Control of Homoclinic Chaos by Weak Periodic Perturbations

CONTROl OF HOMOCN lC l CHAOS BY WEAK PERIODIC PERTURBATIONS

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CONTROl OF HOMOCllNlC tHIOS BY WEAK PERO I DC I PERTURBATIONS Ricardo Chacon University of Extremadura, Spain

vp World Scientific N E W JERSEY

LONDON * SINGAPORE * BElJlNG * S H A N G H A I * HDNG KONG

TAIPEI

CHENNAI

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CONTROL OF HOMOCLINIC CHAOS BY WEAK PERIODIC PERTURBATIONS Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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To my son abrahan who has added a

wonderful, although certainly uncontrollable, chaos to our lives

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PREFACE An exciting and extremely active area of multidisciplinary investigation during the past decade has been the problem of controlling chaotic systems. Indeed there have been a number of books written which have served to review a wide variety of chaos control theories, methods, and perspectives. The main reasons for such interest are the interdisciplinary character of the problem, the implicit promise of a better understanding of chaotic behavior, and the possibility of successful applications in such diverse areas of research as aerodynamics, biology, chemical engineering, epidemiology, electric power systems, electronics, fluid mechanics, laser physics, physiology, secure information processing, and so on. The subject of chaos control exhibits at present a huge spectrum of methods and techniques based on different perspectives. While useful, some of the aforementioned books have suffered from the too ambitious goal of attempting to discuss very concisely every method. On the other hand, many of the papers published on control (suppression/enhancement) of chaos by additional time-dependent excitations (forcing or parametric excitation) have been based on the results of computer simulations. That is why the author’s goal was to write a monograph which would give a reasonably rigorous theory of a particular but highly relevant control technique: the suppression/enhancement of chaos by weak periodic excitations in low-dimensional, dissipative, and non-autonomous systems. Controlling chaos is therefore understood as a procedure which suppresses chaos when it is unwanted, and enhances existing chaos or gives rise to chaos in a dynamical system when it is useful. This book is not meant either to compete with the findings of other authors or to repeat known mathematical tools. Except in a very few places, results published by other authors are not reviewed. This monograph begins with an introduction where the method of controlling chaos by weak periodic excitations is approached from the general idea of the control of nonlinear dynamical systems. Some relevant aspects of the technique, such as its flexibility, robustness, scope, and experimental applicability are also discussed. Emphasis is put on the comparison between harmonic and non-harmonic excitations. Chapter 2 presents an intuitive argument to illustrate how added periodic excitations modify the stability of perturbed generic limit cycles. The class of chaotic, dissipative, and non-autonomous dynamical systems to be controlled is described as well as Melnikov’s method, which is the analytical technique used to obtain the vii

viii

PREFACE

theoretical results. For the sake of clarity, the cases with and without noise are studied separately. Also, it is shown that the maximum survival of the symmetries of solutions from a wide class of dynamical systems, subjected t o both a primary chaos-inducing and a chaos-controlling excitation, corresponds to the optimal choice of the control parameters. For the purely deterministic case, the theory considers separately the cases of subharmonic resonance and non-subharmonic resonance between the chaos-inducing and chaos-controlling excitations. The theorems provide analytical estimates of the ranges of parameters (of the chaos-controlling excitation) for suppression/enhancement of the initial chaos. A generic analytical expression is discussed for the width of the intervals (of initial phase difference between the two excitations) for which chaotic dynamics can be controlled. The rational approach to the case of incommensurability between the two involved driving frequencies is analyzed. Finally, the special case of the main resonance is discussed in detail. Chapter 3 aims at discussing the physical mechanisms underlying the control of chaos by weak periodic excitations in generic systems. The notion of geometrical resonance is shown to provide such a mechanism by means of an almost adiabatic invariant associated with each geometrical resonance solution. The relations between geometrical resonance and both autoresonance and stochastic resonance are also included for completeness. Chapter 4 contains detailed studies of two relevant and interdisciplinary application problems which are ab initio mathematically well described by low-dimensional nonlinear ordinary differential equations: (i) the control of chaotic escape from a pcr tential well; (ii) the suppression of chaos in a driven Josephson junction. Also, the onset and the inhibition of chaos of charged particles in a non-ideal electrostatic wave packet is discussed. In each case, the performance of the control technique is derived theoretically and also tested by computer simulation. Chapter 5 presents a detailed application to physical problems which are well described by high-dimensional equations: (i) the control of chaos in chains of coupled chaotic Duffing oscillators; (ii) the control of chaotic solitons in Frenkel-Kontorova chains; (iii) the suppression of spatiotemporal chaos in perturbed partial differential equations such as the sineGordon and the nonlinear Schrodinger. The book concludes with a brief outline of some important open problems in the present theory and several possible future applications (such as ratchets and Bose-Einstein condensates). I am grateful t o many colleagues for many interesting discussions, exchanges, and lectures. In particular, Francisco Balibrea, who awoke my interest in the field of discrete dynamical systems, has continued to encourage me throughout and more so with respect to the writing of this book. JosC: Diaz Bejarano, my doctoral advisor, stimulated my quest into dynamical systems. His advice deserves my a p preciation. Further thanks go to Robert Chatwin, Jason Gallas, Isaac Goldhirsch, F i m Hyme, Renu Malhotra, Pedro J. Martinez, Niurka R. Quintero, Angel Shchez, Valery Tereshko, and many others for valuable discussions, useful comments, and

PREFACE

ix

their invaluable knowledge. The author thanks Drs G. Chen, L. M. Floria, L. Friedland, N. R. Quintero, S. Rajasekar, M. Salerno, A. Sanchez, S. J. Schiff, and I. B. Schwartz for kindly providing respective reprints of some of their works. The author appreciates the support, cooperation, and patience of the editorial and production stat€ at the World Scientific Publishing company. Last but not least, to my wife Yolanda and son Abrahan who are closest to me in my life, for their love and unconditional support.

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CONTENTS Preface 1 Introduction 1.1 Control of chaotic dynamical system . . . . . . . . . . . . . . . . . . 1.2 Non-feedback control methods . . . . . . . . . . . . . . . . . . . . . . 1.3 Controlling chaos by weak periodic excitations . . . . . . . . . . . . . 1.3.1 Robustness and flexibility . . . . . . . . . . . . . . . . . . . . 1.3.2 Applicability and scope . . . . . . . . . . . . . . . . . . . . . . 1.4 Harmonic versus non-harmonic excitations: the waveform effect . . . 1.4.1 Reshaping-induced strange non-chaotic attractors . . . . . . . 1.4.2 Reshaping-induced crisis phenomena . . . . . . . . . . . . . . 1.4.3 Reshaping-induced basin boundary fractality . . . . . . . . . . 1.4.4 Reshaping-induced escape from a potential well . . . . . . . . 1.4.5 Reshaping-induced control of directed transport . . . . . . . . 1.4.6 Reshaping-induced control of synchronization of coupled limitcycle oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 1 1 2 3 3 4 4 6 14 15 16 20 26 27

2 Theoretical Approach 31 2.1 Dissipative systems versus Hamiltonian system . . . . . . . . . . . . 31 2.2 Stability of perturbed limit cycles . . . . . . . . . . . . . . . . . . . . 32 2.3 Non-autonomous second-order differential systems . . . . . . . . . . . 34 2.4 Basics of Melnikov’s method . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.1 Illustration: A damped driven pendulum . . . . . . . . . . . . 38 2.5 The generic Melnikov function: Deterministic case . . . . . . . . . . . 40 2.5.1 Suppression of chaos . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.2 Enhancement of chaos . . . . . . . . . . . . . . . . . . . . . . 56 2.5.3 Case of non-subharmonic resonances . . . . . . . . . . . . . . 60 2.5.4 The special case of the main resonance . . . . . . . . . . . . . 68 2.6 The generic Melnikov function: The noise effect . . . . . . . . . . . . 80 2.6.1 Additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6.2 Multiplicative noise . . . . . . . . . . . . . . . . . . . . . . . . 84 2.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 xi

xii

CONTENTS

3 Physical Mechanisms 3.1 Energy-based approach . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Geometrical resonance . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Autoresonance . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Stochastic resonance . . . . . . . . . . . . . . . . . . . . . . . 3.2 Geometrical resonance analysis: Chaos, stability and control . . . . . 3.2.1 Geometrical resonance in a damped pendulum subjected to p e riodic pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Geometrical resonance in an overdamped bistable system . . . 3.2.3 Geometrical resonance approach to control of chaos by weak periodic perturbations . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Geometrical resonance and globally stable limit cycle in the van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Geometrical resonance in spatio-temporal systems . . . . . . . 3.3 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 91 92 94 102 106

106 110 113 116 119 121

4 Applications: Low-dimensional systems 125 4.1 Control of chaotic escape from a potential well . . . . . . . . . . . . . 125 4.1.1 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.1.2 Escape suppression theorems . . . . . . . . . . . . . . . . . . . 128 4.1.3 Inhibition of the erosion of non-escaping basins . . . . . . . . 132 4.1.4 Role of nonlinear dissipation . . . . . . . . . . . . . . . . . . . 133 4.1.5 Robustness of chaotic escape control . . . . . . . . . . . . . . 136 4.1.6 Case of incommensurate escapesuppressing excitations . . . . 139 4.2 Taming chaos in a driven Josephson junction . . . . . . . . . . . . . . 144 4.2.1 Model equation . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.2.2 Suppression of homoclinic bifurcations . . . . . . . . . . . . . 145 4.2.3 Comparison withLyapunovexponent calculations . . . . . . . 151 4.3 Suppression of chaos of charged particles in an electrostatic wave packet159 4.3.1 The three wave case . . . . . . . . . . . . . . . . . . . . . . . 159 4.3.2 Case of a general electrostatic wave packet . . . . . . . . . . . 167 4.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5 Applications: High-dimensional systems 5.1 Controlling chaos in chaotic coupled oscillators . . . . . . . . . . . . . 5.1.1 Localized control of spatio-temporal chaos . . . . . . . . . . . 5.1.2 Application to chaotic solitons in Frenkel-Kontorova chains . . 5.2 Controlling chaos in partial differential equations . . . . . . . . . . . 5.2.1 Damped sineGordon equation additively driven by two spatiG temporal periodic fields . . . . . . . . . . . . . . . . . . . . . .

