthetics, Pfliigers Archiv 413, 267-272 (1989).

[13] N.-H. Holstein-Rathlou and D.J. Marsh, A Dynamic Model of the Tubu-loglomerular Feedback Mechanism, Am. J. Physiol. 258, F1448-F1459 (1990).

[14] N.-H. Holstein-Rathlou and D.J. Marsh, Oscillations of Tubular Pressure, Flow, and Distal Chloride Concentration in Rats, Am. J. Physiol. 256, F1007-F1014 (1989).

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[19] D. Casellas, M. Dupont, N. Bouriquet, L.C. Moore, A. Artuso and A. Mim-ran Anatomic Pairing of Afferent Arterioles and Renin Cell Distribution in Rat Kidneys, Am. J. Physiol. 267, F931-F936 (1994).

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[21] H. Bohr, K.S. Jensen, T. Petersen, B. Rathjen, and E. Mosekilde, Parallel Computer Simulation of Nearest-Neighbor Interaction in a System of Nephrons, Parallel Comp. 12, 113-120 (1989).

[22] M. Barfred, E. Mosekilde, and N.-H. Holstein-Rathlou, Bifurcation Analysis of Nephron Pressure and Flow Regulation, Chaos 6, 280-287 (1996).

[23] M.D. Andersen, N. Carlsson, E. Mosekilde, and N.-H. Holstein-Rathlou, Dynamic Model of Nephron-Nephron Interaction in Membrane Transport and Renal Physiology, edited by H. Layton and A. Weinstein (Springer-Verlag, New York, 2001).

[24] N.-H. Holstein-Rathlou, K.-P. Yip, O.V. Sosnovtseva, and E. Mosekilde, Synchronization Phenomena in Nephron-Nephron Interaction, Chaos 11, 417-426 (2001).

[25] D.E. Postnov, O.V. Sosnovtseva, E. Mosekilde, and N.-H. Holstein-Rathlou, Cooperative Phase Dynamics in Coupled Nephrons, Int. J. Mod. Phys. B 15, 3079-3098 (2001).


N.-H. Holstein-Rathlou and P.P. Leyssac, Oscillations in the Proximal In-tratubular Pressure: A Mathematical Model, Am. J. Physiol. 252, F560-F572 (1987).

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Y.-M. Chen, K.-P. Yip, D.J. Marsh and N.-H. Holstein-Rathlou, Magnitude of TGF-Initiated Nephron-Nephron Interactions is Increased in SHR, Am. J. Physiol. 269, F198-F204 (1995).

V.I. Arnol'd, Small Denominators. I. Mapping of the Circumference onto Itself, Am. Math. Soc. Transl. Ser. 2 46, 213-284 (1965).

K.-P. Yip, N.-H. Holstein-Rathlou, and D.J. Marsh, Dynamics of TGF-Initiated Nephron-Nephron Interactions in Normotensive rats and SHR, Am. J. Physiol. 262, F980-F988 (1992).

O. Kallskog and D.J. Marsh, TGF-Initiated Vascular Interactions between Adjacent Nephrons in the Rat Kidney, Am. J. Physiol. 259, F60-F64 (1990).

M.G. Rosenblum, A.S. Pikovsky, and J. Kurths, Phase Synchronization of Chaotic Oscillators, Phys. Rev. Lett. 76, 1804-1807 (1996).

A.S. Pikovsky, M.G. Rosenblum, G.V. Osipov, and J. Kurths, Phase Synchronization of Chaotic Oscillators by External Driving, Physica D 104 219-238 (1997).


[39] L.R. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963).

[40] D.E. Postnov, T.E. Vadivasova, O.V. Sosnovtseva, A.G. Balanov, V.S. An-ishchenko, and E. Mosekilde, Role of Multistability in the Transition to Chaotic Phase Synchronization, Chaos 9, 227-232 (1999).

[41] T.E. Vadivasova, O.V. Sosnovtseva, A.G. Balanov, and V.V. Astakhov Phase Multistability of Synchronous Chaotic Oscillations, Discrete Dynamics in Nature and Society 4, 231-243 (2000).

[42] J. Rasmusen, E. Mosekilde, and Chr. Reick, Bifurcations in Two Coupled Rossler Systems, Math. Comp. Sim. 40, 247-270 (1996).

Chapter 10

C O H E R E N C E R E S O N A N C E OSCILLATORS

10.1 But What about the Noise?

Biological systems are often characterized by an extraordinary degree of inter-connectedness with feedback regulations over many different time scales and via a range of different mechanisms [1] . As discussed in Chapter 1, one can envisage, for instance, how the different hormonal control systems of the human body interact so that the release of one hormone influences the release of a spectrum of other hormones that again influence the secretion of new hormones. At the same time, several hormones may have overlapping effects so that they can substitute for one another with respect to certain functions. Other hormones may have antagonistic effects and suppress the function of each other. It is well-known, for instance, that body-builders who take anabolic steroides and testosteron to increase their muscle mass may develop female attributes. If the body artificially receives surplus testosteron, enzymes in the fat tissue will transform some of the hormone into estrogen, and it is estimated that removal of bosoms ranges between the five most common plastic surgical operations on men in California. In this perspective, one must view the endocrine control as an orchestrated action of the hormonal system as a whole [2]. Similarly, the exchange of genetic material between microorganisms may sometimes proceed so effectively that one may become inclined to consider the microbiological world as one large organism. Even for macroecological systems, the complexity in the

395


interspecies interaction is such that an intervention often has ramifications in completely unexpected directions.

The high degree of interconnectedness is also manifest both for the system of functional units in the kidney, discussed in Chapter 9, and for the insulin producing pancreatic cells, considered in Chapter 5. The individual nephron has a well-defined structure and an associated set of functions and regulatory mechanisms. However, the nephrons interact with one another, and this interaction appears to be strong enough to provide the assembly of nephrons with a set of functions and behaviors of its own. Regulation of the functioning of the kidney may partly involve shifts between different forms of collaboration among the nephrons.

In many respects, this interconnectedness of the biological systems represents one of the most challenging problems to the mathematical and physical sciences in the coming decades. From conventional physics we are used to applying a number of standard approximations of which the separation of modes (for weakly interacting systems) and the separation of time scales (the so-called adiabatic approximation) are some of the most significant. As we have seen, for instance in connection with our discussion of the bifurcation diagram for pancreatic /3-cells, these procedures are not particularly helpful for nonlinear dynamical systems. The essential bifurcations involved in the transition from continuous spiking to bursting behavior cannot be explained in terms of the separated fast and slow subsystems, but involves the full three dimensional dynamics. Moreover the cells interact with one another via a variety of local and global coupling mechanisms, and regulation of the functioning of the pancreas may again involve shifts between different forms of synchronization between the cells.

