ATLANTIS STUDIES IN MATHEMATICS FOR ENGINEERING AND SCIENCE

VOLUME 8

SERIES EDITOR: C.K. CHUI

Atlantis Studies inMathematics for Engineering and Science

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Nonlinear HybridContinuous/Discrete-Time

Models

Marat Akhmet

Middle East Technical University, Ankara, Turkey

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Atlantis Studies in Mathematics for Engineering and ScienceVolume 1: Continued Fractions: Volume 1: Convergence Theory – L. Lorentzen, H. WaadelandVolume 2: Mean Field Theories and Dual Variation – T. SuzukiVolume 3: The Hybrid Grand Unified Theory – V. Lakshmikantham, E. Escultura, S. LeelaVolume 4: The Wavelet Transform – R.S. PathakVolume 5: Theory of Causal Differential Equations – V. Lakshmikantham, S. Leela, Z. DriciVolume 6: The Omega Problem of all Members of the United Nations – E.N. ChukwuVolume 7: Boundary Element Methods with Applications to Nonlinear Problems – G. Chen, J. Zhou

ISBNsPrint: 978-94-91216-02-2E-Book: 978-94-91216-03-9ISSN: 1875-7642

c© 2011 ATLANTIS PRESS

To my family.

Preface

The dynamical systems theory was shaped by the investigation of stellar and planetary

motions. The applications were later extended to isolated mechanisms, electronics and

other processes. Analysis was dominated by assumptions of continuity and extension to

infinity. At present we face new challenges, such as events with a finite life and models of

large dimensions. The processes are interconnected, rather than isolated, and the motions

are alternately continuous and discrete. To meet these challenges we need appropriate

tools: differential equations with a discontinuous right-hand side, differential equations

with impulses and hybrid systems.

Hybrid systems are a recent concept in dynamical systems theory and have important ap-

plications. Hybrid dynamical systems have not been defined in a precise way yet. Roughly

speaking, we call a model as hybrid if it is a combination of continuous and discrete

dynamics. One can say that a system is hybrid if some of the dependent variables sat-

isfy differential, while others - discrete equations. In addition, one may have a variable

that satisfies a differential and a discrete equation alternately. More formal and general

definitions of hybrid systems are given in [279, 280]. Further information can be found

in [55, 80, 96, 97, 101, 128, 145, 194–196, 201, 209, 210, 213, 227, 231, 279, 280, 282, 338].

In the early 80’s, K. Cook, S. Busenberg, J. Wiener and S. Shah started developing a new

type of differential equations. They called these differential equations with piecewise con-

stant argument. Many interesting results and many applications of this theory have been

produced in the last three decades. The existence and uniqueness of solutions, oscillations

and stability, integral manifolds and periodic solutions, and numerous other issues have

been intensively discussed. Besides the theoretical analysis, various models in biology,

mechanics and electronics were introduced through these systems.

The original method of investigation of these equations was based on the reduction to dis-

vii

viii Nonlinear Hybrid Continuous/Discrete-Time Models

crete systems. That is, only the values of solutions at moments that are integers or multiples

of integers were discussed. Moreover, systems must be linear with respect to the values

of solutions, if the argument is not deviated. These requirements significantly limit the

theoretical depth of investigation, as well as the scope of real world problems that can be

modeled using these equations. From the analysis of MathSciNet, we found that the num-

ber of papers on differential equations with piecewise constant argument published over

the last ten and five years is 138 and 69, respectively. This number is 12 in 2009, and 11 in

2010. Thus, it is clear that the rate of publication is quite low and has not changed over the

last ten years, and that a new methodological approach is necessary to develop the theory

further.

In the Conference on Differential and Difference Equations at the Florida Institute of Tech-

nology, 2005, it was proposed [8] to consider non-linear differential equations with a more

general type of piecewise constant argument, and equivalent integral equations, as the basis

of investigation. In our opinion, this line of thought should provide new direction for the

theory. In this book, we will show in detail how this can be done. The reader will see that

not only can the scope of problems studied be expanded through this approach, but one can

also observe entirely new phenomena and deepen the parallelism with the theory of ordi-

nary differential equations, despite the fact that the systems under discussion are functional

differential equations.

Next, a compartmental model of blood pressure distribution is considered. The systemic

arterial pressure is assumed to be dominating. The heart contraction moments are supposed

to be prescribed, and the cases where they behave periodically and almost periodically are

examined.

We also consider the case where the moments are defined recursively, so that chaotic phe-

nomena appear. Additionally, the case where the moments are not prescribed and are vari-

able is investigated.

Finally, a biological model of integrate-and-fire oscillators is examined. The main result

is the solution of the synchronization problem. That is, we find conditions that ensure

that identical or not quite identical units of the system fire in unison. This problem is of

extreme importance for the cardiac pacemaker research. Examples with numerical simula-

tions are provided to validate the theoretical results, and prospects for further investigation

are discussed.

The first chapter serves as the introductory part of the book. We present the description of

the systems and the main definitions, and outline the literature.

Preface ix

In the following chapter, linear and quasilinear systems with argument functions that are

both advanced and delayed, are considered. It is shown that the set of all solutions of the

linear homogeneous equation is a finite-dimensional linear space under certain conditions.

The fundamental matrix of solutions is built. For quasilinear systems, the integral repre-

sentation formula is defined. Moreover, basic concepts of stability theory are provided for

these equations.

In the third chapter, we prove the reduction principle for differential equations with piece-

wise constant arguments. Theorems on the existence of integral manifolds, their stability,

and the stability of the zero solution are proved.