181 181 181 184 190 191

xiii

CONTENTS

5.2.2 5.2.3 5.2.4 5.2.5 5.2.6 5.3 Notes

Damped sineGordon equation additively and parametrically driven by two spatio-temporal periodic fields . . . . . . . . . . Damped sineGordon equation additively driven by two temporal periodic excitations . . . . . . . . . . . . . . . . . . . . . Nonlinear Schrodinger equation subjected to dissipative and spatially periodic perturbations . . . . . . . . . . . . . . . . . 44model additively driven by two spatic-temporal periodic fields . . . . . . . . . . . . . . . . . . . . . 44model additively and parametrically driven by two spatio-temporal periodic fields . . . . . . . . . . . . . . and references . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Further Remarks and Open Problems 6.1 Openproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Beyond the main resonance . . . . . . . . . . . . . . . . . 6.1.2 Reshaping-induced control . . . . . . . . . . . . . . . . . . . . 6.1.3 Amplitude modulation control . . . . . . . . . . . . . . . . 6.2 Further applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Ratchet systems . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Coupled Bose-Einstein condensates . . . . . . . . . . . . . 6.3 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 198 202 204 207 210

213 213 . . 213 213 . . 214 216 216 . . 218 219

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Chapter 1

INTRODUCTION 1.1

Control of chaotic dynamical systems

Nowadays, the diversity of techniques, perspectives, and kind of problems that chaos control deals with contrasts sharply with classical control tasks such as that of stabilizing an equilibrium state. This is a direct consequence of taking into account two generic properties of real-world dynamical systems: nonlinearity and nonintegrability. This book is about control of homoclinic and heteroclinic chaos in nonlinear, non-autonomous and dissapative, oscillator systems by weak periodic (mainly harmonic) excitations. It describes a rapidly growing subfield of chaos control with applications to a great number of problems in engineering and science such as fluid mixing, Josephson junction arrays, and secure information processing, to mention only a few. In this monograph control of chaos is understood in the broadest sense: a procedure to enhance or suppress chaos depending upon the needs. Enhancement of chaos means increasing the duration of a chaotic transient, passing from transient to steady chaos, or increasing the leading (positive) Lyapunov exponent of the initial chaotic state. Suppression of chaos means decreasing the duration of a chaotic transient, passing from steady chaos to a regular state via transient chaos or not, or decreasing the positive Lyapunov exponent of the initial chaotic attractor. Although an exhaustive classification (and discussion) of the different methods of controlling chaos is beyond the purpose of the present work, one can roughly classify them into two kinds: feedback methods and non-feedback methods. Feedback methods inhibit chaos by stabilizing one of the unstable periodic orbits embedded in an existing chaotic attractor by means of weak time-dependent variations of a system parameter. Such control methods are important for two reasons. First, since they are based on fairly generic properties of chaotic dynamics, they are a priori applicable to a broad diversity of dynamical systems. Second, since chaotic systems present an infinite number of unstable periodic orbits, a given dynamical system can a priori exhibit a broad diversity of controlled responses. On the other hand, since feedback methods involve detection of the degree of deviation of the selected unstable periodic orbit from the chaotic state in real time, the control experimental device typically becomes a closed-loop system, and thus tends to be relatively complicated. Other essential limitations of such methods lie in their inability t o control both high-speed chaotic systems (such as Josephson junction arrays or fast electreoptical systems)

2

Introduction

and chaotic systems embedded in highly noisy environments. All this limits the scope and applicability of the feedback methods: they become impractical for the control of fast processes and large systems. 1.2

Non-feedback control methods

Non-feedback methods suppress or enhance chaos by adding a (preferably) weak time-dependent forcing or by perturbing a system parameter with (preferably) small time-dependent excitations. Periodic signals have been by far the most commonly chosen as control excitations, albeit the study of the effect of random and chaotic time-dependent excitations in controlling chaos has also been initiated. In contrast to the procedure of feedback methods, such control excitations are independent of the system’s state. This distinctive characteristic presents both advantages and difficulties. On the one hand, the control experimental device is typically a rather simple high-speed open-loop system which does not require on-line monitoring or processing. Thus, non-feedback methods are attractive due to both their easy applicability to experimental situations and their robustness uis-h-uis noise. On the other hand, the main criticism of such methods focuses on the difficulty of predicting the nature of the regularized state as well as the parameter ranges for control. This is clearly the case of high-dimensional systems such as solid state lasers and neural network models where the absence of a unified theoretical approach implies a tentative application of the control excitations. However, it will be shown in this book that a rigorous control theory is emerging for a certain class of important low-dimensional systems, including the perturbed pendulum as well as other universal models. Such a theory provides analytical estimates of the ranges of parameters of the chaos-controlling excitation for enhancement/suppression of the initial chaos, as well as key information concerning the periodicity of the regularized responses. Various techniques of non-feedback control have been proposed that can be roughly classified into three types: (i) the parametric excitation of an experimentally adjustable parameter; (ii) entrainment to the target dynamics; (iii) the application of an external periodic excitation. It will be shown in the next chapter that techniques (i) and (iii) may be unified in a general setting for the class of dissipative systems considered in this work. There exists numerical, theoretical, and experimental evidence that the period of the most effective chaos-controlling excitations usually is a rational fraction of a certain period associated with the uncontrolled system, although the effectiveness of incommensurate excitations has also been demonstrated in some particular cases. Indeed, resonances between the chaos-controlling excitation and (i) a (periodic) chaos-inducing excitation, (ii) an unstable periodic orbit embedded in the chaotic attractor, (iii) a natural period in a period-doubling route to chaos, or (iv) a period associated with some peak in the power spectrum, have been considered in diverse successful chaos-controlling excitations. This is not really surprising since these types of resonances are closely related to each other. For instance, when

Controlling chaos by weak periodic excitations

3

a damped, harmonically forced oscillator presents steady chaos, the power spectrum corresponding to a given state variable typically presents its main peaks at frequencies which are rational fractions of the chaos-inducing frequency for certain ranges of the chaos-inducing amplitude. 1.3

Controlling chaos by weak periodic excitations

The present book is concerned with the control of chaos in dissipative non-autonomous systems described by the differential equation

(1.1) where U ( x )is a nonlinear potential, -d(x, i)is a generic dissipative force which may include constant forces and time-delay terms, p,(x, k)F,(t) is a chaos-inducing excitation, and p,(x, k)F,(t) is an as yet undetermined suitable chaos-controllingexcitation, with F,(t), F,(t) being harmonic functions of initial phases 0, 0 , and frequencies w , R, respectively. The theory discussed in Chap. 2 (developed on the basis of Melnikov’s method) imposes on Eq. (1.1) some additional limitations: the excitation, timedelay, and dissipation terms are weak perturbations of the underlying conservative system x + d U ( x ) / d x = 0 which has a separatrix. These restrictions will not apply, however, in discussing the mechanisms underlying the control of chaos by weak periodic excitations in Chap. 3. 1.9.1 Robustness and flexibility It is shown in Chap. 4 that the theoretical results demonstrated in Chap. 2, concerning periodic chaos-inducing excitations, are also valid for chaotic chaos-inducing excitations whose power spectra exhibit a strong peak at a certain frequency which is taken as the chaos-inducing frequency. A similar effectiveness is found for quasiperiodic chaos-inducing excitations where the existence of two incommensurate strong peaks in the power spectrum increases the complexity of the control scenario. These results represent a new aspect of the robustness in the control of chaos by weak harmonic excitations, which extends the well-known robustness against external noise (both additive and multiplicative), and thus make this control method highly reliable. Another attractive aspect is its flexibility in the choice of particular chaos-controlling excitations. Indeed, the theorems of Chap. 2 do not impose any restriction on the nature of such excitations except their perturbational (small amplitude) character. Depending upon the specific chaotic system to be controlled, the chosen chaoscontrolling excitation can be a parametric excitation of a potential term, a parametric excitation of a dissipative term, an (additional) forcing term, a parametric excitation of the (periodic) chaos-inducing signal, and so on. However, it is worth mentioning that the phenomenon of parametric resonance diminishes or even completely prevents the control of chaos by certain parametric excitations. An illustrative example of this effect appears in the problem of controlling chaotic escape from a potential well (cf. Sec. 4.1).

4

Introduction

1.3.2 Applicability and scope The aforementioned hyper-robustness of the present control method permits its reliable application to a broad spectrum of experimental realizations. In testing the robustness of the method vis-a-vis experiment one must assume that the parameters of the chaos-inducing excitation (amongst others) can be affected by random fluctuations. The theoretical predictions for the suitable amplitudes and initial phases of the chaos-controlling excitation are in the form of finite intervals whose widths are typically much larger than the error bars associated with the fluctuations. The frequency is a more delicate parameter since the control theorems require an exact resonance condition between the two frequencies involved (chaos-inducing and chaoscontrolling). Numerical simulations show that the effect of a slight deviation from the resonance condition is an adiabatic (very slow) variation of the initial phase of the chaos-controlling excitation. Since the ranges of suitable initial phases are real intervals, the “off-resonance” chaos-controlling excitation could yield, at least, an intermittent control of the dynamics. It should be emphasized that Eq. (l.l),including such paradigmatic dynamical systems as the pendulum and the two-well Duffing oscillator, also appears when modeling key aspects underlying the dynamics of more complex systems like chaotic coupled oscillators and nonlinear wave equations. Thus, the findings of this present book will also apply directly to diverse situations concerning the control of spatiotemporal chaos. 1.4 Harmonic versus non-harmonic excitations: the waveform effect

It is clear at first sight from the literature that it is harmonic excitations which have been overwhelmingly chosen as representative of periodic control excitations. Since harmonic functions are solutions of linear differential equations (rarely of nonlinear equations), this means that the role to be played by the excitation’s shape in controlling chaos has not as yet been fully explored. However, there exist theoretical and numerical results in support of the existence of generic routes of order-chaos by changing only the waveform of a periodic excitation. As will be shown below, such reshaping-induced routes are closely related to those associated with the application of harmonic excitations. As a first example, consider the family of systems

(1.2) where U ( x )is a nonlinear potential, -d(z, i)is a general dissipative force, and F ( t ) is a general periodic function of period T. Figure 1.1 shows an illustrative example of an orderttchaos route induced by solely reshaping a periodic excitation, corresponding to a linearly damped pendulum subjected to a periodic string of symmetric pulses: Z

+ sin z = -qi + y cn (wt;rn) ,

(1.3)

Harmonic versus non-harmonic excitations: the waveform effect

5

where q , y > 0, and cn(wt;m) is the Jacobian elliptic function of parameter m. When m = 0, then cn (wt; m = 0 ) = cos ( w t ) ; i.e., one recovers the well-known case of harmonic forcing. To study the structural stability of the pendulum when only the forcing waveform is varied, one fixes the forcing period T = const, making the frequency w = w (m)= 4 K ( m ) / T , where K is the complete elliptic integral of the f i s t kind. Note that, by increasing m, the pulse becomes narrower and narrower, and for m N 1 one recovers a periodic sharply kicking forcing very close to the periodic two-sided &function, but with finite width and amplitude as in real-world pulses. Figure 1.1 shows the bifurcation diagram for the variable d x / d t versus rn, with q = 0.5, y = 1.1, and T = 37r.

0.8

Figure 1.1: Bifurcation diagram corresponding to the pendulum (1.3).