An approach to the problem of biological interconnectedness that has gained much support in the last couple of decades is to consider large assemblies of simple, more or less identical units in the form, for instance, of neural networks, coupled map lattices or arrays of coupled nonlinear oscillators [3, 4, 5]. The approach that we have chosen in the present book has been to try to maintain a more detailed description of the complex behavior of the individual units and of the physiological interaction mechanisms. This has, of course, to a certain extent been at the expense of the number of units that we have been able to consider.

A direct consequence of the interconnectedness we observe in the living world

Coherence Resonance Oscillators 397

is that delimitation of functional units (or definition of system boundaries) becomes difficult. Time series obtained to elucidate the mechanisms of a particular process become corrupted by uncontrolled external influences. The temporal variation of the concentration of a particular hormone in the blood, for instance, may be modified by a wealth of other rhythms that are not simultaneously recorded. For many years, this type of difficulty has hampered efforts to interprete data on heart rate variability. In the early stages of application of nonlinear dynamics to biological data analysis, attempts were made to associate the heart rate variability with some kind of internal chaotic dynamics. By now, we suppose, most investigators recognize that the control of the heart involves signals from a variety of different sources, and that for many purposes a good part of the heart rate variabily is better described as noise.

Our analyses of nephron pressure and flow regulation in the rat has met with the same type of difficulty. The individual nephron interacts with neighboring nephrons, and this interaction forces us to consider systems of larger and larger dimension. At the same time, the nephrons adjust their function in accordance with variations in the arterial blood pressure associated, for instance, with changes in the activity of the animal. Experimental data for the tubular pressure variations also contain relatively strong heart rate and ventilatory components, components that in the present context are most reasonably treated as random noise.

For the pancreatic /3-cells we have met the problem of noise in a very different form, namely as a consequence of the limited number of potassium channels in a single cell. Because of their relatively low number, the random opening and closing of individual potassium channels influence the function of the cell to such an extent that it cannot display regular bursting dynamics. Stochastic processes associated with the finite number of molecules involved in biological processes at the cellular level are also known to arise in a variety of other problems, including, for instance, cell size regulation [6]. Hence, we conclude that noise is an important problem that must be considered in the description of many biological systems.

From the point of view of nonlinear dynamics, one of the most essential effects of noise is to wash out some of the detailed structures in the bifurcation diagrams [7, 8]. Application of noise to a period-doubling system will truncate the bifurcation sequence by opening a so-called bifurcation gap around the accumulation point for the period-doubling cascade. As soon as the noise


amplitude becomes comparable to the trajectory splitting for a (high-periodic) orbit, the subsequent bifurcations can no longer be observed. And at the same time, the first reverse (or chaotic band merging) bifurcation on the other side of the accumulation point disappear. Considering the noise level at which the individual nephron operates, we conclude that it will never be possible experimentally to verify the details of the bifucation diagram for the single nephron model that we presented in Fig. 9.6.

Crutchfield et al. [9] and Shraiman et al. [10] developed a renormalization group theory for the effect of noise on the period-doubling cascade. They showed that each time the noise intensity is increased by a certain (asymptotically universal) factor, a new pair of bifurcations one either side of the accumulation point become unresolvable. Arecchi et al. [11] studied the forced Duffing equation in a parameter range where five different attractors coexist. They showed how external noise can bridge the otherwise disjoint basins of attraction for the various states and how the noise-induced transitions can be described in terms of simple kinetic equations. Finally, Ott et al. have studied how the volume of a chaotic attractor depends on the noise level [12], and Herzel has shown that application of random noise can stabilize a chaotic orbit [13].

For excitable systems, the situation is quite different. An excitable system, of which many types of nerve cells may serve as examples, is characterized by having a stable equilibrium point close to a separatrix in phase space. If the system is excited beyond this separatrix (the excitation threshold) it will perform an excursion in phase space (a spike) and then return to its stable equilibrium point. Receptor nerves in living organisms may typically have developed in such a way that the excitation threshold exceeds the characteristic noise level in the surroundings by a (small) margin. In the presence of a signal, even of relatively low amplitude, the threshold may be crossed whenever the signal has a maximum, as the noise is superimposed on the signal. This mechanism allows the biological system to detect signals that nearly disappear in the noise background [14]. On the other hand, as long as the amplitude of the signal is subthreshold, it can be detected only in the presence of noise of a proper intensity. This is an example where noise plays a constructive role in the functioning of biological systems, and the observation that such phenomena actually occur in nature [14] has generated an upsurge of interest in the area of stochastic resonance [15, 16, 17].

The purpose of the present chapter is to discuss a related phenomenon that is


called autonomous stochastic resonance [18, 19] or coherence resonance [20, 21]. This phenomenon has also been applied to analyze the dynamics of various nerve models [22, 23]. Here, the emphasis is on the ability of an excitable system to generate a pseudo-regular, oscillatory dynamics in response to a random external excitation. A similar phenomenon has been described by Mosekilde et al. [6] for a model of the co-evolution of bacteria and bacteriophages. Here, the random processes by which the bacterial cells acquire resistance towards a paticular virus strain (a so-called lysogenic response to the phage attack) play the role of noise. The macroscopic (or deterministic) dynamics is controlled by the regrowth of the original, non-resistant cells when the virus infection is over. Such regrowth takes place because of the usually slightly higher growth rate for the unmodified bacterial cells compared to the resistant cells. Under proper conditions, this type of system will also develop a pseudo-regular, oscillatory dynamics of virus infections of the cell culture. In many ways, such systems behave like limit cycle oscillators, and in the following sections we shall study the different types of synchronization that can arise between coherence resonance oscillators [24, 25].

For regular oscillations, when phase locking takes place, a stabilization of the phase shift between the interacting modes occurs, and the frequencies become equal. As discussed in Chapter 6, the classical results for the regular oscillations can be generalized to some classes of chaotic oscillations. It was shown in particular that synchronization in systems demonstrating the period-doubling route to chaos can be described in terms of a locking of the average frequencies [26, 27]. Following a suggestion made by Rosenblum and Pikovsky [28, 29], synchronization of chaotic systems was subsequently extended to involve a locking of the instantaneous phases.

