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Images of synchronized chaos: Experiments with circuitsNikolai F. RulkovInstitute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402

~Received 23 February 1996; accepted for publication 6 May 1996!

Synchronization of oscillations underlies organized dynamical behavior of many physbiological and other systems. Recent studies of the dynamics of coupled systems with combehavior indicate that synchronization can occur not only in case of periodic oscillations, butin regimes ofchaotic oscillations. Using experimental observations of chaotic oscillationscoupled nonlinear circuits we discuss a few forms of cooperative behavior that are related tregimes of synchronized chaos. This paper is prepared under the request of the editors of the sfocus issue ofChaosand contains the materials for the lecture at the International SchoolNonlinear Science, ‘‘Nonlinear Waves: Synchronization and Patterns,’’ Nizhniy Novgorod, Rus1995. The main goal of the paper is to outline the collection of examples that illustrate the stathe art of chaos synchronization. ©1996 American Institute of Physics.@S1054-1500~96!00403-X#

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One of the most significant properties of oscillations gen-erated by nonlinear dynamical systems is their ability tobe synchronized. The synchronization of the oscillationscan be achieved due to either an external forcing or cou-plings between the systems. Although synchronization isoriginally referred to systems with periodic oscillations,recent studies show that phenomenon of synchronizationextends to cases of irregular, chaotic oscillations generated by dynamical systems. The growing interest in systems with chaotic dynamics and their applications in dif-ferent fields of science bring new issues to thesynchronization theory. Despite the fact that many recentpublications discuss examples of synchronized chaos, thgeneral framework of this phenomena has not been quitedeveloped. The main point of this paper is to presentexamples of chaos synchronization that illustrate themodern state of this framework. In order not to be ab-stract we consider the cases of chaos synchronization thawe observed in experiments with electronic circuits. Wealso discuss mechanisms responsible for the onset of sychronization.

TABLE OF CONTENTS

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262A. Periodic Motions and Synchronization. . . . . . . . 263B. Synchronization of Chaotic Oscillations. . . . . . . 263

II. SYNCHRONIZED CHAOS IN FORM OFIDENTICAL OSCILLATIONS. . . . . . . . . . . . . . . . 264A. Synchronization with Resistive Coupling. . . . . . 264

1. Synchronization manifold. . . . . . . . . . . . . . . 2642. Dissipative coupling. . . . . . . . . . . . . . . . . . . . 2643. Chaos synchronization and stability. . . . . . . 2654. Mechanism for chaos synchronization. . . . . 266

B. Stabilization of Unstable Orbits by Means ofSynchronization. . . . . . . . . . . . . . . . . . . . . . . . . . 266

III. GENERALIZED CASES OF SYNCHRONIZEDCHAOS IN DRIVE-RESPONSE SYSTEMS. . . . . 267

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A. The Auxiliary System Method for DetectingSynchronized Chaos. . . . . . . . . . . . . . . . . . . . . . . 2671. Synchronization and predictability. . . . . . . . 2672. The auxiliary system. . . . . . . . . . . . . . . . . . . 267

B. Generalized Synchronization in Systems withDissipative Coupling. . . . . . . . . . . . . . . . . . . . . . 268

C. Synchronized Chaos in Systems with DifferentFrequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2691. Synchronized chaos. . . . . . . . . . . . . . . . . . . . 2702. Synchronization of unstable periodic orbits.. 2

D. Regularization by Chaotic Driving. . . . . . . . . . . 2711. Synchronization and regularization. . . . . . . . 2722. Auxiliary system and multistability. . . . . . . . 273

IV. SYNCHRONIZATION OF CHAOS BYPERIODIC SIGNALS. . . . . . . . . . . . . . . . . . . . . . . 274

A. Phase Locked Chaotic Attractors. . . . . . . . . . . . . 274B. Chaotic and Periodic Orbits. . . . . . . . . . . . . . . . . 275C. Chaos Suppression. . . . . . . . . . . . . . . . . . . . . . . . 275

V. THRESHOLD SYNCHRONIZATION. .. . . . . . . . 275VI. CONCLUSIONS AND OUTLOOK. . . . . . . . . . . . 276VII. ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . 276APPENDIX A: MECHANISMS FOR

SYNCHRONIZATION OFPERIODIC OSCILLATIONS. . . . . . . 276

1. Threshold Synchronization. . . . . . . . . . . . . . . . . . 2762. Synchronization and Beats. . . . . . . . . . . . . . . . . . 277

A. Weak forcing. . . . . . . . . . . . . . . . . . . . . . . . . 277B. Strong forcing. . . . . . . . . . . . . . . . . . . . . . . . . 278

APPENDIX B: IMPLEMENTATION OF THECIRCUIT. . . . . . . . . . . . . . . . . . . . . . . 278

I. INTRODUCTION

Synchronization of oscillations is a well-known nonliear phenomenon that is frequently encountered in natThe ability of nonlinear oscillators to synchronize each otis a basis for the explanation of many processes of naand, therefore, synchronization plays a significant role inence. Numerous applications of the synchronization in m

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263Nikolai F. Rulkov: Images of synchronized chaos

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chanics, electronics, communication, measurements, anmany other fields have shown that the synchronizationextremely important in engineering.

Usually, synchronization is understood as the abilitycoupled self-excited oscillators with different frequenciesswitch their behavior from the regime of independent osclations characterized by beats to the regime of cooperatstable periodic oscillations, as the strength of the couplingincreased. As a result of synchronization, the oscillatochange their frequencies in a such way that these frequenbecome identical or related via a rational factor. Dependiupon the properties of oscillators considered, there areferent explanations as to why these oscillators synchronFor instance, one can distinguish between the mechanismsynchronization of relaxation oscillators that produce shapulses and the mechanism for synchronization of the oscitors that generate smooth waveforms~the main features ofthese mechanisms are outlined in Appendix A!. However,one should keep in mind that the separation of these mecnisms is not precise, and they can be considered as limitcases of a general mechanism, where resonances and acof dissipative forces are very important.

A. Periodic motions and synchronization

The first nonlinear theory for synchronization of quasharmonic oscillations is due to van der Pol.1 Now, there aremany books and reviews that discuss different aspects ofsynchronization theory for periodic oscillators. StudyinRefs. 2–8 can be very useful for understanding the mfeatures of the onset of synchronized behavior.

Despite the complexity of mechanisms involved in thprocess of the onset of synchronization, the phenomenonsynchronization of periodic oscillations has a transparegeometrical interpretation. Indeed, in the phase space ofautonomous system the image of the periodic oscillationsstable limit cycle. Behavior of a network ofn coupled peri-odic oscillators depends on the parameters of couplinamong the oscillators. In the phase space of the network wzero couplings, the post-transient oscillations will correspoto trajectories that fill the surface of a stablen-dimensionaltorus. The phenomenon of synchronization among periooscillators is understood as the ability of the networkswitch its behavior from quasiperiodic oscillations, assoated with aperiodic motions on a stable torus, to periodoscillations, as the strength of the couplings between thecillators is increased. Therefore, the image of the synchnized periodic oscillations is again a stable limit cycle, bnow this limit cycles is in the joint phase space of the nework of coupled oscillators.