It is worth mentioning that, for fixed T ,x(O),i ( O ) , the same qualitative phenomena seen in this figure continued to be seen as the ratio q/y is varied over a certain range (depending upon TIx(O),i ( O ) ) , the difference being the specific values of the waveform parameter m at which the qualitative changes (crisis, bifurcations, and so on) occur. In particular, the threshold value mthreshold such that X ( m 2 rnthreshold) < 0 decreases as q/y is increased. Figure 1.2 depicts the leading Lyapunov exponent X versus the waveform parameter m for T = 3n and two values of q/y: 0.454545 (W) and 0.166666 (*). In such a situation, assume that the pendulum (1.3) is in a chaotic state for q/y = ( ~ / y denoted )~ by A (see Fig. 1.2). Then, increasing m from rnl to m2, and keeping the ratio q/y = (q/y)I constant, the Lyapunov exponent X decreases and, in some cases, becomes negative which means that the pendulum reaches a regular (periodic or equilibrium) state (as in the case indicated by point B). Contrariwise, if B represents a periodic state for example, the route B-iA makes it chaotic (see Fig. 1.2).

6

Introduction

0.3 I

/z

I

0 .o

-0.3 0.0

T

m,

0.5

q r n

I

1.o

Figure 1.2: Leading Lyapunov exponent vs m for the pendulum (1.3). Now, fixing m = m2 (i.e., fixing the waveform) one can diminish the ratio q/y (raising y, lowering q, or both) such that the Lyapunov exponent X increases and, in some case (here for q/y = (q/y)2), becomes positive, and thus the pendulum reaches a chaotic state (as in the instance indicated by point C). This is a well-known route to reach a chaotic state in non-autonomous systems. Observe that the pathways of types A-B and C-B are only ad hoc routes from chaotic (regular) states to regular (chaotic) states, the most common being a simultaneous variation of the excitation waveform and the relative strength of the damping coefficient with respect to the excitation amplitude. This is closely related to the scenario behind the control of chaos by resonant periodic excitations. In particular, this is exactly the case for the family of system x+T-

+

- -qi

+ y cos ( w t ) + a y cos (Rt + 0 ),

(1.4)

where cry cos (Rt 0 ) is the chaos-controlling excitation (0 < a < 1).Indeed, in Eq. (1.4) the resonance between the forcing terms implies R/w = n/m,n, m positive integers, i.e., both cosines have a common period T = 27rm/w and the addition of cry cos (Ot + 0 ) to ycos ( w t ) has the effect of changing both the waveform and the amplitude of the chaos-inducing forcing. Therefore, it is possible that, by choosing suitable values of a and 0 for each resonance R / w , the initial chaotic state may be controlled (i.e., suppressed or enhanced). 1.4.1 Reshaping-induced strange non-chaotic attractors As a second example, a route by which strange non-chaotic attractors arise and ultimately become chaotic by altering solely the shape (of, for instance, only one periodic term) of a two-period excitation will be characterized in the following.

7

Harmonic versus non-harmonic excitations: the waveform effect

For the sake of clarity, consider fist the reshaping-induced appearance of strange non-chaotic attractors by studying the analyzable twcdmensional map z,+~ = [a cn (Re,; m)

en+,

= (0,

+ b]sin z,,

(1.5)

+ 2 7 ~mod ) (27~),

(1.6) where a and b are parameters, and cn(RB;m) is the Jacobian elliptic function of parameter m. For irrational w , the circle map (1.6) defines a quasiperiodic excitation which is multiplicative in the nonlinear equation (1.5). Since we are interested in the case when solely the cn shape is varied, one fixes its period T = const, making R = R (m)= 4K(m)/T, where K ( m ) is the elliptic integral of the first kind. Note that, when m = 0, then cn [R ( m= 0) 8; m = 01 = cos (27rO/T), i.e., one recovers the well-known harmonic limiting case. 0.1

0

3 -0.1

* v

-0.2 - 0.3

0

0.2

0.6

0.4

0.8

1

m Figure 1.3: Lyapunov exponent AT(m)vs m for the map (1.5), (1.6). With increasing m, the shape of the excitation becomes ever narrower, and for m M 1 one has a periodic sharply kicking excitation. In the other limit one has cn[4K(m + l)B/T;m -+ I] = 0, i.e., the pulse area tends to 0 if m + 1, for T = const, so that the z and 0 dynamics decouple in this limit. The nonlinearity in Eq. (1.5) is the same as in its harmonic counterpart, so that one straightforwardly obtains that the one-dimensional invariant subspace is z = 0, while the transverse Lyapunov exponent (for T = 27r) is

For the two limiting values of the shape parameter (0, l}, one has AT(^ = 0) = LnIb[-Ln(2/{1+ [ l - ( ~ / b ) ~ ] } ) i f a < b,AT(rn=l)=LnIbl. W i t h a a n d b c o n stant, one can study the transverse Lyapunov exponent as a function of the excitation

8

Introduction

shape parameter m only. A typical plot of A,(m) is shown in Fig. 1.3. The qualitative form of this function remains the same as a and b are varied. Thus, for the case a > 2 , l > b > 0, there always exists a critical value m = m, = m,(a, b ) such that AT(m m,) 2 0 (a strange non-chaotic attractor appears) and AT(^ > m,) < 0 (the attractor is the line z = 0). Figure 1.4 shows a typical sequence of phase portraits illustrating the evolution of strange non-chaotic attractors in phase space as the shape parameter approaches its critical value m,, the remaining parameters held constant. It is worth mentioning that the particular form of the curve plotted in Fig. 1.3 is closely related to that of (the reciprocal of) the function K ( m ) ,which controls the rate at which the excitation waveform is varied in the map (1.5), (1.6). In other words, the specific form of the corresponding curve AT (as a function of a certain effective shape parameter) should strongly depend, in a general case, on the particular rate at which the excitation waveform is varied, the amplitude and the period being held constant.

<

-3.15

I 3.15

0.00

0

4

z

(b) m 4 . 9

0

4 -3.15

0.00

3.15

0

Figure 1.4: Phase portraits corresponding to the map (1.5), (1.6).

9

Harmonic versus non-harmonic excitations: the waveform effect 4

(c) m4.95

I -3.15

0.00

3.15

e

4

(

4 -3.15 o

(d) m4.9528

e

3 15

0.00

Figure 1.4: (Continued) For a generic excitation f (0; T ,ai) of period T , unit amplitude, and shape controlled by parameters ai,a physically plausible candidate for such an effective shape parameter p,f could be the mean value of its absolute value: p,f = p e f ( a i ) = (l/T) ( & T ai)l , do. For example, one finds p e f ( m )= [(l- m) /m]”2K-’ 0 rn for the above elliptic excitation cn, so that the functions -p,f(l - m),A,(m) have almost identical forms. It is straightforward to see that, when the variation of the (effective) shape parameter is monotonous, there exists a unique critical value for the transition to strange non-chaotic attractor (given a suitable choice of the remaining parameters), while for non-monotonous changes one finds (for the same suitable set of the remaining parameters) a number of critical values depending upon the specific variation. As an illustrative example, consider now K ( g ( m ) )instead of K ( m ) in the map (1.5), (1.6), where g(m) = (32/3)m3- 16m2 (19/3)m is a function with a single maximum and a single minimum. Note that we choose the function g = g(m), with g(m = 0) = 0, g(m = 1) = 1, and 0 g(m) 1,m E [0,1], in order for the excitation cn [2K(g(m))0/7~; m] to have the same period (2n) for all m.

JOT I f

<

+ <

10

Introduction

0

-0.2

E

Figure 1.5: Lyapunov exponent AT (m)vs m for the map (1.5), (1.6). Figure 1.5 shows the corresponding transverse Lyapunov exponent vs the shape parameter, where one sees three critical values for the appearance (disappearance) of strange non-chaotic attractors. Therefore, the appearance or suppression of strange non-chaotic attractors can be induced, at will, by suitably reshaping the excitation waveform. Consider now an elliptic generalization of the two-period quasiperiodically (using harmonic functions) forced damped pendulum:

4+ v$ + sin4 = A + V [cn ( R t ; m )+ cos ( w t ) ],

(1.8)

where 4 is the angle of the pendulum with the vertical axis, v is the dissipation coefficient, A is a constant, V is the excitation amplitude, and R = R ( m ) = 4K(m)/T,,, w = 2.rr/TC,, where T,, and T,,, are two incommensurate periods. After fixing the parameters v = 0.8270429, A = 0.8, V = 0.55, T,,/T,,, = (& - 1 ) / 2 (the reciprocal of the golden mean), T,, = 5.196464 (for which there exists a strange nonchaotic attractor at m = 0), one increases the shape parameter to study the stability of such attractors. Recall that the word strange has the sense that a strange attractor is an attractor which is neither a finite set of points nor is it piecewise differentiable, while the word chaotic has the sense that a chaotic attractor is one for which typical orbits have a positive Lyapunov exponent. Five ranges can be distinguished as m varies from 0 to 1. Over the first range, 0 < m 5 0.9, one only finds strange non-chaotic attractors which are very similar to that at m = 0. Figure 1.6(a) shows the leading Lyapunov exponent X (one Lyapunov exponent is always zero for Eq. (1.8)) vs m over the mentioned range (strange non-chaotic attractor range). To visualize the attractors, the dimensionless variables 4 and $ are plotted on the Poincari! surface of the section defined by wt, = nTcos, n = 0 , 1 , ... .

Harmonic versus non-harmonic excitations: the waveform effect

11 i

-0.01

A. -0.02 -0.03 0.0

(a)

0.6

0.3

0.9

m

0.00

A.

-0.02

-0.04 0.900

0.935

0.970

m

0.015

A. 0.000

-0.015'

0.97

0.98

0.99

m

0.04-

A.

-0.04

-0.08

4

il

1

0.990

\d

(d)

0.995

1 .ooo

m

Figure 1.6: The Lyapunov exponent vs the shape parameter for Eq. (1.8).

12

Introduction

Figure 1.7(a) shows a single long trajectory on the strange non-chaotic attractor for m = O.O17(X M -0.0123) which is typical for the attractors of the strange non-chaotic attractor range.

-3.15

3.15

0.00

4

2 50

125

125

a

& 000

000

-3.15

0.00

3.15

-3.15

0.00

4

3.15

4

1.25

1

-3.15

5

0.00

4

Figure 1.7: Stroboscopic attractors in the

C#I

-

2 plane for the system (1.8).