Synchronization phenomena have also been investigated in nonlinear stochastic systems. Locking of mean switching frequencies and some kind of phase locking have been discovered both in periodically forced and in coupled noise-driven bistable systems [30, 31, 32, 33], and even for noisy signals, the phase description has been found to be useful for the analysis of synchronization, for instance, in the human cardio-respiratory system [34]. These investigations were based on the classic approach to synchronization in the presence of noise as developed by Stratonovich [35]. This implies that the phase locking for stochastic systems is considered as a state lasting for a finite time and described in terms of phase diffusion [33] or via the shape of the distribution function for


the phase difference [34]. Nonlinear systems perturbed by noise display a wide spectrum of addi

tional complex phenomena, ranging from noise-induced chaos [36, 37] and noise-induced order [13, 38] to stochastic ratchets [39, 40].

In this chapter, we shall study the synchronization of coupled nonidentical excitable systems that each operates in a regime of coherence resonance. The noise intensity governs the frequency of the noise-induced oscillations and can, therefore, be considered as a frequency mismatch parameter. The transition from the nonsynchronous to the synchronous state is signaled by the merging of the peak frequencies in the power spectra and also by a strongly uniform distribution of instantaneous phase differences. With a small mismatch, the transition occurs via a frequency locking of noise-induced oscillations. For large mismatch, the transition is related to the suppression of the peak frequency.

10.2 Coherence Resonance

Coherence resonance [18, 19] manifests itself as a more or less regular oscillatory response of an excitable system to the application of noise of a proper magnitude. In this case there is no external forcing involved. However, the excitable system exhibits a characteristic time constant associated with the duration of a spike (or pulse) when the system is excited. Pikovsky and Kurths [20] used this observation to explain the characteristics of coherence resonance in terms of different noise dependences for the activation (or excitation) and excursion (or relaxation) times.

To examine the phenomenon of coherence resonance in some detail we shall consider the Morris-Lecar (ML) model [41] that describes the spiking and refractory dynamics of a nerve cell. The Morris-Lecar model is a simplification of the original Hodgkin-Huxley model [42] that also formed the basis for the /3-cell model considered in Chapter 5. In contrast to the /3-cell model, the ML-model is two-dimensional (and, hence, cannot show bursting dynamics nor period-doublings and chaos), and the parameters have been adjusted such that the equilibrium point is stable. However, application of noise of a proper magnitude can bring the system across a separatrix in phase space, upon which it spikes and returns to the stable equilibrium point. The Morris-Lecar model includes a calcium current generating fast action potentials and a delayed rectifier potassium current. To maintain a constant potential in the resting state,


a leak current is also taken into account. Two diffusively coupled ML models may be written as:

dvi; ,2 = I%on{vi,2, Wlfi) + I + -Dl.2^1,2 W + ff(«n,l ~ Vlfi),

(10.1)

dt dwh2 _ ^00(^1,2) - Wl,2

dt Tooî^)

where

Iion{v,w) = gCain00{v)(vCa-v) + gKU)(VK-v)+gL(vL-v),

m00(u) = [1 + tanh{(u - va)/vb}]/2,

w^v) = [l + tanh{(v-vc)/vd}]/2, Tooiv) = l /cosh{(v-w c ) / (2v d )} .

Here, v denotes the transmembrane voltage of the neuron and w represents the activation of the potassium current. I is the external stimulus current and £1^ denote noncorrelated sources of Gaussian noise with intensities D\2- The last term in the first line of Eqs. (10.1) represents the diffusive interaction between the two cells with a coupling strength g. The parameter set used in our simulations is: I = 0.23, va = —0.01, Vb = 0.15, vc — 0.0, Vd = 0.3, gca = 1-1, gK = 2.0, <7L = 0.5, vca — 1-0, VR — —0.7, VL = —0.5, and the time separation parameter e = 0.02. vca, VK, and vi represent the reversal potentials associated with the different currents, and gca, gK, and <jx are the corresponding conductances. Like the Sherman model discussed in Chapter 5, the conductances are measured in terms of some typical conductance. Potentials are measured in Votts. For a detailed explanation of the remaining parameters we refer the reader to the original literature [41]. The subscript n in Eqs. (10.1) determines the different types of interaction. If n = 1, a unidirectional interaction is realized; the first system being the "master" and the second the "slave". If n = 2, the systems are mutually coupled.

Besides numerical simulations of the ML model, we shall also present experimental results obtained with a monovibrator circuit that generates a single electric pulse whenever the external signal exceeds a threshold level [43]. The wiring scheme of two coupled monovibrator circuits is shown in Fig. 10.1. This system is described by the following dynamical equations:

£~JT = X(î,2 - J/1,2 - (A,2î,2(t) + a a ; i ,2 + 7 ^ ) ) - J / i , 2 ,


Fig 10.1. Electrical scheme for the coupled monovibrator circuit. The two units are identical, but the noise sources (Va and Va) are independent.

dt = X\,2 ~ 2/1,2 + S(Z1,2 - J/1,2 - X2,l + J/2,l), (10.2)

where x\$ a r e voltages at the outputs of the operational amplifiers and 3/ 2 are voltage drops across the capacitors C. The constants a and 7 are positive and defined in terms of the values of the resistors R\, R2, -R3, and Rj. Vb represents the normalized threshold voltage. The function x is a sign-function that takes values of + 1 and —1 for positive and negative arguments, respectively. Two independent noise sources £i;2 with noise intensities Dip are introduced.

Figure 10.2 displays the typical shapes of the power spectra observed in the regime of coherence resonance for the relaxation-type neuron model (10.1) with vanishing interaction (g = 0). Each spectrum possesses a well-defined global maximum that could be interpreted as a natural frequency of oscillation. This type of regularized behavior is observed within a finite range of noise intensities.

It is interesting to see the similar influence of the noise intensity on the features of the averaged power spectra for the monovibrator system (10.2). This is illustrated in Fig. 10.3. With a small noise intensity (D C 0.1 V2), the monovibrator generates pulses of duration r m To — —RC\n{{Vi,/E + l ) / 2} . The time intervals between the pulses are much longer than T. Thus, the resulting power spectrum represents a superposition of appearingly random pulses, and a smooth, broad peak can be observed at low frequencies (curve 1).