B. Synchronization of chaotic oscillations

The progress in studies of nonlinear dynamical systethat was achieved in the past two decades significantlytended the notion of oscillations in nonlinear systems. It wshown that the post-transient oscillations in dynamical sytems can be associated not only with regular behavior sas periodic or quasiperiodic oscillations, but also with ch

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otic behavior. At the same time it was shown that some othe ideas of synchronization can be extended for descriptioof particular types of cooperative behavior in coupled systems with chaotic dynamics. For example, Fujisaka anYamada have demonstrated that two identical systems wichaotic individual dynamics can change their behavior fromuncorrelated chaotic oscillations to identical chaotic oscillations, as the strength of the coupling between the systemsincreased.9

Recent ideas of employing the synchronization of chaoin different applications resulted in intensive studies in thisfield of nonlinear dynamics. Numerous papers, published ithe past six years, employed the ideas of synchronized chafor different techniques of communications with chaoschaos suppression, and monitoring dynamical systems. Athe result of these intensive studies, the framework of synchronized chaos in the form of identical oscillations has beewell understood. However, taking into account the variety oforms of synchronized periodic oscillations, one understandthat the regime of identical chaotic oscillations is just a particular form of synchronized chaos, and other, more compleforms of synchronized chaotic oscillations, can exist. Despitthe fact that the first steps toward the understanding of succomplicated cases of synchronized chaos were carried outAfraimovich et al.10 more than ten years ago, the generaproblem of synchronized chaos has still not been studied.

It is known that even for periodic oscillators, the theo-retical analysis of synchronization in general is a very complicated problem. However, experimental analysis of thsynchronized periodic oscillations is straightforward. Indeedin order to observe the onset of synchronization in periodioscillators, one needs to detect the formation of a stable limcycle in the joint phase space of the coupled systems. Thcan easily be done by analysis of the Lissajous figures otained by direct measurements from the oscillators. The eperiment enables one to study the onset of synchronized priodic oscillations, despite the complexity of the forms of thelimit cycles, and other complicating factors. These advantages of an experimental approach in studies of complicateforms of synchronization make it a very powerful tool for theexploration of synchronized chaos.

In this paper we employ the experimental results observed in coupled chaotic circuits to discuss the state of thart of chaos synchronization. In order to be more specific, wbase the discussions upon the circuits generating smoochaotic waveforms. In Sec. II we consider an example owell-known synchronized chaotic behavior, which is characterized by identical chaotic oscillations. Using this examplewe discuss the issues of the existence and stability of sychronized chaotic motions, and mechanisms that are resposible for the onset of chaos synchronization. In Sec. III wediscuss experiments with directionally coupled circuits inwhich synchronization of chaos leads to richer behavior thathe regime of identical chaotic oscillations. We also discusan experimental approach that we use for studies of theforms of synchronization. In Sec. IV we present experimental results that indicate the possibility of synchronization ochaos with external periodic signals. The specific features

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264 Nikolai F. Rulkov: Images of synchronized chaos

chaos synchronization observed in relaxational oscillaare briefly discussed in Sec. V. Appendix A contains a bdiscussion of mechanisms for synchronization of periooscillators. In Appendix B we give details on implementatiof chaotic circuits used in our experiments.

II. SYNCHRONIZED CHAOS IN FORM OF IDENTICALOSCILLATIONS

In this section we will consider a particular case of sychronized chaos where two coupled systems evolve idecally in time. Synchronized chaotic oscillations of this forhave been studied in detail in the vast literature on the sject. There are two categories of systems where this sobehavior is usually investigated. The first category includcoupled systems that at zero coupling are identical to eother, and where each display chaotic behavior.9–12 Whenthe appropriate coupling is introduced, the systems demstrate identical oscillations with the onset of synchronizatiThe second category, introduced by Pecora and Carroll,13,14

consists of a driving system that exhibits chaotic behavand of a response system. The latter replicates a portioportions of the driving system. The variables that areproduced in the portion are taken from the driving systemthe driving signals.

A. Synchronization with resistive coupling

Let us consider the dynamics of two identical chaocircuits coupled to each other with a resistor; see Fig. 1.details on the implementation and the individual dynamicsthe circuit are briefly discussed in Appendix B. Usually, twtypes of the resistive coupling are considered. The first tis mutual coupling. This coupling is provided by connectithe similar points of the circuit through a resistor. The sond type is directional coupling, where the same pointsconnected through the resistor and a unity-gain OP ampliThe directional dissipative coupling can be considered aproportional feedback control.

1. Synchronization manifold

The model describing the dynamics of two coupled idetical circuits presented in Fig. 1 can be written in the form

x15x21e21~y12x1!,

FIG. 1. The diagram of two dissipatively coupled chaotic circuits. The cof directional coupling is shown. In the case of mutual coupling the poA andB are connected through the resistorRc without the OP amplifier.

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The shape of nonlinearityf (x) and the dependence of theparameters upon the physical values of the circuits elemenare discussed in Appendix B. In the case of mutual couplinge215e12. In the case of directional couplinge2150. Thevalues of coupling parameters are proportional to the conductance of the resistorRc ,

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x15y1 , x25y2 , x35y3 . ~4!

The phase space trajectories of the circuits that are located othe manifold given by~4!, correspond to identical oscilla-tions in both systems. One can easily see, from Eqs.~1! and~2!, that when the systems behave identically, the behavior othe coupled system is the same as that exhibited by the indvidual systems without coupling. Therefore, if the param-eters of the circuits are chosen in a region associated witchaotic behavior, and the manifold~4! is stable, then thecircuits can demonstrate synchronized chaotic oscillations.

2. Dissipative coupling

Figure 2 shows the onset of synchronization in the ex-periment with directional coupling. Due to the use of theunity-gain OP amplifier in the coupling, the response circuitdoes not influence the behavior of the drive circuit, thereforethe chaotic oscillations in the drive circuit do not dependupon the strength of the coupling; see Fig. 1. In the experiment, the inner parameters of the circuits were set to thvalues where both uncoupled circuits generate chaotic oscilations, which correspond to the attractors shown in Fig2~a!. Due to the local instability of trajectories of these at-tractors, the oscillations in the uncoupled circuits (Rc→`)are not correlated. The chaotic oscillations in the drive and inthe response circuits can be considered as two identical, bindependent, modes of oscillation. When the coupling is in-troduced, then the driving mode will tend to suppress theuncorrelated behavior of the initially excited mode of theresponse circuit. The mechanism of this suppression is thdissipation of energy in the coupling resistor. The differencebetween the voltages@x1(t) and x2(t)] induces a currentthough the resistorRc . With a low value of the resistanceRc , a higher current is induced in the resistor. Therefore, fothe same voltage difference@x1(t)2x2(t)#, the energy dis-sipation in the coupling will be larger when the resistancehas a lower value. If the dissipation is sufficient for suppres

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FIG. 2. ~a! The projection of the chaotic attractor onto the plane (x1 ,x3). These are measurements from the driving circuit. The projection of the chaattractor on the plane@x1(t),y1(t)# are plotted in~b! and ~c!. ~b! Unsynchronized oscillations (Rc51.0 kV!; and ~c! synchronized chaotic oscillations(Rc50.2 kV!. The parameter values of the drive and response circuits areC18'C28'220 nF,C1'C2'332 nF,L15L25144.9 mH,r 1'r 2'348 V andR15R254.01 kV, a15a2522.5.

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sion of the inner instabilities in the response circuit, theboth circuits demonstrate identical chaotic oscillations.this case, time evolution of all variables of the response ccuit is identical to the time evolution of similar variables othe drive circuit. Figures 2~b! and 2~c! show the projectionsof the chaotic attractors measured in the circuits, with tvalues of coupling taken below and above the thresholdthe onset of synchronization.