The next range, 0.9 5 m 5 0.97, is characterized by the overwhelming presence of strange non-chaotic attractors except in a few naxrow windows in which chaotic

Harmonic versus non-harmonic excitations: the waveform effect

13

attractors appear (A > 0), as can be seen in Fig. 1.6(b). Note that in this cohabitation range the Lyapunov exponent undergoes larger variations than in the preceding range. Figure 1.7(b) gives an example of a strange non-chaotic attractor in the cohabitation range for m = 0.951 (near the minimum, X -0.0388). Comparison of Figs. 1.7(a) and 1.7(b) shows how the strangeness of the strange non-chaotic attractors varies as the shape parameter changes. Over the range of the transition to chaos, 0.97 5 m 5 0.99 (cf. Fig. 1.6(c)), there exist many shape parameter intervals for strange non-chaotic attractors which are interspersed with shape parameter intervals for chaotic attractors. A feature of the central part of this range is that the leading Lyapunov exponent remains very close to zero (the small amplitude fluctuations are artifacts of the numerical computation). Nonetheless, the leading Lyapunov exponent passes through zero linearly with respect to the shape parameter near the transition to chaos, thus confirming the scaling behavior found for other control parameters (see Fig. 1.8 where the critical value is m, x 0.97309).

h -0.004 -

i

i/

inc

m Figure 1.8: The Lyapunov exponent vs the shape parameter for Eq. (1.8). The fourth range, 0.99 5 m 5 0.999233, is chaotic (see Fig. 1.7(c) for a chaotic attractor at m = 0.99923). The final range, 0.999233 5 m < 1 (cf. Fig. 1.6(d)), is twofrequency quasiperiodic, as in the example depicted in Fig. 1.7(d) at m = 0.999235. This quasiperiodic range is a consequence of the drastic shrinkage of the pulse width as m + 1. Observe that the leading Lyapunov exponent presents a minimum at m ‘v 0.999883 which is apparently related to the beginning of the appearance of symmetry of the two-frequency quasiperiodic attractor. It is worth mentioning that similar results were found for other parameter values. In conclusion, it was shown that there typically exists a wide range of the shape parameter in quasiperiodically driven dynamical systems where only strange, chaotic and non-chaotic, attractors appear, so that certain control tasks, such as switching

14

Introduction

between the two types of strange attractor, can be more easily performed than for other control parameters. 1.4.2 Reshaping-induced crisis phenomena As a third example, a route by which crises occur by humpdoubling of a parametric excitation which is initially formed by a periodic string of single-humped symmetric pulses will be characterized in the following by studying the analyzable t w e dimensional map Z,+I

=

On+,

=

+ +

az, z: y N ( m )sn (00, 20,mod(2~),

+ a;m)dn (00, + CP; m),

(1.9) (1.10)

where a , y,0 = 4K(m)/T, and CP = 4K(rn)'p/T ('p E [O,T]) are parameters, sn (.; m) and dn (.; m) are Jacobian elliptic functions of parameter m ( K ( m )is the complete elliptic integral of the first kind), and N ( m ) is a normalization function (Boltzmann form), (1.11) N ( m ) = { u b [1+ exp ( ( m- c) /d)]-'}-',

+

with a N 0.43932, b N 0.69796,~N 0.3727,d N 0.26883, which is introduced for the excitation function to have the same amplitude, y,and period, T, for any waveform (i.e., Vm E [0,1]). The excitation function p (0; m,T) = N ( m )sn (00; m)dn (06;m) is used as an example to illustrate the crisis induced by hump-doubling. One has p (0; m = 0, T ) = sin (27r0/T), i.e., one recovers the widely studied case of a singlehumped (harmonic) excitation. This is relevant to comparing the structural stability of the map when solely the excitation shape is varied from a single-humped to a double-humped shape. Since dn (00;m) represents a periodic string of asymmetric pulses, whose effective width decreases as m increases from m = 0, at the limiting value m = 1the excitation function p (0;m = 1,T) vanishes except on a set of instants that has Lebesgue measure zero, i.e. the variables z and 0 decouple. Numerical simulations indicated that, for certain parameter values, the map (1.9),(1.10) presents two attractors: z = 00 (which, for the present purposes, should be regarded as a generic non-chaotic attractor) and a chaotic attractor located in the region z E [-0.1,0.1]. By looking at the fixed points of Eqs. (1.9) and (l.lO), one deduces the critical parameter values (a,, y,,T,, p ' ,, m,) for which the two fixed points ( z , 0) = { (z,,, 0) , ( z b b , 0)) touch (i.e., a crisis occurs), where z~ is the smallest z value on the upper ( z > 0) basin boundary of the attractor z = 00 while z, is the largest z value on the chaotic attractor. Setting 0 = 0 in (1.9) and assuming that z, is independent of n, one has 2% =

and

{ 1 - a f [(l

zbb = z+,

-

a ) 2- 4yN(m) sn (4K'pIT; m)dn (4K'pIT;m)]'I2} /2,

z,, = z-. Therefore, a crisis occurs when z+ = z-, i.e., for

a = a, = 1 - 2 {y,N(m,) sn [4K(m,)'p,/T,; m,] dn [4K(m,)'p,/T,; m , ] } ' / 2 . (1.12)

15

Harmonic versus non-harmonic excitations: the waveform effect

Let us suppose that for fixed y = y,,T = T,, and 'p z 'p, (> 0), and m 2 0 (nearly harmonic excitation), one has that the chosen a > a, = a,(m). As m is increased from m 2 0, a,(m) increases so that the two fixed points move towards each other and, in some case that depends upon the choice (y,,T,, p ' ,, a ) , coalesce at m = m, for which a,(m = m,) = a. Thus, a reshaping induced crisis occurs. 1.4.3 Reshaping-induced basin boundary fractality As a fourth example, a generic route is described for the modification of fractal basin boundaries in nonlinear systems by changing only the shape of a periodic term in the dynamics equations. To demonstrate the new mechanism in the simplest possible context, consider the following two-dimensional map: zn+1

=

(1.13)

Xzn+cn

(1.14) where one assumes 1 < X < 2,O 6 O < 27r, m E [0,1[, and where cn is the Jacobian elliptic function of parameter m and (real) period 4K(m), with K ( m ) the complete elliptic integral of the first kind. When m = 0, then cn[4K(m = 0)8/7r;m = 01 = cos (28). Increasing m makes the pulse given by cn [4K(m)8/7r; m] progressively narrower. The Jacobian matrix of the map (1.13),(1.14)has eigenvalues 4K(m)/7r and X which are greater than 1 so that there can be no attractors with finite z . In fact, there exist only two attractors ( z = co and z = -co) and one wishes to characterize the e v e lution of the fractality of their basin boundary, z = f ( 0 ) , as rn varies over the range [0,1[. To find this boundary set, one notes first that On = [4K(m)/7r]" Oo mod (27r). The map (1.13),(1.14) is two-to-one, i.e., given O n + l , it is not possible to find On uniquely since there are two possible solutions of (1.14),

and On

+

= r 2 / 2 K ( m ) t'n+1/

[4K(m)/~].

However, one can select any zn and find one orbit that ends at above On and taking

znPl = x-'z, For the given

( Z N , 0,)

( Z N , 8,)

- X-' cn { [ 4 ~ ( m ) / ~00; ] "m} .

one finds that this orbit started at

by using the

16

Introduction

The boundary between the two basins are those (zo,Q,) such that N + co,so these z and Q are related by

ZN

is finite as

(1.15) Since X has

> 1 and m

E [0, I[, this sum converges absolutely and uniformly. One also

(1.16)

where dn and sn are the Jacobian elliptic functions. The latter sum diverges Vm E [O, 1[ because X < 2. Hence f (Q)is non-differentiable. Figure 1.9 shows approximate plots of the curve (1.15) for three values of the shape parameter m = {0,0.5,0.99}. It can be shown that the box-counting dimension of the curve (1.15) is

d=d(m)=2-

In X In [4K(m)/.rr]

(1.17)

'

For m = 0, one recovers the value d ( m = 0 ) = 2 - (1nX) (ln2)-', while one obtains d ( m + 1) = 2 as the symmetric pulses modeled by the function cn [4K(m)O/a; m]become narrow. Figure 1.10 shows the normalized box-counting dimension d ( m )/ d ( 0 ) versus m for X = const. One sees that the increase in the normalized box-counting dimension is especially noticeable for very narrow pulses ( m 5 l), which is a consequence of its dependence on K ( m ) . It is worth mentioning that similar results can be obtained for other periodic functions instead of cn and for other general systems, i.e., the fractality of a basin boundary can be varied by reshaping a suitable periodic term in the dynamics equations. 1.4.4 Reshaping-induced escape from a potential well As a fifth example, the reshaping-induced chaotic escape of a damped oscillator excited by a periodic string of symmetric pulses of finite width and amplitude from a cubic potential well that typically models a metastable system close to a fold is described. Consider the chaotic escape of the following universal model

2

+

5 - /3z2=

-Pi

+ r N ( m )sn [@ ( t );m]dn [@ ( t ); m],

(1.18)

when only the excitation shape is varied from single-humped to double-humped. Here CP ( t )EE 4K(rn)t/T,sn (.; m) and dn (.; rn) are Jacobian elliptic functions of parameter m (K(rn)is the complete elliptic integral of the first kind), and N ( m ) is given by Eq. (1.11)with the aforementioned meaning.

17

Harmonic versus non-harmonic excitations: the waveform effect

m=O

1.5 t 1 0.5

I

z s o - 0.5

-1

-3-2-1

fx s

0 1 2 3 8

m=0.5

1.5 1 0.5

o

- 0.5

-1 -3 -2-1

0 1 2 3 8

m=0.99 I

-3-2-1

0

I

1 2 3

e Figure 1.9: Plots of the curve (1.15) for three m values. For the universal escape model (1.18),the initial conditions will determine, for a fixed set of its parameters, whether the system escapes to an attractor at infinity, or settles into a bounded oscillation. As is well known, there can exist a rapid and dramatic erosion of the safe basin (union of the basins of the bounded attractors) due to encroachment by the basin of the attractor at infinity (escaping basin). Numerically, one finds that the erosion of the safe basin is maximal as a singlehumped excitation transforms into a doublehumped excitation, the remaining parameters being held constant.

18

Introduction

1.5

1

0

0.2

0.4

0.6

0.8

1

m Figure 1.10: Normalized box-counting dimension d ( m ) / d ( O )vs m (cf. Eq. (1.17)).

To generate the basins of attraction numerically, a grid of (uniformly distributed) 300 x 300 starting points in the region of phase space z(t = 0) E [-0.7,1.3] , i ( t = 0) E [-0.8,0.7] was selected. From this grid of initial conditions, each integration is continued until either x exceeds 20, at which point the system is deemed to have escaped (i.e., to the attractor at infinity), or the maximum allowable number of cycles, here 20, is reached. In the case of a singlehumped harmonic excitation ( m = 0), one assumes that the system presents a very slight erosion of the non-escaping basin. For a fixed set of parameters (p,6, y,T), the escape probability normalized to that of the case with m = 0, P ( m ) / P ( m= 0) was calculated. An illustrative example is shown in Fig. 1.11 for the parameters p = 1 , b = O.1,T = 27r/0.85, and three y values: 0.071 (circles), 0.072 (triangles), and 0.073 (stars). One sees that the normalized escape probability presents a maximum at mmaxN 0.65. Figure 1.12 shows the corresponding basin erosion sequence for six m values. The white region represents the non-escaping basin and the black region the escaping basin. One sees that the erosion and stratification of the basin is maximal at m 1: mma. In order to physically explain the origin of the aforementioned maxima, the impulse transmitted by the excitation over a fixed half-period as a function of the shape parameter is calculated:

=I

TI2

I(m,T)

TN(m) N(m)sn[@(t);m]dn[@((t);m]dt=2K(m)

(1.19)

'

One finds that I ( m , T ) presents (for each T value) an absolute maximum at m = mLax N 0.71718, which is significantly near the maximum of the normalized escape probability mmaxN 0.65.