For an optimal noise strength D pa 0.1 V2 the pauses between pulses are approximately equal to the pulse duration. The corresponding peak in the power spectrum is sharp and relatively high (Fig. 10.3, curve 2). Finally curve 3 illustrates what happens when the noise intensity becomes too high. The peak


,§. 20

D=0.001

D=O.O001 L J \ )F D=0.01

^ g ^ r r

CO

Fig 10.2. Power spectra for the noise-driven Morris-Lecar model in the regime of coherence resonance. D is the noise intensity.

f(Hz)

Fig 10.3. Electronic experiment: Evolution of the power spectrum for the noise-driven monovibrator circuit. Curves 1, 2, and 3 correspond to D = 0.015, 0.1, and 0.6 V2, respectively.

is now much smaller relative to the level of the background noise, and it has been shifted to a relatively high frequency. Figure 10.4(a) illustrates how the peak frequency grows from 0 to approximately 1.5/RC as the noise strength increases. Thus, we observe a noise-induced time scale of the system, which is not a noisy precursor of deterministic behavior.

To characterize the degree of regularity in the coherence resonance we shall use a quantity that can be interpreted as the signal-to-noise ratio [18, 19]:

13 = hup/Aw. (10.3)

Here, up is the peak frequency in the power spectrum of the noise excited system, Au is the width of the peak, and h is the peak height normalized with respect to the noise background. wp/Au> is the familiar quality factor Q for a resonance circuit [35]. In the following sections, /3 will be referred to as the measure of regularity.

The ;0 vs. D curve in Fig. 10.4(b) clearly demonstrates the maximum in regularity at a finite noise intensity that is characteristic of coherence resonance. As discussed above, this maximum can be explained in terms of an optimal balance between the mean duration of a pulse generated by the monovibrator and the mean duration of the interpulse interval [20, 43]. For strong noise the pauses between pulse onsets tend to zero because the monovibrator is imme-


K

60

50

40

CO.

30

20

10

ri °

A> /o

t TO

1° O

O

Co

o\

\ o ° v

(b)

-

D (V2) 0.10

D (V2)

Fig 10.4. Electronic experiment, (a) Peak frequency and (b) measure of regularity /? vs. noise intensity D for C = 0.03/J.F and Vb = -1.016V.

diately pushed out from the equilibrium state. Strong noise can also disrupt the recharging process of the capacitor C. Thus, the pulse duration attains a random value. At the same time, the additive component of the noise manifests itself in the power spectrum. This leads to a decrease of the measure of regularity j3 when the noise intensity D increases beyond 0.1 V2 (Fig. 10.4(b)).

In view of the above results, a noise-driven excitable system can be considered as a "coherence resonance oscillator" whose behavior is characterized by a peak frequency governed by the noise intensity and by a phase defined as the position on a "stochastic limit cycle" [44, 45]. Hence, the question naturally arises [24]: Is it possible that interacting, nonidentical coherence resonance oscillators can adjust their motion in accordance with one another so as to attain a form of synchronization?

10.3 Mutual Synchronization

In this section, the synchronization of two diffusively coupled coherence resonance oscillators will be characterized in terms of locking of the peak frequencies as well as locking of the instantaneous phases. The transition from a non-synchronous to a synchronous state is diagnosed through changes in the Poincare sections and in the distribution of phase differences. We recall from the discussion in Sec. 10.2 that the noise intensity plays the role of a control


parameter governing the onset and the peak frequency of oscillations in the excitable system. To investigate the effect of a frequency mismatch on the synchronization of two CR oscillators, we therefore choose the noise intensity of the second oscillator to be different from that of the first system, and we refer to D-i as the mismatch parameter.

Fig 10.5. Frequency locking observed in the electronic experiment: (a) evolution of the normalized power spectrum for D\ = O.ll^2 (gray color) and D2 = 0.221^2 (black color) as the coupling strength g is varied; (b) the ratio of the peak frequencies (winding number) stabilizes near 1.0 for a range of D% with Dx = 0.086y2.

In Fig. 10.5(a), the evolution of the power spectra is plotted as a function of the coupling strength g for the coupled electronic monovibrators. It is clearly seen how the peak frequencies of the two oscillators approach each other to coincide at some value of g « 0.01. If the noise intensity D<i is varied (while D\ and g are fixed), a frequency-locked region is easily identified within a certain range of the noise intensity D% (Fig. 10.5(b)). Moreover, the frequency-locked interval tends to become broader as the coupling strength is increased.

Similar results were observed in our numerical simulations of the coupled ML system (Eqs. (10.1) with n = 2). In Fig. 10.6, we have ploted the phase diagram in the two-dimensional parameter space spanned by the coupling strength g and the mismatch parameter D%. The synchronization region, which clearly resembles the ArnoPd tongue known for coupled limit cycle oscillators, was obtained by requiring the peak frequencies to coincide within a small margin wi— u>2 < const = 0.0002. In the following paragraphs, the instantaneous phases of the two ML oscillators will be analyzed to provide an alternative diagnostics

406 Chaotic Synchronization; Applications to Living Systems

M)

synchronous

behavioi

Fig 10.6. Synchronization region for two coupled ML models. The noise intensity JD2 effectively plays the role of a frequency mismatch ( A = 0.001).

D,

of synchronization.

Fig 10.7. The phase difference in the coupled ML model as a function of time for non-synchronous (g = 0.02), nearly synchronous (g = 0.035) and synchronous (g = 0.08) states. D2 = 0.00075. The phase slips of 2ir for the nearly synchronous regime are clearly seen in the enlarged inset.

Already in the mid 1980's Treutlein and Schulten [44, 45] introduced the term stochastic limit cycle to describe noise-induced neural impulses. By connecting the most likely escape trajectory out of a stationary point with the most likely return trajectory back to that point, the system's state on this circular trajectory could be described in terms of phase-like variables. Neiman et al. [33] and Rosenblum et al. [34] showed how the instantaneous phases of stochastic oscillations can be locked. Similarly, the concept of instantaneous phases can be applied to discuss the synchronization of coupled CR oscillators The instantaneous phase is now defined as: 6(t) — 27r-*~"r*—I- 2ixk, where n is the time of the fc-th firing. (For additional details see, e.g., the paper on phase synchronization of chaotic oscillators by Rosenblum et al [28].) Based


on the phase variable for each ML system, the instantaneous phase difference is specified as A<j> = <j>\ — <f>2- As the coupling is increased, for a given frequency mismatch ACl, we observe a transition from a regime where phases rotate at different rates (A</> ~ Afit) to a synchronous state where the phase difference remains bounded, but oscillates around some mean value. Hence, there is no average (or long term) phase drift. This is illustrated in Fig. 10.7.