3. Chaos synchronization and stability

In order to define the threshold for chaos synchroniztion, one needs to study the stability of synchronized trajetories as a function of the coupling parameter. Since in ocase the synchronized oscillations are chaotic, these trajeries are always unstable. However, in the analysis of sychronized chaotic motions one has to distinguish betweeninstability for perturbations tangent to the manifold~4! andtransverse to it. The regime of identical chaotic oscillatiois stable when the synchronized trajectories are stable forperturbations in the transverse direction to the synchronition manifold ~4!.

Two most frequently used criteria for stability of synchronized chaotic motions are the Lyapunov function critrion ~see, for example, Refs. 15–18! and the analysis oftransversal ~conditional! Lyapunov exponents calculated

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from the linearized equations for the perturbations transvsal to the synchronization manifold~see, for instance, Refs.9, 13, 19, and 20!. The criterion based on the analysis oLyapunov functions for the vector field of perturbationtransversal to the manifold enables one, in some casesprove that all trajectories in the phase space of the coupsystems are attracted by the manifold of synchronized mtions. Despite the fact that this criterion guarantees the onof synchronization, it is not a general method since thereno procedure for constructing the Lyapunov function for aarbitrary system. In many practical cases, Lyapunov funtions cannot be found, even for systems that possess a stmanifold of synchronized motions for a broad range of prameters of coupled systems, and of the coupling itself.

In contrast with Lyapunov functions, the analysis otransversal Lyapunov exponents is quite straightforward acan be easily employed, even for rather complicated systeHowever, it has been pointed out21–25 that, in practice, thenegativeness of Lyapunov exponents does not always guantee the onset of synchronized motions. Even in the cawhere all transversal Lyapunov exponents are negative, thmay exist atypical trajectories in the immediate vicinity othe manifold of synchronized motion that depart from thmanifold exponentially fast. The appearance of such trajetories is responsible for destabilization of the synchroniz

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chaotic motions. The synchronized chaotic motions becounstable, in the sense that vanishing noise in the coupland/or internal noise in the coupled systems and/or smmismatch between the parameters of the systems lead toloss of the synchronization and to the bubbling behavior cosidered in Refs. 22, 24, and 25.

The applications of the Lyapunov function criteria anthe analysis of transversal Lyapunov exponents for the sbility of the synchronized motions in the model of couplecircuits ~1! and ~2! can be found in Ref. 16 It has been alsshown in the same paper that for strong enough couplingsynchronized chaos in these circuits is robust. The robustnimplies that small perturbations of the parameters of the stems lead to small deviations from the identical oscillationThe stability and robustness of synchronized chaotic motiois easily determined in experimental studies. Indeed, duethe natural noise and small parameter mismatch that areways present in the experiment, the regime of practicaidentical oscillations is observed only in the case where tsynchronized solutions are stable against transversal pebations and robust. Therefore, the experimentally obtainFig. 2~c! shows the chaotic attractor located on thestablemanifold of synchronized motions.

4. Mechanism for chaos synchronization

The onset of the synchronization between chaotic osclators of smooth waveforms differs from the case of periodoscillation. Unlike the case of synchronization between idetical periodic oscillators, the synchronization between idetical chaotic oscillators is, generally, characterized by a ctain threshold value of the coupling that is required for thonset of the synchronization. The reason for the existencethis threshold value is the local instability of the chaottrajectories. In order to achieve synchronization in directioally coupled circuits, the instability in the response circuhas to be suppressed. In other words, the amplitude ofmode of the oscillations induced in the response circuitthe driving has to be sufficient to compete and suppress‘‘independent’’ mode in the response circuit. In the casemutual coupling, the dissipation induced by the couplincauses the onset of the energetically preferred regime of mtual behavior. Depending upon the parameters of the cpling, and the nonlinearity and dissipation in the circuits, thpreferred regime of oscillations can be the mode of identicchaotic oscillations, nonidentical stable periodic and quaperiodic oscillations, and, even, fixed states. Discussionthe bifurcations in the mutually coupled circuits~1! and ~2!that provide the alternations between of these regimes canfound elsewhere.16

It is clear that in some cases the dissipation of energythe coupling between chaotic systems can fail to supprtransversal instability. For example, there are systemwhich some of the variables do not sufficiently influence thinstability, which is provided by the dynamics associatewith the other variables of the system. In these cases einfinitely strong dissipation in the coupling that uses the firgroup of variables may seem incapable of suppressingtransversal instability in coupled systems. Moreover, the

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are known examples where the synchronization of chaosobserved only within an interval of moderate values of thcoupling parameter; see, for example, Ref. 26. This intervof the parameter values is restricted both from below anfrom above. The role of coupling in this case is not only toprovide an extra dissipation, but also to change the innproperties of the synchronized system in order to providebetter interaction between different portions of the system

B. Stabilization of unstable orbits by means ofsynchronization

Considering the case of the directionally coupled circuitone can notice that the synchronized motions on the mafold ~4! correspond not only to identical chaotic motions bualso to identical unstable periodic orbits embedded in thsynchronized chaotic attractor. Since the manifold is stabthe instability of these periodic orbits is caused only by thinstability of the corresponding orbits in the phase spacethe driving circuit. Therefore, if one takes a waveformx1p(t) that corresponds to a unstable periodic orbit embeded in the chaotic attractor of the driving circuit, and drivethe response circuit with the periodic signalx1p(t), then, dueto the stability of the synchronization manifold~4!, the re-sponse and driving circuit will operate in the stable regime oidentical periodic oscillations.

It is important that when the systems operate in the rgime of identical oscillations the term of the system~2!,which describes the coupling, becomes equal to zero.means that when the response circuit reaches the desiredgime of oscillations, the current through the coupling resisto

FIG. 3. An example of time seriesy1(t), y3(t), and I coupl(t) during thetransition to the fixed pointOR after applying the driving at timetstart512.2 ms.

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Rc becomes equal to zero. In practice, due to noise and sdeviation of the parameters of the circuits, the system alwrequires some current through the coupling resistor, butcurrent can be very small. Figure 3 shows an exampletransition from chaotic motions to a stabilized fixed poiafter the appropriate driving is applied. In this case, thestable fixed point does not belong to the chaotic attracHowever, since the fixed point is a solution of uncouplresponse circuit, the currentI coupl through the coupling resistor becomes exponentially small after the system approacthe fixed point. Details of this experiment can be fouelsewhere.27

The technique of chaos suppression by driving the stem with matched periodic signals was studied by Pyraga26

Due to simplicity of implementation and transparent theorecal background, this technique is easily applicable in diffent experiments; see, for example, Refs. 27 and 28.