19

Harmonic versus non-harmonic excitations: the waveform effect

1.8 -

s II E

v

1.6

-

1.4 -

--. 1.2 L

E a

v

1.0 0.81

"

0.0

"

0.2

"

0.4

"

0.6

"

0.8

'

1.0

I

in Figure 1.11: Normalized escape probability vs shape parameter. One can understand such a coincidence by analysing the variation of the system's energy. Indeed, note that Eq. (1.18) can be put into the form,

dE _ - -SIC2 ( t )+ yx ( t )N ( m )sn [@ ( t ); m]dn [@ ( t ); m],

dt

+

(1.20)

where E ( t ) = f i z ( t ) U [x( t ) ] [U(x)= i x 2 - $x3] is the energy function. Integration of Eq. (1.20) over any interval [nT,nT T/2], n = 0 , l, 2 , ..., yields

AE

-6

=

r2

+

x2 ( t )dt

s, +

nTfTf2

+?

Z ( t )N ( m )sn [@ ( t ); m]dn [@ ( t );m]dt,

(1.21)

where A E = E (nT T / 2 ) - E (nT). Now, if one considers fixing the parameters (p,6 , y,T ) for the system to lie on a periodic orbit (i.e., inside the well) near the underlying separatrix at m = 0, the application of the first mean value theorem to the second integral on the r.h.s. of Eq. (1.21) gives

A E = -6

+

x2 ( t )dt + yTx (t*)N(m) 2K(m)'

(1.22)

where t* E [nT,nT T/2].Since we are considering that the initial state is a steady (periodic) state, t* will depend solely on the excitation function but not on n. In this situation, one increases m while holding the remaining parameters constant. For values m > 0 such that the system state is still a periodic orbit (which will be necessarily near the initial periodic orbit in the phase space), one expects that both the dissipation work (integral in Eq. (1.22)) and x (t*)will maintain approximately their

20

Introduction

initial values (at m = 0) while the impulse I (m,T ) will rise from its initial value, so that, in some case depending upon the remaining parameters, the energy increment A E could be enough to surpass the threshold escape energy, i.e., the threshold oscillation amplitude to allow escape from the potential well, Clearly, the probability of this event is maximal at m = miax where I (m,T) presents an absolute maximum, which explains that mmaxN miax. For fixed P , S , and y,since the transmitted impulse depends on both the shape parameter and the period, its critical value yielding the aforementioned escape event can in some case be reached at m # miax provided that T is sufficiently large according to Eq. (1.19).

m=O

m = 0.4

rn = 0.55

m = 0.65

rn = 0.75

in = 0.85

Figure 1.12: Basin erosion sequence for six m values. 1.d. 5 Reshaping-induced control of directed transport As a sixth example, consider the control of directed transport in general systems. Originally motivated by stochastic models of biomolecular (Brownian) motors, the idea of rectifying transport with the aid of fluctuations has also been discussed in other contexts such as voltages in Josephson junction coupled systems and electrical currents in superlattices. It is worth mentioning that the fluctuations have zero mean value, i.e., the dc component is absent. A fundamental result is that there exists

Harmonic versus non-harmonic excitations: the waveform effect

21

a clear relationship between directed transport and broken space-time symmetries, which has been generalized from one-particle models to the case of interacting manyparticles models. An important consequence is that the symmetries may be broken either by violating the temporal shift symmetry of the ac force or by violating the selection symmetry of the potential in space. Consider a general system (classical or quantum, dissipative or non-dissipative, uni- or multidimensional, noisy or noiseless) where the sc-called ratchet effect is induced by solely violating temporal symmetries of a T-periodic zerc-mean ac force f ( t ) which drives the system. A popular choice would be the simple case of a biharmonic force,

(1.23) where harl,o represents indistinctly sin or cos, and pwl = qwz, p , q coprime integers. In this case, the aforementioned symmetries are the shift symmetry

s, : f ( t ) = -f(t + T/2),

(1.24)

with T = qT1 = pT2 (Ti = 2 ~ / w i ) and , the time-reversal symmetries

(1.25) Now the general unsolved problem is to find the regions of the parameter space ( f i r wi, 9,) where the ratchet effect is optimal in the sense that the average of relevant observables is maximal, the remaining parameters being held constant. Is is shown in the following that such regions are those where the effective degree of symmetry breaking is maximal. Without loss of generality, this degree of symmetry breaking mechanism is discussed by using the following working model for the driving force:

fellip

( t )= E f ( t ;T, m, 0) = E sn (Rt + 0 ;m)cn (Qt+ 0 ),

(1.26)

where cn (.; m) and sn (.; m) are Jacobian elliptic functions of parameter rn, R = 2K(m)/T, 0 G K(m)O/.lr,K ( m ) is the complete elliptic integral of the first kind, T is the period of the force, and 0 is the (normalized) initial phase (0 6 [0,2x]). Fixing E , T and , 8, the force waveform changes as the shape parameter m varies from 0 to 1, as can be appreciated in Fig. 1.13for E = 1 , 0 = 0, and three shape parameter values: m = 0 , l - lop6 (gray), and 0.96 (black). Note the increasing symmetry-breaking sequence as the pulse narrows, i.e., as m + 1.

22

Introduction

0.0

0.2

0.4

0.6

0.8

1.0

t/T Figure 1.13: Force f e l l i p ( t ) (Eq. (1.26)) for m = 0 , l -

(gray) and 0.96 (black).

Physically, the motivation of choice (1.26) is that f ( t ;T, m = 0,6) = sin (27rt/T + 6) /2, and that f ( t ; T ,m = 1,6) vanishes except on a set of instants that has Lebesgue measure zero, i.e., in these two limits directed transport is not possible, while it is expected for 0 < m < 1. Thus, one may expect in general the average of any relevant observable R! to exhibit an extremum at a certain value m = me as the shape parameter m is varied, the remaining parameters being held constant. Clearly, two competing fundamental mechanisms allow one to understand the appearance of such an extremum: the increase of the degree of breaking of the shft symmetry as m is increased, which increases the absolute value of the average, and the effective narrowing of the force pulse as m is increased, which decreases the absolute value of the average. The former mechanism arises from the fact that a broken symmetry is a structurally stable situation (Curie’s principle) and can be quantitatively characterized by noting that -f (t

+ T/2; T, m,6 ) --

f ( t ;T, m,6 )

J1-m = D ( t ;T, m, 6) , dn2 (nt + 0 ;m)

(1.27)

where dn (.; m) is the Jacobian elliptic function of parameter m. Equation (1.27) indicates that the degree of deviation from the shift symmetry condition (D(t;T, m, 6) = 1) increases as m + 1, irrespective of the values of the amplitude, period and initial phase. A plot of the asymmetry function D (t;T, m, 6 = 0) is shown in Fig. 1.14. Thus, while increasing the shape parameter m (0 < m < me) improves the directed transport yielding a higher average, it simultaneously narrows the pulse force (see Fig. 1.13), lowering the driving effectiveness of the force. Indeed, the latter becomes the dominant effect for sufficiently narrow pulses ( m > me). Also, one chooses the function (1.26) to satisfy the requirement that me be sufficiently far from 1 for the

23

Harmonic versus non-harmonic excitations: the waveform effect

elliptic force to be effectively approximated by its first two harmonics. One thus obtains a relationship between the amplitudes of the two harmonics in parametric form: c1,2 = e1,2 (m). This relationship does not depend on the initial phase 8, and hence neither does it depend on the breaking of time-reversal symmetries of the biharmonic approximation corresponding to the elliptic force. , ’

/

,

.--. - --__

- --_. -.

Figure 1.14: Deviation function D ( t ;T ,m, 8 = 0) (Eq. (1.27)) For a general biharmonic force (1.23),this means according to the degree of symmetry breaking mechanism that the relationship €2 = E Z ( E I ; ~ q, ) should control solely the degree of breaking of the shift symmetry. Note that this symmetry is not broken when p , y are both odd integers. Consequently, if the degree of symmetry breaking mechanism is right, the relationship €2 = €2(e1;prq ) ( p q = 2n 1,n = 0,1, ...) controlling the degree of breaking of the shift symmetry should be independent of whichever is the particular system where directed transport is induced. This implies that any averaged observable < 92 > should be proportional to certain function g (€1, €2) = g (el, c2;p , q ) which is ~ ~ ( E I ) P ~ ( E in Z ) leading order, with P I ( € ] ) E ; , p 2 ( ~ 2 ) E:, T , s positive integers. Since the aforementioned extremum me is scale-independent , one defines = E (1 - a ) ,€2 = ~a ( a E [0, l]),so that g ( € 1 , € 2 ) (1 - CY)~CY’ taking E = 1 without loss of generality. Since the extremum me is independent of the driving period, one has the symmetry relationship g (el, E Z ; ~ y) , = g (€2, €1; q , p ) . The problem thus reduces to finding the relationship between ( T , s) and ( p ,y). From Maclaurin’s series, one straightforwardly obtains that the only function satisfying all these requirements in leading order is (1 - a)”aq, and hence g ( € 1 , ~ ; pq ), EYE;. Indeed, previous theoretical analyses of every kind on a great diversity of systems have found that the averaged observable is always proportional to such a factor in leading order. One

+

N

+

N

N

N

N

24

Introduction

thus obtains

(1.28) for the biharmonic approximation corresponding to the elliptic force (i.e., p = 2, q = 1). Therefore, the shape function S ( m ) is a universal function which controls the breaking of the shift symmetry in leading order for the resonance ( p , q ) = (2,l). It presents a single maximum at m = 0.960057 21 me for which €2 = E ~ ( E I ) = 0.39650461 (note that €2 = ~ 1 / 2for m = 0.983417; see Fig. 1.15). Since the ratchet effect is scaleindependent, the critical value me could well be defined by a purely geometric condition which takes into account the two aforementioned competing mechanisms (degree of symmetry breaking and narrowing of the force pulse): A(m = m,)/A(m = 0) = @/2, where CP = (&+ 1) /2 is the golden ratio and A(m)= fellzp ( t ; T , m19 ,' = 0) d t . This gives me= 0.9830783... .

":s

161

!

'

!

.

!

'

!

'

!