Phase locking for noisy systems may be observed during a long, but finite time only [33, 35]. Therefore, it has to be determined a priori for how long the phases should be locked (on average) to assert that a noisy system is effectively synchronized. We can decide, for instance, that the stochastic oscillations are synchronous if no 27r phase slip occurs during 50000 periods.

Fig 10.8. Distributions of the phase difference and Poincare sections (insets) for two coupled ML oscillators: (a) inside the synchronization region (g = 0.09), (b) near the boundary (g = 0.045), and (c) outside this region (g = 0.01). £>! = 0.01 and D2 = 0.0015. The Poincare section is specified by the condition Ui = 0.35. From these plots, one can clearly draw an analogy to the transition from a torus to a limit cycle in the deterministic case.

- n o t A<t>

Figure 10.8 illustrates the distribution function of the phase differences (measured during 50000 time periods) and the Poincare section for three discernible regimes (corresponding to the points A, B and C in Fig. 10.6, respectively). Inside the synchronization region (point A), the cross section is concentrated in a small area (Fig. 10.8(a)) and the distribution function appears to be limited to a finite range near a vanishing phase difference. But outside the synchronization region (point C), the Poincare section is completely different and takes

- (b)

X \

(c) a


the form of a ring in the phase space of the system (Fig. 10.8(c)). Moreover, the distribution of the phase differences is nearly homogeneous over 2TT. At the boundary of synchronization (point B), the Poincare section indicates a closed curve, but it is not equally dense everywhere (Fig. 10.8(b)). These results clearly allow us to draw an analogy between the transition from an ergodic torus to a limit cycle in the deterministic case and the evolution observed in the stochastic oscillations. On this background, we can complement the term "stochastic limit cycle" [44, 45] with the notion of a "stochastic torus".

10.4 Forced Synchronization

Let us now focus on the synchronization phenomena observed in the unidi-rectionally coupled ML model (n = 1 in Eqs. (10.1)). In this case, the two subsystems play the role of "master" and "slave", respectively. The noise intensity of the master system is taken at the optimal value {D\ = 0.001) while D<i is used as a mismatch parameter.

Fig 10.9. Synchronization region for the unidirectionally coupled ML models. Directions A and B indicate different transitions to synchronized behavior. Compare this figure with our discussion in Chapter 6 for the chaotic case.

As one readily observes, unidirectionally coupled CR oscillators have many features in common with the behavior observed in forced self-sustained systems. When the coupling strength is changed, various patterns of output response that depend on the interaction between the time scales of the two subsystems are elicited. Figure 10.9 displays the 1 : 1 synchronization region. At the boundary of this region the mean frequencies of the noise-induced oscillations coincide (u)\ — u>2 < const = 0.0002), and the frequencies remain identical within a range of mismatch parameters (i.e., inside the synchronization region). Transition to the synchronous regime can be realized either via mutual locking of the peak frequencies of the interacting units (direction A in Fig. 10.9) or via suppression

0 0 1 1 1 1 1 1 1 1 1 1 1 1

0 0.001 0.002 0.003 0.004 0.005 D2


g=0.04 g=0.06 g=0.08 g=0.20

0.05 0.07 w

Fig 10.10. Evolution of the power spectrum of the driven system (thick curve) along the direction A (Fig. 10.9). Locking of noise-induced frequencies. Thin and dashed curves correspond to the driving and driven systems without coupling.

g=0.05 g=0.08 g=0.10 g=0.40 4 0 I i i i i i i i I i i i i i i i i I I i i i i i i i i I i i i i i i i

30

^ 2 0 OH

10

1 1 1

-

-

1 1 1 1

1 i<

If!\ \ -J If V »*

* P T I i i i T -

1 1

-1 .

:

- 1 t *

«*p-i'

1 1 1 1

1 Ji -* i > i _

11 1 !' " i i T V

0.04 0.08

w Fig 10.11. Evolution of the power spectrum along the direction B in Fig. 10.9. This figure illustrates a transition via the suppression of the noise-induced frequency in the driven system.


of the noise-induced oscillations in one of the coupled subsystems by the signal of the other subsystem (direction B in Fig. 10.9) [46, 47, 48]. These are the same two synchronization scenarios we discussed for deterministic systems in Chapter 1 and for chaotic systems in Chapter 6.

Let us compare the evolution of the power spectrum and the measure of regularity in the two cases. Figure 10.10 clearly demonstrates how the peak frequency of the driven system approaches the frequency of the driving system. The height and width of the frequency peak of the "slave" system vary only slightly. The frequencies of the coupled systems coincide at the moment of synchronization (at the boundary of the synchronization region) and remain equal with increasing coupling strength (inside the synchronization region). In this case, when the peak frequencies coincide, the measure of regularity of the driven system becomes equal to its value for the driving system (Fig. 10.12(a)).

500 I

400 • I -

200 - \ I

100 - \ l

o.ooi o.oi 61 T ~~fo o.ooi o.oi o.i I 10 g g

Fig 10.12. The measure of regularity vs. coupling strength for unidirectionally coupled Morris-Lecar models: (a) D2 = 0.0015 and (b) D2 = 0.003 along the direction A and B, respectively. Horizontal solid and dashed lines indicate the regularity level for the "master" system and for the "slave" system without coupling, respectively.

Figure 10.11 shows how the transition to synchronous behavior for large mismatch parameters occurs in a different way. The peak frequency of the driven system keeps its constant value while its height decreases and the width becomes broader until it becomes difficult to resolve the peak from the noise background. At this moment suppression of this frequency has occurred, and the measure of regularity in the driven system reaches a minimum value (Fig. 10.12(b)). When the coupling is further increased, a frequency peak that coincides with the frequency of the driving system is observed in the power spectrum. This peak grows, and its width becomes narrower. The measure of regularity is in-


creased up to the constant value that corresponds to /? in the driving system with optimal noise intensity (Fig. 10.12(b)).

The behavior of the collective response is shown in Fig. 10.13. Here, we have ploted the dependence of/3 on the mismatch parameter for different values of the coupling strength. It is clearly seen that, at the boundary of the synchronization region, the regularity of the "slave" system approaches the regularity obtained for the respective single system and grows further inside this region. For some coupling strengths (g = 0.03 and 0.05, for instance), the regularity exceeds the maximal level observed for the single system with optimal noise intensity (the dashed line). We conclude that the connection between gain of regularity and synchronization for coupled coherence resonance oscillators requires further investigation.