III. GENERALIZED CASES OF SYNCHRONIZEDCHAOS IN DRIVE-RESPONSE SYSTEMS

In the previous section we considered synchronized cotic oscillations in the form of identical oscillations. In manrecent papers, the onset of identical oscillations is useddefinition of synchronization. It is clear that such an intepretation of synchronized chaos significantly narrowsmeaning of this phenomena. Indeed, taking into accountvariety of different forms of synchronized behaviorcoupled periodic oscillators, one can expect that synchrozation of chaos in the form of identical oscillations is onlyparticular case of synchronized chaos. More general casethe synchronization include the synchronization betwechaotic systems with different parameters. In such casessynchronized chaotic trajectories are located on a stablechronization manifold, which is no longer a hyperplane~4!,but a more complex geometrical object. In order to distguish these ‘‘nontrivial’’ cases from the cases of identicchaotic oscillations, which are usually used in the literatuto define synchronized chaos, we call such synchronizageneralizedsynchronization of chaos.29

The main difficulty of theoretical analysis of the synchronization between different chaotic system is to definset of common features of the dynamical behavior that wenable one to clearly distinguish the synchronized chamotions from unsynchronized ones. Indeed, dependingthe coupling parameters, two chaotic system can, generdemonstrate different forms of chaotic, quasiperiodic, aperiodic oscillations. It is clear that the periodic oscillatiocan be considered as the onset of periodic synchronizaHowever, since the individual behavior of the systemschaotic, then these systems can demonstrate synchronchaotic oscillations. In the general case, the synchronichaotic oscillations are different from the chaotic oscillatiogenerated by the uncoupled systems. Therefore, the simity between the synchronized chaotic attractor and the cotic attractors of uncoupled systems cannot be considerea requirement for the synchronized chaos. The first maematical definition of the synchronized chaos, which

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cludes the case of nonidentical oscillations, has given inpaper by Afraimovich, Verichev, and Rabinovich.10 Thisdefinition is based on the idea of a homeomorphous transfmation, which is required for linking the projections of thsynchronized chaotic trajectories onto phase subspaces ocoupled systems. However, this mathematical definition cotains a number of conditions that cannot be shown tosatisfied in real experiments with nonidentical chaotic sytems. Moreover, we believe that in some cases the requment of the homeomorphism between the projections excsively restricts the meaning of synchronized motions.

In this section we will discuss simplified cases of nonidentical synchronized chaotic oscillations that are observin directionally coupled circuits with different parameter vaues. In these cases the dynamics of the driving systems dnot depend upon the behavior of the response system, thfore the synchronized oscillations should be related to tattractor in the driving system.

A. The auxiliary system method for detectingsynchronized chaos

1. Synchronization and predictability

In order to define a common feature that allows onedistinguish synchronized behavior, we refer to general caof synchronized periodic oscillations in drive-response sytems. Despite the complexity of the synchronized periodoscillations, they always satisfy the property that is the abity to predict the states of the response system from obsvations of the drive system. Due to the stability of the sychronized motions, the response behavior is not sensiblesmall perturbations of the initial conditions of the systems

It was shown in Ref. 29 that the synchronization btween chaotic driving and response systems can also betected through the analysis of such predictability. The abilto predict the current state of the response system fromchaotic data measured from the driving system can be uas a definition of synchronized chaos in a generalized senThe predictability indicates the existence and stability ofchaotic attractor located in the synchronization manifolThe behavior of the system on the manifold is controlleonly by the motions in the phase space of the driving systeNote that, in general, the synchronization manifold can haa very complicated shape and cannot be detected frsimple observations of its projections, even when the mafold is stable. When the systems lose synchronization,driving system does not provide complete control of the bhavior of the response system and small perturbations inresponse system will grow.

2. The auxiliary system

In the experiments we use a replica of the response stem as a predicting device. This auxiliary system is driventhe same way as the response system and has no connewith the response system. Therefore, the regime in whichauxiliary system is synchronized with the driving systemstable whenever the response system is synchronized. N

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that both response and auxiliary systems may be syncnized to the driving system in the generalized sense. Inregime due to the identity between the parameters ofresponse and auxiliary systems they demonstrate idenoscillations. Therefore, the auxiliary system can be conered as the ideal predictor that is able to indicate the curstate of the response system by processing the driving sigAt the same time, these identical chaotic oscillations caneasily detected. The onset of identical chaotic oscillationresponse and auxiliary systems in the presence of small nand parameter mismatch~which are unavoidable in the experiments! guarantees the stability of the synchronizatimanifold, and therefore guarantees synchronization betwthe driving and response systems.30

The auxiliary system approach can also be usefultheoretical analysis of the generalized synchronizationchaos. This approach enables one to reduce the study otransversal stability of very complicated synchronized m

FIG. 4. The circuit diagram of the experiment with driving, response, aauxiliary nonlinear electrical circuits. The parameters values of the dcircuit are set to beC185230 nF,C1'337 nF,L1'140 mH,r 1'334V, andR1'4.21 kV. The parameter values of the response and auxiliary circareC28'225 nF,C2'342 nF,L2'145 mH,r 2'348V, andR2'4.97 kV.

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tions, to the analysis of transversal stability for identical cotic oscillations. However, we have to emphasize thatissues of the robustness, which are briefly discussed inII, are crucial for this analysis.

B. Generalized synchronization in systems withdissipative coupling

To demonstrate the generalized synchronization of chin an experiment with electronic circuits, we built two almoidentical electronic circuits that were driven by a chaosignal from a third circuit. The circuit diagram of the expement is shown in Fig. 4. The chaotic signal generated bydrive circuit was applied to both the response and the auiary circuits through the resistorsRc . The strength of thecoupling was controlled by the values ofRc in each circuit.This was adjusted to have the same value for both circuitsthe experiment we tuned the parameters of the drive cirto correspond to the regime of chaotic oscillations. Thistractor is shown in Fig. 5~a!. The parameters of the responand the auxiliary circuits were tuned to values, which, wiout coupling, namely,Rc→`, lead these circuits to generachaotic oscillations corresponding to the attractor shownFig. 5~b!.

Synchronization of the chaotic oscillations was obserfor values of the coupling withRc,630V. It was easilydetected with the analysis of the projections of the synchnization manifold onto the planes (y1 ,z1) and (y3 ,z3). Forthe synchronized oscillations, the trajectory is projected ithe diagonalsy15z1 andy35z3 on these planes. These idetities guarantee the identity of the currentsJy(t)5Jz(t),which one can see in Fig. 4. Synchronized behavior,served withRc5604V, is shown in Fig. 6. The synchronizechaotic attractor measured in the response circuit is presein Fig. 6~a!. The fact of the synchronization is confirmed bthe stability of the ‘‘diagonal manifold’’ in the state spaceresponse1auxiliary system@see Fig. 6~b!#, from which thestability of the manifold of synchronized motions in thphase space ofdrive1responsesystem follows. Looking atthe projections of the synchronized chaotic attractors othe plane of the variables (x1 ,y1) and (x3 ,y3), it becomes

ndive

its

FIG. 5. The experimentally measured chaotic attractors of the uncoupled drive~a! and response~b! circuits. The parameters in the nonlinear convertersN inthe drive isa'22.85, and in the response it isa'24.62. The attractors in the two systems are not the same, as the systems are different.

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FIG. 6. The projections of synchronized~a!–~c! and unsynchronized~d! chaotic motions measured withRc5604V andRc5731V, respectively. Samplingrate 20ms. ~a! The synchronized chaotic attractor measured in the response circuit.~b! The projection of the chaotic motions onto the plane of the variable(y3 ,z3) measured from response and auxiliary circuits.~c! The projection of the chaotic motions onto the plane of the variables (y3 ,x3) measured fromresponse and driving circuits.~d! The projection of the chaotic motions onto the plane of the variables (y3 ,z3) measured from response and auxiliary circuits.

re-

ith

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clear that oscillations in the driving and response circuits anot identical, see Fig. 6~c!. Therefore, these circuits are synchronized in the generalized sense.

Unsynchronized chaotic oscillations, measured wRc5731V, are shown in Fig. 6~d!. The projection of thesechaotic oscillations onto the plane (y3 ,z3) clearly indicatesthat these oscillations are not synchronized. Indeed, onesee that trajectory frequently leaves the close vicinity of tsynchronization manifold. This means that the manifoldtransversely unstable.