The fourth chapter is devoted to periodic solutions of perturbed linear systems. The method

of the small parameter is used as an instrument in this chapter. Both non-critical and critical

cases are investigated. Continuous and differentiable dependence of solutions on initial data

is examined to prove the main results. Simulations are provided to illustrate the theoretical

analysis.

The stability of nonlinear systems is analyzed in the fifth chapter. We use the Lyapunov-

Razumikhin technique, as well as the method of Lyapunov functions, to investigate stabil-

ity. We apply the obtained theorems to prove the stability of the zero solution of the logistic

equation.

Differential equations with state-dependent piecewise constant argument are described in

the sixth chapter. We analyze the conditions of existence and uniqueness of the solutions of

these systems in a general case, and for quasilinear equations. Then periodic solutions and

the stability of the zero solution are discussed. The theory of differential equations with

discontinuous right-hand-sides and with variable moments of impulses shows the way to

developing the theory of systems with state-dependent piecewise constant argument.

The existence of almost periodic solutions is the main subject of the seventh chapter. Using

Bohner type discontinuous almost periodic functions and the technique developed in [117],

we prove that the exponential dichotomy is a sufficient condition for the existence of these

solutions.

The following chapter deals with the stability of neural networks. The biological explana-

tions of advance and delay arguments are discussed. The methods of Lyapunov functionals

and functions are applied to obtain the main results. Several examples are provided to

illustrate the theorems.

Blood pressure distribution is investigated in the ninth chapter. A model where the sys-

x Nonlinear Hybrid Continuous/Discrete-Time Models

temic arterial pressure dominates is developed. The existence of periodic, almost periodic

motions, stability, and chaotic behavior are examined. We consider both cases where the

moments of jumps of the pressure are prescribed, and where they are variable. In the last

case the threshold concept is formalized.

The last chapter of the book is devoted to the integrate-and-fire model of biological os-

cillators. This model is appropriate for cardiac pacemaker analysis, as well as for neural

networks research. Mathematical models of integrate-and-fire biological oscillators, as far

as we know, were initiated in [260,262]. In this book, a method of analysis of integrate-and-

fire models that consist of pulse-coupled biological oscillators is developed. The method

is based on a thoroughly constructed map and the technique of investigation of differential

equations with discontinuities at non-fixed moments.

Synchronization and existence of periodic motions of identical and not quite identical os-

cillators are investigated. The second Peskin’s conjecture, [262], has been solved. The syn-

chronized regime of identical oscillators admits moments of fire that are equally-distanced.

Their discrete dynamics can be represented as a solution of simple differential equations, if

needed. Consequently, the model belongs to the class of systems considered in our book.

This book contains very recent results, and any comments and suggestions would be greatly

appreciated.

The author would like to thank Duygu Arugaslan, Cemil Buyukadalı, Mehmet Onur Fen,

Mehmet Turan and Enes Yılmaz for discussions of the problems considered in the book

and for collaboration on several joint papers.

Contents

Preface vii

1. Introduction 1

2. Linear and quasi-linear systems with piecewise constant argument 17

2.1 Linear homogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Quasi-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3. The reduction principle for systems with piecewise constant argument 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Existence of integral surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Stability of the zero solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Stability of the integral surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.5 Reduction principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4. The small parameter and differential equations with piecewiseconstant argument 49

4.1 Periodic solutions: the non-critical case . . . . . . . . . . . . . . . . . . . . . . . 494.2 Dependence of solutions on parameters . . . . . . . . . . . . . . . . . . . . . . . 534.3 Periodic solutions: the critical case . . . . . . . . . . . . . . . . . . . . . . . . . . 59Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5. Stability 71

5.1 The Lyapunov-Razumikhin method . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.1 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 The method of Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2.2 Applications to the logistic equation . . . . . . . . . . . . . . . . . . . . 86

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6. The state-dependent piecewise constant argument 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Quasilinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xi

xii Nonlinear Hybrid Continuous/Discrete-Time Models

6.4 Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.5 Stability of the zero solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7. Almost periodic solutions 105

7.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Wexler sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.3 Bohr-Wexler almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . 1127.4 Almost periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8. Stability of neural networks 121

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.4 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

9. The blood pressure distribution 133

9.1 Systemic arterial pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339.1.1 Background on systemic arterial pressure . . . . . . . . . . . . . . . . . . 1339.1.2 The regular and irregular behavior of systemic arterial pressure when mo-

ments of the heart contraction are prescribed . . . . . . . . . . . . . . . . 1359.1.3 Regular behavior of systemic arterial pressure with non prescribed

moments of the heart contraction . . . . . . . . . . . . . . . . . . . . . . 1409.1.4 Investigation of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.2 The global model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 1439.2.2 Bounded, periodic, eventually periodic and almost periodic solutions . . . 1479.2.3 Stability and positiveness . . . . . . . . . . . . . . . . . . . . . . . . . . 1529.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.3 Chaotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.3.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 1579.3.2 Stability and positiveness of bounded and periodic solutions . . . . . . . . 1589.3.3 Blood pressure dynamics as a special initial value problem: chaotic behavior1609.3.4 Periodic solutions revisited. Eventually periodic solutions . . . . . . . . . 1619.3.5 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1649.3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

10. Integrate-and-fire biological oscillators 175

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17510.2 The prototype map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17710.3 Non-identical oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18410.4 The Kamke condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18910.5 The delayed pulse-coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.5.1 The couple of identical oscillators . . . . . . . . . . . . . . . . . . . . . . 191

Contents xiii

10.5.2 Non-identical oscillators: the general case . . . . . . . . . . . . . . . . . 19510.5.3 The simulation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Bibliography 201