Fig 10.13. Regularity /? as a function of the frequency mismatch parameter D2 for different coupling strengths.

0 0.0005 0.001 0.0015 0.002

D2

In the above sections, we considered a noise-driven excitable system in the regime of coherence resonance as a stochastic oscillator that generates a rather regular signal within a certain range of noise intensity. For such interacting CR oscillators we investigated the phenomenon of mutual synchronization both in terms of the locking of the peak frequencies in the power spectrum and in terms of phase locking.

Drawing on analogy to the deterministic case and building on the concept of a stochastic limit cycle [44, 45], we interpreted the nonsynchronous stochastic oscillations as "stochastic quasiperiodicity" which can be characterized by the presence of two different time scales (two independent peaks in the power spectrum) and, geometrically, as an object involving two stochastic limit cycles in the phase space of the system. In the case of forced synchronization, two variants of the transition from nonsynchronous to synchronous behavior were observed. They are related to locking and suppression of the frequency


of the noise-induced oscillations, respectively. With optimal noise intensity, the maximal value of the regularity observed inside the synchronization region sometimes exceeded the value reached in the uncoupled case. Hence, both noise and cooperative dynamics can contribute to the gain of regularity in coupled excitable systems.

10.5 Clustering of Noise-Induced Oscillations

The subject of our last investigation in this book is clustering in a population of excitable systems driven by Gaussian white noise and with randomly distributed coupling strengths. This may be viewed a complementary analysis to our discussion of globally coupled chaotic maps in Chapter 8. The cluster state is a frequency-locked state in which all functional units run at the same noise-induced frequency. Cooperative dynamics of this regime is described in terms of effective synchronization and noise-induced coherence.

When self-sustained oscillators are coupled, their cooperative behavior reveals a set of dynamical patterns of which the most interesting are clustering, coherent structures and synchronization. Synchronization effects have also been observed in stochastic systems where noise controls the characteristic frequency of the system [30, 31, 24] and may enhance synchronization [33, 49]. Array-enhanced coherence behavior in extended noisy systems has been studied, for instance, by Lindner et al. [50], by Postnov et al. [43], and by Hu and Zhou [51].

Clustering, i.e. formation of groups of functional units with similar properties (amplitudes, phases or frequencies), is an important phenomenon which is assumed by some to underly perception and processing of information in the brain [52]. The problem of clustering has been formulated and analyzed in a general context using the frameworks of phase equations [53, 54], self-sustained periodic oscillators [55], chaotic dynamical networks [56] or chains of bistable elements [57]. Vadivasova et al. [58] showed that cluster synchronization is structurally stable to small fluctuations.

In this section we shall investigate how clustering occurs in a population of inhomogeneous excitable systems with randomly distributed coupling strengths [25]. As discussed in the previous sections, excitable systems are particularly sensitive to external noise and can demonstrate coherence resonance [18, 19, 20] at appropriate amounts of random forcing. Bearing this in mind, we shall


consider the collective response of an array of such functional units in terms of effective synchronization and regularization of noise-induced oscillations with distinct eigenfrequencies.

Let us take the FitzHugh-Nagumo model as the unit in the array. Originally suggested as a simplified description of nerve pulses [59], the FitzHugh-Nagumo model has subsequently been used to describe excitable dynamics in a variety of different fields, ranging from chemical reactions to biological processes [60].

With x and y being a fast and a slow variable, respectively, the dynamics of our system reads

x3

Xj - y - Vj + gj(xj+l + Xj-i ~ 2Xj), (10.4)

Xj + aj + D^it), j = l,...,N.

Here, e — 0.01 is the ratio of the time scales for the two variables, the parameter a governs the character of the solutions and is responsible for the excitatory properties of the individual dynamics. We use free boundary and random initial conditions.

As discussed in several of the previous chapters of this book, the collective dynamics of an assembly of coupled oscillators is of significant importance in many biological problems. A population of identical units with the same coupling properties may serve as the simplest model of such systems. In nature, however, to assume complete identity in properties and operating conditions would be too much of an idealization. Hence, in contrast to previous studies, we shall investigate ordering effects in assemblies of elements that are

(i) inhomogeneous, i.e., the activation parameters a,j are random numbers distributed uniformly on [1.0; 1.1];

(ii) subjected to stochastic forcing by Gaussian white noise £j(i), which is statistically independent in space and has zero mean value, i.e., < £j(£)&(*') > = 5ij8{t-t') and <£,-(*) > = 0;

(iii) coupled with strengths gj that have a random uniform distribution in some range A. More precisely, the distribution interval for the coupling strength is determined as A = Qmax — 9min> where Qmin is fixed at 0.005, but Qrnax

and, hence, the mean level (gmin + gmax)/2 can be varied.

Thus, our model accounts for disorder between the interacting units in several different ways. The interesting question is now whether such elements can

dxj

~dT

dt


adjust their motions in accordance with one another to reach some kind of coherent behavior.

In the case of a single excitable system driven by moderate amounts of noise, the trajectory can become quite regular as we have seen above. The Fourier power spectrum then possesses a well-defined global maximum. A quantitative measure of regularity in coherence resonance can be calculated as Rj =< Tj > /y/cr2(Tj), with a2 denoting variance [20]. The pulse duration Tj is specified as the sum of the activation time needed to excite the system from the stable fixed point and the excursion time needed to return from the excited state to the fixed point. (Note that this definition of the regularity measure differs a little from the definition of /3 given before.) The time-averaged pulse duration controls the mean period and, hence, the mean frequency < fj >= 1/ < Tj > of the noise-induced oscillations. Thus, when exhibiting coherence resonance effects, a noise-driven excitable system can be considered as a stochastic oscillator whose behavior can be described in terms of a noise-induced eigenfrequency [24] and a phase defined as the position on a stochastic limit cycle [44, 45].

100

Fig 10.14. Spatiotemporal evolution and eigenfrequencies < fj > for an array of 100 excitable units with different widths of the coupling range A = 0.002 (a,b) and A = 0.1 (c,d). A sequence of clusters is clearly seen in the latter case. Black dots indicate firing events. D = 0.025.