C. Synchronized chaos in systems with differentfrequencies

In this section we present an example of synchronizchaos in which the coupled systems exhibit oscillations wdistinct characteristic frequencies. The circuit diagram of texperiment is shown in Fig. 7. The synchronizing signx1(t) from the driving circuit is applied to the response cicuit, where it is mixed with the signala f @y1(t)#.

To explore the generalized synchronization of chawith frequency ratio 1:2, in the experiment we set the paraeters of the circuits to the values shown in the Fig. 7 captioValues of the control parameters,a, for the nonlinear con-verters in the driving and response circuits werea1522.86and a2514.0, respectively. As the coupling parameter w

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FIG. 7. The diagram of the experimental setup. The parameters values of tdrive circuit are set to beC185221 nF, C1'336 nF, L1'144 mH,r 1'358 V, andR1'4.69 kV. The parameter values of the response andauxiliary circuits areC28'105 nF,C2'160 nF,L2'75 mH,r 2'161V, andR2'4.32 kV.

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FIG. 8. The projections of the chaotic attractors measured in uncoupled circuits with sampling rate 20ms. ~a! (x1 ,x3) projection of the attractor in the drivingcircuit. ~b! (y1 ,y3) projection of the attractor in the response circuit,g50.

et.

ts

t

use the amplification,g, of the driving signal,x1(t), in thevoltage mixtures a f @y1(t)#1gx1(t) and a f @z1(t)#1 gx1(t); see Fig. 7.

1. Synchronized chaos

The chaotic attractors measured from the driving circand the response circuit without driving (g50) are shown inFigs. 8~a! and 8~b!, respectively. Despite the similarity of thshapes of the chaotic attractors the oscillations in the circ

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uit

euits

are generated with different characteristic frequencies. Thresponse circuit oscillates twice as fast as the driving circuiThe attractor in the auxiliary circuit looks exactly the sameas the attractor in the response circuit shown in Fig. 8~b!, butthe oscillations between the auxiliary and response circuiwithout driving are not correlated.

Let us consider the synchronized chaotic oscillations thawere observed withg52.0. Different projections of the syn-

e
FIG. 9. The projections of synchronized chaotic motions measured withg52.0. Sampling rate 20ms. ~a! The projection of the chaotic motions onto the planof the variables (y1 ,z1) measured from response and auxiliary circuits.~b! The projection of the chaotic motions onto the plane of the variables (y1 ,x1)measured from response and driving circuits.~c! The synchronized chaotic attractor measured in the response circuit.~d! The Lissajous figure of the unstableperiod-1 orbit retrieved from the chaotic data.
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chronized chaotic attractor are shown in Fig. 9. The factthe driving and response circuits are synchronized follofrom the identity of chaotic oscillations measured fromauxiliary and response circuits; see Fig. 9~a!. For the sameoscillations, the projection of the synchronized motions othe plane of variables of the drive and response circuits hrather complicated form, which is shown in Fig. 9~b!. Thesynchronized attractor measured in the response circuitfers from the attractor of the driving circuit; compare F8~a! and Fig. 9~c!.

The frequency spectra of the signalsx1(t) andy1(t) areshown in Fig. 10. From these figures one can easily seethe synchronized oscillations in drive and response circhave different characteristic frequencies.

2. Synchronization of unstable periodic orbits

We found that the pairs of unstable periodic orbits ebedded in the chaotic attractors of the driving and respo

FIG. 10. ~a! The spectrum of the driving signal,x1(t). ~b! The spectrum ofthe signal,y1(t), measured from the synchronized response circuit.

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ntoas a

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circuits are phase locked with the same frequency ratvd /v r5

12, wherevd andv r are the main frequencies of the

orbits in driving and response circuits. Figure 9~d! shows theLissajous figure for the simplest unstable periodic orb~period-1! retrieved from the experimental chaotic data. Oncan see that moving along the period-1 orbit the responsystem makes two rotations while the driving system makonly one. Similar Lissajous figures were found for other unstable periodic orbits embedded in the synchronized chaoattractor. From this analysis it is reasonable to assume ththe transition to synchronized chaos is accompaniedphase locking of pairs of unstable limit cycles existing in thchaotic attractors of uncoupled driving and response sytems. Of course, we cannot demonstrate experimentally thall pairs of saddle periodic orbits in driving and responscircuits are phase locked when the chaotic oscillations in thcircuits are synchronized. However, this seems quite plasible and this phase locking can be considered as the mecnism of the onset of synchronized chaos. Since in our expement the synchronization of periodic orbits is characterizeby the frequencies ratio 1:2 this ratio can be considered ascharacteristic of the synchronized chaotic oscillations.

D. Regularization by chaotic driving

In this section we briefly discuss phenomenon of regularization by chaotic driving, which is observed in directionally coupled chaotic systems synchronized with frequencratio 2:1. The essence of this phenomena is that the behavof the chaotic system driven by another chaotic system bcomes more regular than it was in either of these two sytems when they were uncoupled. In order to achieve chasynchronization with frequency ratio 2:1 we use the samexperimental setup as shown in Fig. 7, but the parametersthe drive, response and auxiliary systems are set toC185108 nF, C1'160 nF, L1'75 mH, r 1'162 V, andR1'4.32 kV. The parameters values of the response anauxiliary circuits areC28'224 nF, C2'340 nF, L2'145mH, r 2'357 V, andR2'4.69 kV. Values of the controlparameters,a, for the nonlinear converters in driving andresponse circuits area1'14 anda2'21.4. Chaotic oscilla-

FIG. 11. The projections of the chaotic attractors measured in uncoupled circuits with sampling rate 20ms. ~a! (x1 ,x3) projection of the attractor in the drivingcircuit. ~b! (y1 ,y3) projection of the attractor in the response circuit,g50.

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FIG. 12. Projections of chaotic attractor onto the plane of phase variables (y1 ,y3) in the response circuit~left! and onto plane of variables (y1 ,z1) of auxiliaryand response circuits~right!. The parameters of the coupling are~a! and ~b! g52.0, ~c! and ~d! g51.6, ~e! and ~f! g50.5.

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tions in uncoupled circuits (g50) correspond to attractorshown in Fig. 11. In this case the driving circuit oscillattwice as fast as the response circuit.

1. Synchronization and regularization

When the coupling parameter is set to the valg52.0, we observe synchronized chaotic oscillations wfrequency ratio 2:1. The fact of the onset of synchronizatwas detected via analysis of oscillations in the auxiliary aresponse circuits. The projections of the synchronized cotic attractor are presented in Figs. 12~a! and 12~b!. Compar-ing this synchronized chaotic attractor@Fig. 12~a!# with theattractors of the uncoupled circuits~Fig. 11!, one can see thasynchronized chaotic oscillations are more regular than

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both the uncoupled response circuit and the driving circThe fact of the regularization is also confirmed by analysisspectra of chaotic signals; see Fig. 13.

Studying this regime of synchronized oscillations, wfound that regularization is typical within the synchroniztion zone with frequency ratio 2:1. A naive explanation fchaos regularization observed in the experiment can befollowing. The chaotic driving signalx1(t) contains dc andperiodic components; see Fig. 11~a!. These components cabe considered as additional parameters of the responsecuit. The values of these parameters are proportional tovalue of coupling parameterg. When the coupling is strongenough these parameters change the dynamics of thesponse circuit toward a regime of stable periodic motion.