'

!

m Figure 1.15: Shape function S(m) (Eq. (1.28)). One finds that the degree of symmetry breaking mechanism confirms and explains all the previous experimental, theoretical, and numerical results from a great diversity of systems subjected t o a biharmonic force (1.23). This is the case of the ac driven, damped sineGordon equation, 4tt

- 4m

+ sin ( 4 ) = -P4t + f ( t )

(1.29)

where directed energy transport requires a non-zero topological charge, implying the existence of sineGordon solitons (kinks) in the system. It is worth mentioning that

Harmonic versus non-harmonic excitations: the waveform effect

25

the sineGordon equation has important applications to superconducting devices such as long Josephson junctions. In the simple case of a biharmonic force,

j ( t ) = el sin (6t + 6 0 ) +

sin (m6t

+ h0 + e ) ,

(1.30)

the kink velocity may be controlled by changing both the initial phase 60 and the relative phase B (with the remaining force parameters held constant). In the following, control of kink-mediated directed energy transport by using the elliptic force (1.26) is discussed. One firstly obtains the Fourier series of the force: (1.31) (1.32) Now a collective coordinate approach with two degrees of freedom, X ( t ) and l ( t ) (respectively, position and width of the kink), can be directly applied to obtain the dynamics of these two collective coordinates in the presence of the elliptic force:

p 1

=

-PP .2

-

= 1 /(21)

qfellip(t),

+ 1/(2al) - pz

- (R2,1/2) (1

+ P 2 / M i ),

(1.33)

is the Rice frequency, where the momentum P ( t ) = M0loX/l(t), QR = f i / ( d o ) a = n2/12, and MO = 8, q = 27r, and 10 = 1 are, respectively, the dimensionless kink mass, topological charge, and unperturbed width. From Eq. (1.31), one sees that, even for m values very close to 1, the force may be reliably approximated by its two first harmonics. For this biharmonic approximation, one straightforwardly obtains the following estimate for the average velocity of the kink,

(1.34) with (1.35)

26

Introduction

where

(1.36) and where, as expected, ( X o ( t ) ) = 0, and S ( m )is the shape function (1.28). From Eqs. (1.34)-(1.36), one sees that the average velocity is independent of the initial phase 0, while a nonzero velocity exists for 0 < m < 1 according to the degree of symmetry breaking mechanism. 1.4.6 Reshaping-induced control of synchronization of coupled limit-cycle oscillators As a seventh example, consider a system of N coupled oscillators with phases 4iE [0,27r]. In particular, the dynamics of the system is described by the following N

differential-difference equations: (1.37) with T = 2n,O = 0, and where E is the coupling constant, r is the delay, wO is the intrinsic frequency of the oscillators, and fellip (.; T, m,0) if the elliptic force (1.31). It is straightforward to obtain that the lowest stable frequency associated with the synchronization states (& = Rt + Q0) for the biharmonic approximation corresponding to the elliptic coupling is given by (1.38) where n is the number of neighbours (four in the case of a square lattice with nearestneighbour interaction) and

d m )=

sech [nK(1- m ) / K ( m ) + ] 4sech [27rK(1- m ) / K ( m ) ]

mK2(m)

(1.39)

Figure 1.16 shows the symmetry-breaking-induced frequency suppression (Rmin/w0 versus m ) for the parameters n = 4 , = ~ 0.1, and E = 3. Note that the minimum ( m = 0.9845 N me)is the same for all values of T , n, and E .

27

Notes and references

0.2

0.4

0.6

0.8

1.0

m Figure 1.16: Lowest synchronization frequency vs shape parameter (Eq. (1.38)).

1.5

Notes and references

In this chapter the main features, advantages, and difficulties of two basic control approaches (feedback and non-feedback) were briefly discussed. A much more detailed description of a wide spectrum of such control methods, including classical control methods of engineering, can be found in Chen and Dong [l].Although control of chaos represents nowadays one of the most active areas of research in the field of nonlinear dynamics, it is remarkable that classical control theory was originally developed for industrial and military applications [a]. For feedback methods, the interested reader is referred to Ref. [3]. At present, the literature concerning theoretical, numerical, and experimental studies of non-feedback methods is frankly unapproachable in a monograph of the present type. Therefore, only pioneering key work (from the author’s viewpoint) is mentioned in the following. The effectiveness of periodic parametric excitations in suppressing chaos was shown by Alekseev and Loskutov in Ref. [4]. Hubler and Liischer [5] discussed how a nonlinear oscillator can be driven towards a given target dynamics by means of resonant excitations. Braiman and Goldhirsch [6] provided numerical evidence to show the possibility of inhibiting chaos by an additional periodic external excitation. Salerno [7] showed the possibility of suppressing chaos in long biharmonically driven Josephson junctions by the analysis of a phase-locked map. Comments on references containing the application of Melnikov’s method to the problem of control of chaos by small-amplitude harmonic excitations are included in the next chapter for the sake of completeness. Experimental control of chaos by

28

Introduction

weak periodic excitations has been demonstrated in many diverse systems [ll-241. The ratchet effect [25] presents promising applications in a great diversity of problems, such as electronic transport through molecules [26], smoothing surfaces [27],controlling vortex density in superconductors [28], separating particles [29], controlling directed current in semiconductors [30], and rectifying the phase across a SQUID [31]. Synchronization phenomena are of great interest in diverse fields, such as science, engineering, and social life, where apparently different phenomena can be understood within a common framework. The synchronization of periodic oscillators, chaotic systems, large ensembles, and oscillatory media has attracted constant interest for many decades [32]. In particular, globally coupled oscillators are a simple class of many-body dynamical systems, in which each oscillator is coupled to all the others [33-351. The discussion of many of the results of this chapter originated in work by the present author and coworkers [8-10,361.

[1] Chen, G. and Dong, X., (1998) &om Chaos to Order: Perspectives, Methodologies and Applications, World Scientific, Singapore. Sontag, E. D., (1998) Mathematical Control Theory: Deterministic Finite[2] Dimensional Systems, Springer, Berlin, 2nd ed. [3] Ott, E., Sauer, T., and Yorke, J. A,, (1994) Coping with Chaos, Chapter 12: “Control: Theory of Stabilization of Unstable Orbits,” Wiley, New York. Alekseev, V. V. and Loskutov, A. Y., (1987) “Control of a system with a strange [4] attractor through periodic parametric action,” Sow. Phys. Dokl. 32, pp. 13461348. [5] Hubler, A. W. and Luscher, E., (1989) “Resonant stimulation and control of nonlinear oscillators,” Naturwissenschaft 76, pp. 67-69. [6] Braiman, Y. and Goldhirsch, I., (1991) “Taming chaotic dynamics with weak periodic perturbations,” Phys. Rev. Lett. 66, pp. 2545-2548. [7] Salerno, M., (1991) “Suppression of phase-locking chaos in long Josephson junctions by biharmonic microwave fields,” Phys. Rev. B 44, pp. 2720-2726. [8] Chac6n, R. and Diaz Bejarano, J., (1993) “Routes to suppressing chaos by weak periodic perturbations,” Phys. Rev. Lett. 71, pp. 3103-3106. [9] C h a c h , R. and Martinez Garcia-Hoz, A., (2002) “Route to chaos via strange nonchaotic attractors by reshaping periodic excitations,” Europhys. Lett. 57 (l),pp. 7-13. [lo] Chac6n, R., (2002) “Modifying fractal basin boundaries by reshaping periodic terms,” J . Math. Phys. 43 (7), pp. 3586-3591. [ll]Ditto, W. L., Rauseo, S. N., and Spano, M. L., (1990) “Experimental control of chaos,” Phys. Rev. Lett. 65, pp. 3211-3214. [12] Azevedo, A. and Rezende, S. M., (1991) “Controlling chaos in spin-wave instabilities,” Phys. Rev. Lett. 66, pp. 1342-1345.

Notes and references

29

[13] Hunt, E. R., (1991) “Stabilizing high-period orbits in a chaotic system: The diode resonator,” Phys. Rev. Lett. 68, pp. 1953-1955. [14] Roy, R., Murphy, T. W., Maier, T . D., Gills, Z., and Hunt, E. R., (1992) “Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, pp. 1259-1262. [15] Petrov, V., Gaspar, V., Masere, J., and Showalter, K., (1993) “Controlling chaos in the Belousov-Zhabotinsky reaction,” Nature (London), 361, pp. 240-243. [16] Meucci, R., Gadomski, W., Ciofini, M., and Arecchi, F. T., (1994) “Experimental control of chaos by means of weak parametric perturbations,” Phys. Rev. E 49, pp. R2528-R2531. (171 Ding, W. X., She, H. Q., Huang, W., and Y u , C. X., (1994) “Controlling cham in a discharge plasma,” Phys. Rev. Lett. 72, pp. 96-99. [18] Schiff, S. J. et al., (1994) “Controlling chaos in the brain,” Nature 370, pp. 615-620. (191 Vohra, S. T., Fabiny, L., and Bucholtz, F., (1995) “Suppressed and induced chaos by near resonant perturbation of bifurcations,” Phys. Rev. Lett. 75 (l),pp. 65-68. [20] Chizhevsky, V. N. and Corbalan, R., (1996) “Experimental observation of perturbation-induced intermittency in the dynamics of a loss-modulated COz Laser,” Phys. Rev. E 54 (5), pp. 4576-4579. [21] Dangoisse, D., Celet, J.-C., and Glorieux, P., (1997) “Global investigation of the influence of the phase of subharmonic excitation of a driven system,” Phys. Rev. E 56 (2), pp. 1396-1406. [22] Uchida, A., Sato, T., Ogawa, T., and Kannari, F., (1998) “Nonfeedback control of chaos in a microchip solid-state laser by internal frequency resonance,’’ Phys. Rev. E 58 (6), pp. 7249-7255. [23] Schwartz, I. B., Triandaf, I., Meucci, R., and Carr, T. W., (2002) “Open-loop sustained chaos and control: a manifold approach,” Phys. Rev. E 66 (2), pp. 026213/17. [24] Alonso, S., SaguBs, F., and Mikhailov, A. S., (2003) “Taming Winfree turbulence of scroll waves in excitable media,” Science 299 (5607), pp. 1722-1725. [25] Reimann, P., (2002) “Brownian motors: noisy transport far from the equilibrium,” Phys. Rep. 361, pp. 57-265. [26] Lehmann, J., Kohler, S., Hanggi, P., and Nitzan, A., (2003) “Rectification of laserinduced electronic transport through molecules,” J. Chem. Phys. 118, pp. 32833292. [27] DerBnyi, I., Lee, C.-S., and BarabLi, A.-L., (1998) “Ratchet effect in surface electromigration: smoothing surfaces by an AC field,” Phys. Rev. Lett. 80, pp. 1473-1476. [28] Lee, C.-S., Jank6, B., DerBnyi, I., and BarabLi, A.-L., (1999) “Reducing vortex density in superconductors using the ratchet effect,” Nature 400, pp. 337-340. [29] Rousselet, J., Salome, L., Ajdari, A., and Prost, J., (1994) “Directional motion of Brownian particles induced by a periodic asymmetric potential,’’ Nature 370, pp. 446448.