In our experiments with different distribution intervals for the coupling strength, three basic types of space-time behavior in the one-dimensional array (10.5) of 100 units can be observed. For a vanishing or very narrow interval


of the coupling strength A, the behavior is totally incoherent, as reflected in the irregular patterns of black spots (firing events) in Fig. 10.14(a). The firing events in the individual units occur at frequencies that are randomly spread over the range [0.05;0.27] (Fig. 10.14(b)). In this case, no stable frequency-or phase-locked groups can be detected. A qualitatively different behavioral pattern is encountered for a broader range of coupling strength (Figs. 10.14(c), and (d)). Now synchronized groups, i.e., clusters of stochastic elements, appear. Within each cluster the frequency difference between any two oscillators vanishes or is small compared with the difference between neighboring clusters. With increasing distribution interval for the coupling, the number of clusters decreases (Fig. 10.15), approaching the global synchronous state (one-cluster state), where all units fire simultaneously. Since both the incoherent behavior and the totally synchronized behavior are well understood [49, 51], we shall focus our attention on the clustering of noise-induced oscillations.

0.35

0.30

«£- 0.25

0.20

0.15 0 20 40 60 80 100

J

Fig 10.15. Reduction of the number of frequency-locked clusters with increasing width of the coupling range (D = 0.025). For A = 1.0 (thin curve), nearly all oscillators show the same average frequency.

Let us consider now the individual cluster as a spatial meta-unit of the array and describe its main properties [25]. Because of the given distribution of system parameters, the elements in the cluster have different randomly scattered frequencies for vanishing coupling, and there is no correlation between the firing events of the different cells. With interaction, a frequency locking effect takes place (Fig. 10.16(a)), and the elements of a particular cluster fire in synchrony. However, the deviation of pulse duration a1, = < r? > — < Tj > 2 changes within

J i I i L


the cluster. The variance is minimum at the center of the cluster and increases near the ends (Fig. 10.16(b)). Thus, with frequency entrainment, the stochastic oscillators demonstrate different degrees of mutual synchronization.

0.29

0.26

0.23

0.20

58 61 58 61

Fig 10.16. Eigenfrequency < fj > (a), variance of pulse duration a? (b), effective cross-diffusion coefficient D3

e^ (c) and noise-induced regularity Rj (d) within a single cluster. Width of coupling interval and noise intensity are fixed at 0.1 and 0.025, respectively.

Frequency-locking entrainment is closely related to the phase conditions. In the presence of Gaussian noise (or another random process with unlimited distribution function) the phase-locked state will inevitably be broken at some moment. Thus, the system is deemed to be effectively synchronized if phase locking is observed during a finite but sufficiently long period of time. An appropriate measure of stochastic synchronization is the cross-diffusion coefficient D3

eff = \ft[< <j>](t) > - < (j)j(t) >2] [35]. This quantity describes


how an initial distribution of phase differences <f>j(t) between neighboring elements spreads in time. As before, we use the instantaneous phase introduced as $j(t) = 2irt

f~^t + 2-Kk, where tk is the time of the A;-th firing defined in simulations by the threshold crossing of Xj(t) at x = 1.0. In the present example, the cross-diffusion coefficient vanishes within each cluster (Fig. 10.16(c)) to take nonzero values at inter-cluster units. This satisfies the stronger condition of phase synchronization which provides a high degree of collective entrainment within a cluster of stochastic oscillators. Hence, the notion of effective synchronization can be generalized to a spatially extended group of elements. Similar effects have recently been observed for coupled Van der Pol oscillators with fluctuations [58].

What are the coherence properties of such frequency-locked clusters? It is clearly seen that the regularity exhibits a maximum value within the synchronized state (Fig. 10.16(d)), while the outer-cluster elements demonstrate a lower level of coherence. Comparative analysis of the regularity and pulse deviation functions allows us to suppose that a high coherence behavior within a cluster is related to synchronization phenomenon.

Fig 10.17. Illustration of synchronization-enhanced coherence resonance for the system (10.5) demonstrating cluster structure for A = 0.1. The regularity R averaged over the spatial coordinate is plotted versus noise intensity for individual clusters (curve 1 and 2) and for the whole array with a cluster structure (curve 3). Dashed curve corresponds to the

""6.00 0.02 0.04 0.06 0.08 0.10 uncoupled array. D

In general, the collective response of a cluster is characterized by two aspects. The first aspect is synchronization that leads to frequency and phase entrainment. The second is the regularity of each functional unit due to coherence resonance effect. Remarkably, the regularity averaged over the spatial coordinate can be maximized within each cluster by tuning the noise (Fig. 10.17). At weak external noise, a cluster considered as a functional unit demonstrates weak


coherence, although firings in the individual elements can occur simultaneously. This is related to the relatively large fluctuations of the pulse duration of each of the composing elements. With increasing noise intensity D, the coherence of the temporal and spatial structure of the firing processes is enhanced and reaches a maximum. At large noise level, the frequency and phase fluctuations grow rapidly and this leads to the destruction of the coherence properties of the involved units and, hence, of the spatial coherence structure. Because of the phenomenon of array-enhanced coherence resonance [51], the regularity of the whole cluster is much higher than that of the uncoupled elements (compare the curves 1 and 2 with the dashed curve in Fig. 10.17).

3.0 r—•—r

2.0 -

* 1 0 ; , |

0.0 !^J^M

-1 .0 I—-0 20 40 60 80 100 0 20 40 60 80 100

J J

Fig 10.18. Synchronous (a) and coherence (b) properties along the array with cluster structure for varying levels of noise. The width of the coupling interval is fixed at 0.1.

Let us return now to the whole system. An array composed of excitable elements can be considered at the macro level as a sequence of clusters whose size and structure are determined by a random distribution of firing properties and by the degree of interaction. Fig. 10.18 illustrates the ordering effect caused by the stochastic synchronization and the resulting high coherence within each cluster at the optimal level of noise. The coherence of net output is averaged over a set of clusters. Because of the frequency difference between the clusters, the regularity of the array output is lower than the maximum value of each cluster (curve 3 in Fig. 10.17).

To conclude, we have considered the coherence properties of an array of diffusively coupled excitable systems in the regime of cluster synchronization.

1 • 1 '

i i . 1

D=0.025 D=0.1

I . I ,

1

(a)

\kJi

•

-

u i


A random distribution of the system parameters responsible for the excitatory properties and the strength of interaction led to self-organization (in the form of clustering) that manifests itself as stochastic phase locking and as a mean frequency entrainment between groups of cells. Composed by a number of elements with different properties, each cluster can be considered as a "spatial" excitable unit exhibiting coherence resonance. Its degree of coherence can be enhanced by tuning the noise intensity. Gain of regularity within each cluster is associated with the effect of stochastic synchronization. We believe that such effects can be of importance for biological systems where the background noise can play a constructive role in the emergence of ordering phenomena in large networks of excitable elements.