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this periodic motions is a ‘‘strong’’ attractor, then irregulcomponents of the driving signal will be substantially supressed in the response system. However, despite thethat such changes in the response circuit are responsiblthe onset of synchronization, this naive explanation canaccount for regularization by chaos observed in our expment. First, the amplitude of chaotic component of the dring signal is also proportional to the value of coupling prameter. Second, comparing the values of dc componand the amplitudes of periodic components in the drivsignals of this experiment@see Fig. 11~a!# and the experi-ment of synchronization with the frequency ratio 1:2@seeFig. 8~a!# one can see that these parameters of the drivsignals are almost the same in both cases. However, incase of chaos synchronization with frequency ratio 1:2,phenomena of regularization by chaos is not observTherefore, these experiments indicate that relation betwcharacteristic frequencies of driving and response systemcrucial for such regularization.

FIG. 13. ~a! The spectrum of the driving signal,x1(t). ~b! The spectrum ofthe signal,y1(t), measured from the uncoupled response circuit~dashedline! and the synchronized response circuit withg52.0 ~solid line!.

FIG. 14. The diagram of the experimental setup.

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Due to the low-dimensionality of the chaotic attractor inthe phase space of the driving system, the attractor canconsidered as a closed band of the trajectories flow. Durinone rotation along the attractor this band increases its widtbecause of local instability of the trajectories, and then, thband folds to fit the initial width. Since the response circuit issynchronized with the frequency ratio 2:1, the band of thchaotic attractor in the response system is subject to doubfolding. As the result of the double folding the trajectories othe attractor in the response system are more twisted thanis in the driving system. We believe that regularization in thesynchronized response system is associated with this mutiple folding induced by the topology of the fast driving at-tractor. However, in order to support this statement mordetailed numerical studies have to be done.

2. Auxiliary system and multistability

When the value of coupling parameterg decreases, weobserve ‘‘period doubling bifurcations’’ associated with syn-chronizedchaoticmotions; see Figs. 12~a!, 12~c!, and 12~e!.After the first period doubling bifurcation the response cir-cuit has two regimes of synchronized motions. In the phasspace of the response system both regimes correspond tosame attractor@see Fig. 12~c!#. The only difference betweenthese regimes is the ‘‘phase’’ of the synchronized oscillations. These regimes of oscillations can be distinguished bmeans of an auxiliary system. In the phase spaceresponse-auxiliary systems, these synchronized oscillatiocorrespond to different attractors. Figure 12~d! shows theprojections of these synchronized attractors onto the planevariables of the response and auxiliary circuits. ‘‘In-phase’motions ~shown by dots! correspond to the trajectories lo-cated on the diagonal; see Fig. 12~d!. The trajectories of‘‘out-of-phase’’ oscillations are shown by a dashed line. Thisexample illustrates that the onset of identical oscillations iauxiliary and response systems is sufficient, but not necesary condition for chaos synchronization between drive anresponse systems.

The transition from ‘‘period-1’’@Fig. 12~a!# to ‘‘period-2’’ @Fig. 12~c!# motions occurs not at a bifurcation point butwithin a certain interval of the coupling parameter valueswhere the intermittent behavior is observed. This intermittency is characterized by switching the ‘‘phases’’ of synchronized ‘‘period-2’’ motions. From the viewpoint of predict-ability these intermittent chaotic oscillations are notsynchronized. Outside this interval of the parameter valuethe trajectories flows of the ‘‘period-2’’ attractor are wellseparated, and the behavior of the response circuit can agbe predicted.

In the experiment we clearly detect two consequent period doubling bifurcations from the chaotic motions. Everysuch bifurcation doubles number of coexisting synchronizechaotic attractors in the phase space of response-auxiliasystems. After destabilization of ‘‘period-4’’ motions the re-sponse system forms chaotic attractor presented in Fi12~e!. This attractor is characterized by the intermittency thais caused by ‘‘phase’’ switching within the synchronized

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FIG. 15. The images of periodically synchronized and unsynchronized chaotic oscillations. Synchronized attractors are shown in~a! and~c!. Unsynchronizedattractors are shown in~b! and~d!. Intersections of the trajectories with the Poincare´ cross sections are shown by pluses~bright dots! against the backgroundof the attractor projections, which are plotted by small dots. The parameters of the external driving are~a! A5304 mV,F51.155 kHz;~b! A5304 mV,F51.169 kHz;~c! A5304 mV,F51.727 kHz;~d! A5304 mV,F51.723 kHz.

.

t-

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chaotic set. Due to this switching the behavior of the rsponse systems cannot be predicted; see Fig. 12~f!.

IV. SYNCHRONIZATION OF CHAOS BY PERIODICSIGNALS

As was indicated in the previous section, the onsetsynchronized chaos is accompanied by phase lockingpairs of periodic orbits embedded in the chaotic attractorsdriving and response systems. This phase locking betwthe periodic orbits can be considered as a possible bifurtion scenario for the transition from nonsynchronized chaooscillations to synchronized chaos. The phase lockingsaddle periodic orbits can also be used to ‘‘synchronize’’ tchaotic oscillations to an external periodic signal. This ‘‘sychronized’’ chaotic behavior can be achieved if the externperiodic signal can provide phase locking for the periodorbits embedded in the chaotic attractor of autonomous stem.

In order to demonstrate this type of synchronized behaior we will consider an experiment where the response ccuit is driven by a periodic signaludr(t)5A sin(2pFt). Thecircuit diagram of the experiment is shown in Fig. 14. In thexperiment the signaludr(t) is added to the signala f (y),which is produced by the nonlinear converterN. The param-eters of the response circuit are selected to have the s

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values as the parameters of the driving circuit shown in Fig7. As a result, the uncoupled response circuit~with A50)generates the chaotic oscillations that correspond to the atractor shown in Fig. 8~a!.

A. Phase locked chaotic attractors

We consider two examples of ‘‘synchronized’’ chaoticoscillations. In the first example the oscillations are synchronized with frequency ratio 1:1. In the second, the frequencratio is 2:3. The chaotic attractors measured inside and ouside the synchronization zone are shown in Fig. 15. In ordeto study the phase locking, the experimentally measured atractors were examined on the cross sections conditionedthe period of the external driving. In the plots of the attrac-tors, the intersections of the trajectory with such Poincar´cross sections are marked by small pluses. In order to swhere these intersections are located on the attractor, we pthese stroboscopic observations against the backgroundthe projection of the attractor shown by small dots. Comparing the attractor in Fig. 15~a! with the attractor in Fig. 15~b!,one can see that the chaotic oscillations of the first attractoare phase locked with the external periodic signal. Indeed, aintersections of the trajectory of the phase locked attractoare located in a narrow domain on the top of the attracto@see Fig. 15~a!#, and the attractor itself looks very similar to

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FIG. 16. ~a! Projection of a stable periodic motion appeared inside the synchronization zone.A51.1 V,F51.723 kHz.~b! The Poincare´ cross section of theunsynchronized chaotic attractor appeared after the periodic motion becomes unstable.A51.52 V,F51.723 kHz.

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the chaotic attractor observed in the unperturbed syst@compare Figs. 8~a! and 15~a!#. When the parameters of thedriving signal are taken outside the synchronization zone,points of the intersections are scattered all over the chaoattractor; see Fig. 15~b!. Qualitatively similar transitions be-tween the periodically ‘‘synchronized’’ and ‘‘nonsynchronized’’ chaotic oscillations are observed in the second eample, which is presented in Figs. 15~c! and 15~d!.