30

Introduction

[30] Alekseev, K. N., Erementchouk, M. V., and Kusmartsev, F. V., (1999) “Direct current generation due to wave mixing in semiconductors,” Europhys. Lett. 47, pp. 595-600. [31] Zapata, I., Bartussek, R., Sols, F., and Hanggi, P., (1996) “Voltage rectification by a SQUID ratchet,” Phys. Rev. Lett. 77,pp. 2292-2295. [32] Pikovsky, A. Rosenblum, M., and Kurths, J., (2001) Synchronization. A universal concept in nonlinear sciences, Cambridge University Press, Cambridge. [33] Niebur, E., Schuster, H. G., and Kammen, D. M., (1991) “Collective frequencies and metastability in networks of limit-cycle oscillators with time delay,” Phys. Rev. Lett. 67, pp. 2753-2756. [34] Strogatz, S. H., Mirollo, R. E., and Matthews, P. C., (1992) “Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping,” Phys. Rev. Lett. 68, pp. 2730-2733. [35] Zheng, Z., Hu, G., and Hu, B., (1998) “Phase slips and phase synchronization of coupled oscillators,” Phys. Rev. Lett. 81, pp. 5318-5321. [36] Chacbn, R. and Quintero, N. R., (2005) “On the ratchet effect,” arXiv:physics/0503125 (preprint).

Chapter 2 THEORETICAL APPROACH 2.1

Dissipative systems versus Hamiltonian systems

The dynamics of chaotic Hamiltonian systems, in both their classical and quantum versions, has attracted great interest in the last two decades or so. In particular, classical Hamiltonians play a central role in plasma physics and celestial mechanics, where dissipative forces can sometimes be neglected on the time scales of interest. Deterministic stochasticity is often unwanted in diverse situations appearing in such contexts. Thus, the problem of suppressing (or, in general, controlling) deterministic stochasticity naturally arises, so that the consideration of periodic stochasticitycontrolling excitations seems pertinent as in the dissipative case. However, dissipative and Hamiltonian flows are markedly different, as is well-known. Consider, for instance, the Hamiltonian limiting case associated with the system (1.1). The control theorems discussed for dissipative systems in the present chapter predict (under certain conditions) effective chaos-controlling excitations having amplitudes appreciably less than those of the associated chaos-inducing excitations. Furthermore, the theory provides analytical generic expressions for the intervals of initial phase difference between the two excitations involved for which chaotic dynamics can be controlled (eliminated or enhanced). Such intervals have finite measure. These two remarkable properties are ultimately a consequence of the existence of attractors and hence of basins of attraction. Additionally, chaotic behavior is generally found in dissipative systems for a certain range of parameters, with the remaining range giving rise to regular behaviour, while for Hamiltonian systems stochastic behaviour is generally present over the entire parameter range. In particular, a generic Hamiltonian perturbation always yields stochastic motion in a layer surrounding a separatrix, while the motion near the separatrix is not necessarily chaotic for a dissipative system. Since in Hamiltonian systems transients never decay, i.e., the orbits never settle down to a lower-dimensional attractor, one should expect a rather different control scenario for Hamiltonian systems in view of the aforementioned properties, and this is beyond the scope of the present work. This book deals with the family of dissipative systems described by Eq. (1.1)which are capable of being studied by Melnikov’s method. 31

32

2.2

Theoretical Approach

Stability of perturbed limit cycles

It was mentioned in the previous chapter that feedback methods present the attraction of using small-amplitude chaos-controlling excitations as a result of two important properties of chaotic systems: (i) chaotic orbits are closures of the set of (infinite) unstable periodic orbits, (ii) the dynamics in the chaotic attractor is ergodic. Thus, sooner or later a chaotic trajectory will fall into the vicinity of the chosen unstable orbit. Then, the associated unstable fixed point in the Poincar6 section is stabilized by moving its stable manifold to the system state point. On the other hand, the basic idea underlying non-feedback methods is the application of periodic resonant cham-controlling excitations which are suitably chosen to drive the system into periodic orbits (unstable for certain parameter ranges) such that the resonant effect is enough to change dramatically the system dynamics, even for very small amplitudes of the cham-controlling excitation. Thus, both types of control have the common characteristic of using weak excitations t o reliably suppress or enhance the initial chaotic state by changing the stability properties of periodic orbits. Therefore, the method of control of chaos by weak periodic excitations can be heuristically discussed by analytically studying a one-dimensional model of an unstable limit cycle affected by two weak resonant excitations xn-tl =

[a+ E (fn

+ PG)] x n ,

(2.1)

with (Y > 1,0 < p < l , f n = a s i n n , c , = &sn[2K(m) ( n + 0)/7r;m] (i.e., for the sake of clarity, by choosing the main resonance (Tsin= T,, = 27r)). Here sn is the Jacobian elliptic function of parameter m, K ( m ) is the complete elliptic integral of the first kind, and 0 is the initial phase (0 0 27r). Of course, the foldzngand-stretchzng mechanism, which is typically the source of chaos in one-dimensional maps, is absent in map (2.1). It is designed to only capture the effect of weak nonautonomous excitations on the stability of a generic unstable limit cycle. The elliptic function sn is chosen to be the control excitation to take into account in a simple form the effect of the excitation waveform on the control scenario. When m = 0, then sn [2K(n + 0)/7r; m = 01 = sin(n + 0 ) .In the other limit, m = 1, one readily

< <

(2.2) which is the Fourier expansion of the square wave function of period 27r. Note that the normalization factor 2K(m)/7t permits one to solely change the waveform from a sine to a square wave by varying the elliptic parameter from 0 to 1, respectively. Thus, one has a unique waveform parameter, m, t o solely change this aspect of the time-periodic excitation, while its amplitude and period are held fixed. To characterize the effect of the two weak excitations in the map (2.1), one calculates the Lyapunov exponent X for E # 0: A

+ E (fn + PG)]) ,

= Re (Ln [a

(2.3)

33

Stability of perturbed limit cycles

where angle brackets () denote the average over n. For small E , the Lyapunov exponent becomes (fn

+ pcn)

-

1 E 2 a

-

(-)

2

(f,"+ 2pfncn + p 2 c i ) + 0 ( E ~ .)

(2.4)

It is straightforward to see that

=

(CK)

f [I

-%I, (2.5)

where E ( m ) is the complete elliptic integral of the second kind. Substituting (2.5) into (2.4) one finally obtains

(2.6) To clarify the effect of the second resonant excitation, cn, on the enhancement or reduction of instabilities (positive or negative A, respectively), let us consider that, in the absence of the second excitation ( p = 0), the map (2.1) presents a weakly unstable initial state L n a 2 f such as X N Xf ( p = 0) = L n a - (&/a)'2 0. For p # 0, the Lyapunov exponent (2.6) can be put into the form

a

: (3

X = X+(m= 0) - -

+

~ % ~ ([Pm %2(m) ) cosO] + O ( E ~ )

(2.7)

with

(2.8)

( m )(X2(m)) is a monotonously increasing (decreasing) function with limiting values Wl(m= 0) = 1,S 1 ( m= 1) = 2 (%4m= 0) = 2, %2(m= 1) = 1.34144... .) Then for fixed waveform m and small fixed p # 0, the Lyapunov exponent X decreases when ,b + W2(m)cos 0 0 and, in some range of 0, may become negative, thus stabilizing z -the optimal value of 0 for stabilization being 0 = = 0. Contrarily,

0Sb

34

Theoretical Approach

+

@y= n yields

X increases when ,B %2 (m)cos 0 < 0 so that the initial phase 0 = the largest positive Lyapunov exponent. Observe that for P M ,Bthreshold,

(2.9) one has maximum-range intervals of suitable initial phase for stabilization [ O z A@,,,, A@,,] and strengthening - A@,,,, A@,,] of instabilities (A@,,, = ~ 1 2 ) Similarly, . for P > Pthresholdr one sees that the respective ranges have shrunk, i.e., A@,, < ~ 1 2 It. is worth mentioning that the discussed control mechanism is valid for any waveform of the second resonant excitation, i.e., one expects the role played by the initial phase of the control excitation to be independent of its particular waveform.

@Zb+

2.3

[@Tb @y+

Non-autonomous second-order differential systems

It was already mentioned in Chap. 1 that the present book mainly deals with the control of chaos by weak harmonic excitations in model systems given by Eq. (1.1).The dynamics of many well-known real-world system fit reasonably ab initio Eq. (l.l), the damped driven pendulum being a paradigmatic example among many others. However, a broad variety of dynamical equations reduce to Eq. (1.1) under diverse conditions : (i) suitable changes of variables, as in the interaction of charged particles with an electrostatic wave packet; (ii) investigation of separable solutions in nonlinear dissipative partial differential equations such as the one-dimensional Schrodinger equation subjected to spatio-temporal excitations; (iii) study of the synchronization manifold in arrays of chaotic coupled oscillators; (iv) study of the dynamics of the centre of mass of solitons in Frenkel-Kontorova chains and in sine-Gordon equations. Since the problem of controlling spatio-temporal chaos is pertinent and important in each of the aforementioned physical contexts, the applicability of the results of the present work extends from the control of temporal chaos to (certain aspects of) the control of spatietemporal chaos. To illustrate this point, diverse specific examples will be studied in Chap.5. 2.4

Basics of Melnikov’s method

Melnikov’s method is one of the few analytical methods available for determining the onset of chaotic instabilities in near-integrable systems. It is a global perturbation technique that utilizes the geometrical properties of integrable system as a framework in which to develop the analysis of associated perturbed systems. Generally, the unperturbed system is an integrable Hamiltonian system having a normally hyperbolic invariant set whose unstable and stable manifolds intersect non-transversely. The global geometry corresponding to the integrable system serves to develop coordinates which are used to determine if any of the homoclinic orbits in the normally hyperbolic invariant set survives under perturbation.

35

Basics of Melnikov's method

For the sake of clarity, and to illustrate the basic approach, the version of the method for homoclinic orbits in two-dimensional, time-periodic vector fields will be considered here. Consider the family of systems

x = fo (x)

+ Ef' (x,t ),

(2.10)

where x,fo,flE Rz and fi is periodic in t with (fixed) period T . The unperturbed equation ( E = 0) represents an integrable Hamiltonian system, i.e., f0l

=

dH 652, foz =

aH -->

(2.11)

8x1

while Efl (x,t ) is generally a dissipative small perturbation (i.e., it need not be Hamiltonian itself). It is also assumed that the unperturbed system possesses a homoclinic orbit xo (t - t o ) to a hyperbolic saddle point XO, such that the interior of {XO (t - t o ) I t E R } U {Xo} is filled with a continuous family of periodic orbits. Now, it is useful to consider the Poincari! map P,"" : Cto -+ Cto,where Cto= {(x,t) 1 t = t o E [O,T]}c R2 x S' is the global cross section at time to, to simplify the phase space visualization. Note that the unperturbed Poincari! map P$ has a hyperbolic saddle point %O whose stable and unstable manifolds W" (20), W" ( j z o ) are smoothly joined forming a closed curve which is filled with non-transverse homoclinic points for P$ . One can expect that under a small perturbation E f i (x,t)the perturbed stable and unstable manifolds no longer join smoothly, perhaps yielding either transversal homoclinic points or no homoclinic points at all. For E sufficiently small, the perturbed system has a unique hyperbolic 7'-periodic orbit x ( t ,t o ) = xo O(E). Correspondingly, the Poincari! map P,"" has a unique hyperbolic saddle point X = XO + O(E). The orbits xs (t,to), x u(t,to) lying in the manifolds W" (X) , W" (X), respectively, and based on Cto can be represented as

+

xs ( t ,to) xu( t ,t o )

= =

xo (t - t o ) xo (t - to)

+ EX; (t,to) + 0 (2),t E [ t o , + EX: ( t ,t o ) + 0 ,t E (E2)

00)

(-00,

to] .