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INDEX

absorbing area, 51, 79 absorbing areas for

intermittency maps, 108 logistic maps, 51, 88 partial synchronization, 63 two-cluster dynamics, 298

animal gaits, 1 antiphase synchronization, 371 Arnol'd tongues, 9, 230 arrays of

population pools, 280 Rossler systems, 116

asymmetric clusters, 334 asymptotic stability, 46, 126 asynchronous orbits for

logistic maps, 96 Rossler oscillators, 139

attractor bubbling, 180

bacterie-virus interaction, 260 basin of attraction, 48 basins of attraction, for

intermittency maps, 110

logistic maps, 48, 90 Rossler systems, 132

bifurcation gap, 397 bifurcation scenarios for

chaotic hierarchy, 244 logistic maps, 96 pancreatic cell model, 181 Rossler systems, 134 single nephron model, 360

blowout bifurcation, 39, 127, 194 boundary crisis, 101 bounded rationality, 18 bubbling transition, 35 bursting dynamics, 177

capital-output ratio, 19 capital self-ordering, 15 cascaded population pools, 260 cell models, 178, 400 cell inhomogeneities, 203 chaotic hierarchy, 239 chaotic phase synchronization, 212 chaotic saddle, 101, 139, 179, 310

425

426 Index

chaotic spiking, 183 cluster formation, 319, 338 clustering of

excitable systems, 412 logistic maps, 319, 338 Rossler systems, 162

cluster splitting bifurcation, 340 Cobb-Douglas function, 17 coexisting attractors, 66, 247 coherence resonance, 400 complete synchronization, 74 contact bifurcation, 36, 90, 109 continuous invariant measure, 38 coupling asymmetry, 145 criticallity of bifurcation, 152

decision rule, 18 diffusive coupling 126, 193, 248

economic cycles, 13 economic long wave, 15 elasticity of substitution, 17 embedded Arnol'd tongues, 230 escapement mechanism, 14 excitable cells, 177, 402

Floquet multipliers, 238 fractal boundaries, 191 full synchronization, 33

gap junction, 178 global coupling, 240, 291 global riddling, 35, 74 global riddling for coupled

intermittency maps, 111 logistic maps, 48, 97

Rossler systems, 132, 141 pancreatic cells, 200 two-cluster dynamics, 301

hard riddling bifurcation, 92, 304 high-dimensional chaos, 249 Hilbert transformation, 216, 372 Hodgkin-Huxley model, 178 homoclinic bifurcation for

logistic map, 37 pancreatic cell model, 179

homoclinic mechanism, 274 hot flashes, 3 hyperchaos, 51, 116, 198, 247 hypertensive rats, 350

insulin secretion, 177 intermingled basins, 54, 114 intermittency

on-off, 44, 51 type-Ill, 103

invariant area, 79 invariant manifold, 38 ion pumps, 177 islet of Langerhans, 177

jet lag, 3

K-cluster state, 318 kidney flow regulation, 349

local riddling, 35, 75 local riddling for coupled

logistic maps, 48 pancreatic cells, 196

logistic map, 37

Index 427

low-periodic orbits for logistic maps, 44 pancreatic cells, 192 Rossler systems, 125

Lyapunov exponents, 39, 127, 194 Lyapunov function, 126 Lyapunov value 35, 84

master-slave configuration, 56 mean field coupling, 240 membrane potential, 181, 400 microbiological reactors, 259 microperfusion, 352 Milnor attractor, 35, 134 mixed absorbing area, 77, 88 momentary phase, 215 monovibrator circuit 401 Morris-Lecar model, 400 multistability, 222

nearly symmetric clusters, 326 Nernst potential, 181, 401 nested Arnol'd tongues, 232 nephron structure, 351 noise-induced oscillations, 400 non-resonant condition, 81

on-off intermittency, 44 on-off intermittency in coupled

intermittency maps, 116 logistic maps, 51 pancreatic cells, 198 two-cluster dynamics, 313

pancreatic cells, 177 parameter mismatch in

Rossler systems, 140 pancreatic cells, 203 transcritical riddling, 313

partial synchronization of logistic maps, 56 Rossler systems, 159

pendulum clock, 14 period-adding transitions, 189 period-orbit threshold theory, 44,

125, 192 piecewise-linear maps, 35 population dynamics, 259 preimage, 43, 99, 112, 133 phase diagram for

intermittency maps, 107 logistic maps, 45, 61, 102 pancreatic cell model, 187 Rossler systems, 224 single nephron model, 364

propagating waves, 178

random ion channels, 178 regularity parameter, 411 repelling orbit, 116 resonant torus, 232, 246 return time, 215 riddled basin, 41 riddled basin for

intermittency maps, 111 logistic maps, 49, 64 pancreatic cells, 196, 200 Rossler systems, 130

riddling bifurcation, 35 riddling scenarios for

logistic maps, 49, 117 partial synchronization, 65

428 Index

Rossler systems, 159 two-cluster dynamics, 307

rotation number, 214 Rossler system, 9, 126, 212

secure communication, 56 Sherman model, 181 single pool system, 265 soft riddling bifurcation, 87 spiral chaos, 213 stability conditions, 39 stochastic dynamics, 400 subcritical

period-doubling, 104, 188 pitchfork bifurcation, 45

subharmonic saddle-node bifurcation, 99

supercritical period-doubling, 45 suppression of dynamics, 233 synchronization error, 77 synchronization manifold, 38, 123 synchronization measures, 7, 211 synchronization of

chicken heart cells, 8 economic sectors, 21 hormonal systems, 3 nephrons in the kidney, 370 noise-induced oscillations, 404 pancreatic cells, 192 Rossler systems, 123

synchronization region, 77

Takens-Bogdanov point, 267 torus birth bifurcation, 9, 237 torus destruction, 137 transcritical riddling, 296

transverse destabilization, 39 transverse eigenvalues, 39, 60, 136 transverse Lyapunov exponents,

39, 47, 60 trapping zone, 34 tubuloglomerular feedback, 352 two-cluster dynamics, 293 two-dimensional maps, 36 two-pool system, 270

unfolding procedure for logistic maps, 81 two-cluster dynamics, 300

vascular coupling, 353

weak attractor, 35 weak stability, 35, 59, 124, 138 winding number, 405

chaotic synchronization application to living

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