B. Chaotic and periodic orbits

Let us discuss a mechanism that explains this typesynchronization of chaotic oscillations. Since a chaotictractor contains an infinite number of saddle periodic orbithe motion of the system along the chaotic trajectory canconsidered as switching between nearest saddle periodicbits. Therefore, the set of unstable periodic orbits forms‘‘skeleton’’ of the chaotic attractor and define time charateristics of the chaotic oscillations. External periodic forcinapplied to the system can lock a sufficient number of theperiodic orbits. When this ‘‘skeleton’’ becomes phase lockwith the periodic forcing, the chaotic trajectory switches btween phase locked saddle periodic orbits. A well-knowexample of a chaotic attractor that contains the period‘‘skeleton,’’ whose phases are related to the phase of exnal periodic signal, is the chaotic attractor in the periodicadriven Duffing’s equation. In other words, the qualitativdifference between the nonsynchronized and periodica‘‘synchronized’’ chaotic oscillations is the same as betwelow-dimensional chaotic oscillations in an autonomous sytem and the chaotic oscillations appearing due to a periodriving of a two-dimensional nonlinear system, which is na self-excited oscillator.

In the type of synchronization considered, the periodforcing is needed only for locking the phases of the periodorbits, but not for suppression of the instability. Due to thfact, such synchronization does not require large forcing aplitude. The existence of certain threshold value of the aplitude for this type of synchronization is caused, mainly, bthe spread of the main frequencies of the saddle perioorbits embedded in the synchronizing attractor.

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C. Chaos suppression

Inside the synchronization zone the phase locked attrtor is subject to bifurcations that can lead to the onsetstable periodic motions. For example, when the amplitudedriving increases, the circuit demonstrates the inversequence of the period doubling bifurcations from the phalocked chaotic attractor shown in Fig. 15~c!. These bifurca-tions lead to the formation of different stable periodic orbiOne of these periodic orbits is presented in Fig. 16~a!. Theonset of stable periodic oscillations in the chaotic systembe considered as a mechanism for chaos suppression. Incase, the regime of stable periodic oscillations is obserwithin a certain interval of the amplitude valuesA. When thedriving becomes too strong the periodic motion losesstability and the system switches to unsynchronized chabehavior. The cross section of the attractor that correspoto this nonsynchronized behavior is shown in Fig. 16~b!.

The suppression of chaos in dynamical systemsmeans of external periodic forcing is a well-known phenoenon, see, for example, Refs. 31–35. Usually the suppresof chaos is provided by the onset of stable periodic oscitions. The possibility of the appearance of stable periomotions in the phase space of the driven system is notprising. Indeed, any static or periodic influence applied tononlinear system changes the parameters of the sysWhen parameters of a chaotic system change the sysgoes through various bifurcations. In many cases these bcation sequences are characterized by alternation of chaand periodic regimes of oscillations. By selecting the paraeters of external forcing inside a periodic window, one csuppress chaos in the system.

V. THRESHOLD SYNCHRONIZATION

In this paper we have illustrated only the regimessynchronized chaos that we observed in the coupled osctors of smooth waveforms. It is clear that these regimesnot capture all possible forms of synchronized chaos. Finstance, a different form of synchronized chaotic osciltions is observed in coupled relaxation oscillators. This tyof chaos synchronization has a more transparent mechan

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and its dynamical features are different from the cases csidered here. The nonlinear theory of synchronizationtween periodic relaxation oscillators was intensively studin applications to electronic systems36–38 and in biologicalsystems; see, for example, Refs. 39 and 40 and referetherein. The ability of relaxation oscillators to demonstraindividual chaotic dynamics was shown in many papers; sfor example, Refs. 41 and 42. The theoretical studies ofonset of synchronized chaotic pulsations and experimeobservations of synchronization between chaotic relaxacircuits can be found in the papers.43,44It was experimentallydemonstrated in the paper44 that the synchronization and suppression of chaos in relaxation oscillators can be providedexternal pulses. It was shown that there are chaotic relaxaoscillators that can, theoretically, be synchronized by aquence of external pulses of arbitrary small amplitude.course, this sequence of pulses has to be perfectly matcha chaotic or an unstable periodic regimes of pulsations inunperturbed oscillator. This indicates the difference in fetures of the chaos synchronization between this class oflaxation oscillators and the oscillators of smooth waveform

VI. CONCLUSIONS AND OUTLOOK

Using the results of experimental observations, we hdiscussed a few forms of synchronized chaotic behavior.spite the fact that we did not review all studied casessynchronized chaotic behavior, we hope that the imagesynchronized chaos presented in this paper give an idethe variety of different forms of chaos synchronization.

Since the general framework for the phenomenon of schronization of chaotic oscillations is not available, we rstricted our considerations to the case of identical chaoscillations, and the cases of synchronized chaos in drresponse systems. In the case of identical chaotic osctions, the synchronized chaotic attractor has a rather simimage. In the phase space of the coupled systems, the trtories of this attractor are located on the integral manifowhich has the form of a hyperplane.

The cases of drive-response systems are simplified insense that the motions on the synchronized chaotic attraare related to the motions on the attractor in the phase spof the driving system. To study the onset of chaos synchnization in these systems we use the fact of stable predability of the response behavior from the driving chaotic snal. Using the auxiliary system as the predicting device,reduce the problem of the investigation of complex synchnized trajectories in the phase space of the drive-resposystem to the analyses of stability and robustness of idenoscillations in response-auxiliary systems.

Analysis of chaotic signals measured from synchronizsystems indicates phase locking between saddle periodicbits embedded in the chaotic attractor in the phase spacthe driving system, and the saddle periodic orbits in the schronized attractor in the phase space of the response sysIt seems plausible that such phase locking between the pof saddle periodic orbits in the coupled system is a bifurtion mechanism for the onset of synchronized chaotic os

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ncesteee,thentaltion

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yn-e-oticive-illa-pleajec-ld,

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lations. We believe that the idea of phase locking betweesaddle periodic orbits can be useful for building a generframework for synchronized chaotic oscillations.

In all examples of synchronized chaos considered in thpaper, the interacting systems are physically separated. Iobvious that the phenomenon of synchronization can taplace in the high-dimensional systems in which this separtion is not clear. There are many known examples where tlow-dimensional chaotic attractor appears in a system wihigh-dimensional phase space. This low-dimensional behaior of the system can also be explained by synchronizatiobetween different portions of the system. Due to this synchronization a small portion of the system can be consideras a driving subsystem, while the other part of the systeoperates as a synchronized response subsystem. It is cthat the partition of the system into the synchronized susystems may be a very difficult problem. We believe thadetailed studies of typical routes of the onset of synchronizechaos will give a background not only for detection of chaosynchronization but also for the partitioning of high-dimensional systems.

ACKNOWLEDGMENTS

The author is very grateful to his colleagues at the Insttute for Nonlinear Science for fruitful discussions. Stimulating discussions and the encouragement of M. I. Rabinovicwere especially helpful. This work was supported in part bGrants No. DE-FG03-95ER14516 and No. DE-FG0390ER14148 from the U.S. Department of Energy.

APPENDIX A: MECHANISMS FOR SYNCHRONIZATIONOF PERIODIC OSCILLATIONS

In this appendix we discuss basic principles of synchronization of periodic oscillators by external periodic signals

1. Threshold synchronization

At first, we consider the mechanism for synchronizatioin relaxation oscillators. Being a dissipative system, a relaation oscillator contains a positive feedback or negative rsistance that destabilizes static states. It also has a nonlinelement and a time defined element. The nonlinear elemedefines a threshold level and determines the amplitudegenerated pulsations. The time interval between the pulstions is controlled by the time defined element. This elemesets the time that is required by the system to recover froprevious firing and to become ready for the next firing. During the recovery phase the system ‘‘slowly’’ approachesthreshold level where it fires by itself; see Fig. 17~a!. Imme-diately before the threshold state the system is extremesensitive to small external perturbations, which may cauthe firing. Therefore, if small external pulses appear at thtimes when they are able to push the system through tthreshold level, the firings will be in synchrony with thepulses; see Fig. 17~b!. In addition to this, these externalpulses change the period of the relaxation oscillations bshortening the process of slow evolution. It is clear that thsynchronization with external pulses of small amplitude ca

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be achieved only if the period of these pulses is related toduration of the recovering process. However, mention shobe made that, in some cases, the external driving can signcantly influence the initial conditions and/or the dynamicsthe recovering process.

2. Synchronization and beats

Synchronization in generators of smooth waveformmainly results from the action of dissipative forces. To illutrate the synchronization of periodic oscillations, we will usa simplified description of the synchronization mechanisBeing a dissipative system, any generator of periodic oslations contains a positive feedback or negative resistathat destabilizes stationary states, and a nonlinear elemthat controls the amplitude of the oscillations. Unlike threlaxation oscillator, in the generator of smooth waveformthe frequency of the oscillations is usually defined by a resnant element. Due to dissipation, the resonant element renates within a certain frequency band. The nonlinearity adissipation in the system select a particular form of periodoscillations~mode!. The stability of this mode indicates thathis mode is preferred energetically and, therefore, the mocorresponds to a minimum of the energy functional. Due

FIG. 17. Sketches of phase portraits~left! and waveforms~right! of anautonomous~a! and a synchronized~b! periodic relaxation oscillator.~a!Fast and slow motions along the stable limit cycleP correspond to the firingand recovering, respectively. Transitions from slow to fast motions occuthe threshold point.~b! The external pulses interrupt the slow motions aninduce firing. When the period of the external pulses is a little less thantime interval of slow motions, and the amplitude of the pulses is sufficientpush the system through the threshold, then the external pulses synchrothe firings.SP is the image of the synchronized periodic oscillations.

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the existence of a band of resonant frequencies, the chateristic frequencies of this mode change with a small vartion of the parameters of the oscillator.

An external periodic forcing applied to the oscillator wiperiodically modulate the ‘‘environment’’ in which the intrinsic mode of oscillations lives. As a result of this modlation, the nonlinearity and dissipation will again tendchange the intrinsic mode of oscillations to a mode thatpreferred energetically. A significant role in this process cby played by resonance between the perturbations andmode. Therefore, the frequencies of the mode tend tocome related to the frequencies of external forcing. If tperturbations introduced by the external forcing are not sficient for the proper change of the intrinsic mode, then tsystem will be out of synchronization and will demonstraquasiperiodic oscillations that are known as beats. Inphase space of the system these quasiperiodic oscillatcorrespond to the unlocked trajectories on the stable tosee Fig. 18~c!.

A. Weak forcing

There are two well-known mechanisms that descrchange of the intrinsic mode of oscillations. One of themechanisms is self-tuning of the frequencies of the unp

r atdthetonize

FIG. 18. Fragments of tori, periodic motions, and their Poincare´ cross sec-tions illustrating two mechanisms for the onset of synchronized periooscillators. The trajectories on the Poincare´ cross sections are sketched ouby thick curves. Synchronized periodic motion is marked byPS . ~a!, ~b!,and ~c! illustrate the transition from synchronized to unsynchronized oslations in a system with weak periodic driving. The transition from synchnized oscillations that correspond to stable periodic motionPS on the torus~a! to the quasiperiodic motions~c! due to the tangential bifurcation assocated with stablePS and unstablePU periodic motions. These periodic motions merge with each other~b! and then disappear~c!. The onset of syn-chronization caused by strong periodic driving is illustrated by the transitfrom ~c! to ~d!. The torus merges into the unstable periodic motionP0 ~c!and as a result, the periodic motion becomes stable~d!.

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turbed intrinsic mode. This self-tuning is caused by dissitive forces, therefore it is directed toward one of the minimof the energy functional. Being an integral characteristic tfunctional depends upon the phase relation betweenmode and the perturbations induced by the external sigAs a result of it, this dependence has a periodic structDue to this periodicity, a stable mode of locked frequencalways appears together with an unstable mode of locfrequencies. In the phase space of the oscillator, this ccorresponds to the formation of stable and unstable perimotions on a torus; see Figs. 18~a!–18~c!. This mechanism istypical for external forcing with small amplitude. In orderachieve synchronization with small amplitude, the frequcies induced by the forcing have to be very close to renance with the frequencies of the unperturbed mode ofcillations. It is clear that the amplitude of perturbatiorequired for locking depends on how much the spectrumthe inner mode has to be changed in order to provide a rnance with the perturbations induced by the external forcTherefore, when the frequencies of the perturbations arresonance with frequencies of an unperturbed intrinsic mothen the amplitude of the external forcing that is sufficiefor locking can be infinitely small. Of course, this is correonly for the synchronization of stable periodic oscillationssystems without noise.

B. Strong forcing

The second mechanism for the change of the intrinmode is the competition between the inner and indumodes excited in the oscillator. Due to nonlinearity and dsipation one mode of oscillations causes strong dissipafor the other modes. As a result of this competition the strgest mode will tend to eliminate the other modes. Tmechanism is observed for forcing of large amplitude. In tcase the independent intrinsic mode simply disappears.mechanism can also be distinguished from the viewpoinbifurcation theory. It is characterized by the merging of ttorus into unstable periodic motion; see Figs. 18~c! and18~d!. As the result of this bifurcation, the torus disappeand the periodic motion becomes stable.

APPENDIX B: IMPLEMENTATION OF THE CIRCUIT

In order to study chaos synchronization in experimenwe employ electronic circuits whose individual dynamics aknown to be chaotic. The chaotic circuit consists of a nolinear converterN and linear feedback; see the diagramthe driving circuit in Fig. 1. Chaotic circuits of such archtecture was studied by Dmitrievet al.45 In our experimentswe used a nonlinear converter whose diagram is showFig. 19. The converter transforms input voltagex into anoutput, which has nonlinear dependencea f (x) on the input.The parametera stands for the gain of the converterx50. The shape of the nonlinearity produced by the cverter,N, is shown in Fig. 20.

The linear feedback contains the low pass filter (RC8)and the resonant circuitrLC. Depending on the parameteof the linear feedback and the parameter of the nonlinea

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a, the circuit can generate various regimes of periodic anchaotic oscillations. It can be easily shown that dynamics othe circuit is described by the following system of differen-tial equations:

x15x2 ,

x252x12dx21x3 , ~B1!

x35g@a f ~x1!2x3#2sx2 .

x1(t) is the voltage across the capacitorC,x2(t)5AL/CJ(t), with J(t) the current through the inductorL. x3(t) is the voltage across the capacitorC8. Time hasbeen scaled by 1/ALC. The parameters of this system havethe following dependence on the physical values of the circuit elements:g5ALC/RC8, d5rAC/L ands5C/C8.

FIG. 19. The diagram of the nonlinear converter.Rn152.7 kV,Rn25Rn457.4 kV, Rn35100V, Rn55186 kV, Rn652 kV. The diodesD1 and D2 are of the 1N4148 type. The operational amplifiers OP1 and OPare both the TL082 type and the operational amplifier OP3 is of the LF356Ntype.

FIG. 20. The shape of the nonlinear functionf (x) measured in the experi-ment.

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