(2.12)

For a Hamiltonian perturbation such orbits generically intersect, giving rise to an infinite number of homoclinic points and stochastic phenomena. However, there exist three possibilities for a generic dissipative perturbation: (i) the unstable orbit always lies inside the stable orbit; (ii) the unstable orbit always lies outside the stable orbit; (iii) the unstable and stable orbits intersect transversely giving rise to chaotic phenomena. One next defines the distance d ( t 0 ) between the two points x" (to, t o ) and xs ( t o , t o ) along the direction of the vector n, which is orthogonal to the tangent vector f",as (2.13) where n=

(7:).

(2.14)

36

Theoretical Approach

Introducing the wedge operator a A b = albz - azbl and substituting (2.14) in (2.13), one can write (2.15) Inserting (2.12) into (2.15) one obtains

(2.16) where D(t0) is the Melnikov distance at to. Its corresponding expression for an arbitrary value of t is defined as

o(tjt o )

=

(XO(t - t o ) ) A (x;" ( t ,t o ) - X: (t,t o ) ) D"(t, t o ) - D"(t,to).

fo

=

(2.17)

Taking the time derivative of D", one has

D s ( t ,t o )

= =

+

(XO(t - t o ) ) A Xs ( t ,t o ) fo (XO(t - t o ) ) A $ ( t ,t o ) M (XO(t - t o ) ) Xo (t - t o ) A X? ( t ,t o )

fo

+fo (xo (t - t o ) ) A xs (4 t o ) I

(2.18)

where (2.19) is the Jacobian matrix of f o evaluated at xo (t - to). To calculate ( t ,t o ) and x o (t - t o ) one substitutes (2.12) into (2.10) and expands the right-hand side in a series with respect to E

x o (t - t o )

+

€*

(t,t o )

=

=

+

fo (xo(t - t o ) EX; ( t ,t o ) ) +&fl (xo ( t - t o ) + EX; (t,t o ) , t ) fo (xo (t - t o ) ) + EM (xo(t- t o ) ) x; (4 t o ) +O(E2).

(2.20)

From (2.20) one straightforwardly obtains

x o (t - t o )

= fo

and XI

( t ,t o )

= M (XO(t - t o ) ) ~f

(xo (t - t o ) )

(2.21)

( t ,t o ) + fi (xO ( t - t o ) ,t ) .

(2.22)

37

Basics of Melnikov's method

Using (2.21) and (2.22) in (2.18),

D" ( t ,t o )

=

M (XO( t - to)) fo (XO( t - t o ) ) A xi ( t ,t o ) +fo (XO(t - t o ) ) A M (XO(t - t o ) ) xi ( t ,to) +fo (XO( t - t o ) ) A fi (XO(t - t o ) ,t ) .

(2.23)

Note that the first two terms in (2.23) combine to give

D" ( t ,t o )

=

=

TrM (XO(t - t o ) ) fo (XO(t - t o ) ) A XS ( t ,t o ) +fo (Xo (t - to)) A fi (Xo (t - t o ) , t ) TrM (XO( t - to)) D" fo (XO( t - to)) A fi (XO(t - t o ) ,t ) , (2.24)

+

where Tr M is the trace of the Jacobian matrix of fo, and where the identity Pa A b + a A Pb = Tr P (aA b) with P a 2 x 2 matrix has been used. For the present case of an unperturbed Hamiltonian system, Tr M = 0. Integrating (2.24) then yields

(2.25) However, limt,, xo (t - t o ) = Xo, and hence limt,, fo (XO(t - t o ) ) = 0 because 0. Moreover xi ( t ,t o ) is bounded. Therefore, one gets D " ( m ,t o ) = 0. fo (xo) = Thus.

fo

(XO(t - t o ) ) A fi (XO(t - t o ) , t ) d t .

(2.26)

Proceeding similarly to calculate D", one obtains

D" (to, t o ) =

s_,

to

fo (xo (t - t o ) ) A f l (xo (t - t o ) , t )d t .

(2.27)

The Melnikov function M ( t o )= D(t0) = D"(t0, t o ) - D"(t0,t o ) is then

M(to) =

[I

fo

(XO(t - t o ) ) A f1 (XO(t - t o )

, t )d t ,

(2.28)

which allows the distance function (2.16) to be put into the form (2.29)

Remarks. First, the Melnikov function M ( t o ) provides a first-order approximation for the distance between the unstable and stable manifolds of Xi. In computing &'(to) one is effectively standing at a fixed point xo(0) (and the vector n (x(0)) is also fixed) on a moving cross section CtO and watching the perturbed manifolds oscillate as to

38

Theoretical Approach

varies. If M ( t o )has simple zeros at to,%,i.e., M(t0,i)= 0 and (dM/dto) (to,i) # 0 , then the unstable and stable manifolds will intersect transversally, creating a homoclinic point. This process is called a homoclinic bifurcation and indicates the onset of chaotic motion. It is worth mentioning that this chaotic motion is local in the sense that it is expected only for trajectories based at points suficiently near the separatrix of the unperturbed system. Second, the Melnikov theory discussed for homoclinic orbits also applies to the breakup of heteroclinic orbits: the Melnikov function is the same and has exactly the same interpretation for both kind of orbits connecting hyperbolic saddle points. Third, Melnikov’s method predictions are solely concerned with homoclinic (and heteroclinic) bifurcations. This meam that in most cases they provide an approximate threshold (in the parameter space) for transient chaos but not, in general, for steady chaos (strange chaotic attractor). This is ultimately a consequence of the Smale-Birkhoffhomoclinic theorem which states that the existence of transverse homoclinic orbits implies the existence of a highly complicated, chaotic orbit structure nearby (a Smale horseshoe). However, the horseshoe scenario provides chaos solely on a set of zero measure, while the appearance of a chaotic attractor means steady chaotic behaviour over a global domain of phase space. Fourth, the Melnikov function M(t0) is T-periodic in t o , so that it is enough to analyze its behaviour over a single time period. Fifth, by using the change of variables t + t t o the Melnikov function can be put into the form

+

M(t0) =

1:

fo (XO( t ) )A f1

(XO( t ), t

+ t o ) dt,

(2.30)

in order to facilitate its calculation. Sixth, Melnikov’s method discussed above is also valid for perturbations which represent time-delay terms, i.e., for the family of systems x = fo (x) E f i (x,t) Eg [x(t - T ) - x ( t ) ]where , T is the time delay and g E R is a~matrix. ~ In ~ this case the Melnikov function remains given by Eq. (2.30) with the substitution fi (x,t) -+ fl b , t ) g [x(t - ). - x (t)l. 2.4.1 Illustration: A damped driven pendulum To illustrate the calculation of the Melnikov function with a paradigmatic example, consider the nonlinearly damped, biharmonically driven pendulum given by the equation (2.31) z + sina: = I - sz1iln-l + F cos ( w t ) V Fcos (at O) ,

+

+

+

+

+

where 17, R and 0 are the normalized amplitude factor, frequency, and initial phase, respectively, of the chaos-controlling excitation (0 < q << l ) , and I , 6, n, F, and w are the normalized parameters of the constant force, damping coefficient, d a m p ing exponent, chaos-inducing amplitude, and chaos-inducing frequency, respectively (0 < I , 6, F << 1 , n = 0 , 1 , 2 , ...). It is worth mentioning that Eq. (2.31) has been

39

Basics of Melnikov's method

studied by very many workers, partly for historical reasons and partly because it provides a useful model' for the dynamics of a Josephson junction. The application of Melnikov's method to Eq. (2.31) involves calculating the Melnikov functions (cf. Eq. (2.30)),

1:

J

J-00

+F

+qF

--03

+

zo,+(t)cos [w (t t o ) ] dt

/a

50,*((t)cos [R (t

+ t o ) + Q] dt,

(2.32)

--03

where the negative (positive) sign refers to the bottom (top) homoclinic orbit of the underlying integrable pendulum ( I = b = F = 0) : zo,*(t) =

+ (4arctanet -

io,*(t)

f2s~ht.

=

,

7 ~ )

(2.33)

After substituting Eq. (2.33) into Eq. (2.32) and computing the resulting integrals using standard integral tables, Eq. (2.32) can be written as

Mi ( t o )

= D*

f A cos (wto) f C cos (Rto + 0 ),

(2.34)

with

D*

=

f 2 ~ I

A

=

27~Fsech - ,

C

=

2~qFsech

-

-

>

(3

(F)

(2.35)

where B(m, n ) is the Euler beta function. Remarks. First, A and C are positive functions, while D- is a negative function, over the complete range of their parameters. Second, the sign of Df depends upon the relative strength of the constant force with respect to the damping coefficient, I / b 2 E ( n ) H Df 3 0, where the threshold function is (2.36) 7T

which is a monotonously increasing function. For Coulombian ( n = 0), Stokesian ( n = 1, i.e., lineal), and Newtonian ( n = 2) dissipative forces one obtains E ( n = 0) = 1,E ( n = 1) = 4 / ~ and , E ( n = 2) = 2, respectively.

40 2.5

Theoretical Approach

The generic Melnikov function: Deterministic case

Let us assume that the family of systems modeled by Eq. (1.1) satisfies the aforementioned requirements of Melnikov's method. Then, the application of the method to Eq. (1.1) provides the generic Melnikov function

bf&

(To)

=D

fA h a r

(LLTO)

-t Bhar' ( f h o

+ "&)

,

(2.37)

where h a r ( 7 ) means indistinctly sin (7) or cos ( T ) , and A is a non-negative function, while D , B can be non-negative or negative functions, depending upon the respective parameters for each specific system. In particular, D contains the effect of the damping, time-delay terms, and constant forces. In the absence of time-delay terms and constant drivings, D < 0, while one has the three cases D 2 0 when a constant driving and a time-delay term act on the system. Also, A and B contain the effect of the chaos-inducing and chaos-controlling excitations, respectively. Note that changing the sign of B is equivalent to having a fixed shift of the initial phase:

(2.38) where the two signs before T apply to each of the sign superscripts of KP. Therefore, B will be considered (for instance) as a positive function in the following. Observe that, as phase and initial time 70 are not fixed, one can study the simple zeros of M t h , ( 7 0 ) by choosing quite freely the trigonometric functions in Eq. (2.37). One thus avoids the unnecessary complications involved in the abstract form of Eq. (2.37). Therefore, consider, for instance, the nonlinearly damped, biharmonically driven pendulum given by Eq. (2.31). Its corresponding Melnikov function Mi( t o ) (cf. Eq. (2.34)) will be used to establish the general control results. Observe that the Melnikov functions M t h , ( T O ) , Mi(to) (cf. Eqs. (2.37) and (2.34), respectively) are connected by linear relationships which are known for each specific system (1.1):

(2.39) Therefore, the control theorems corresponding to any Melnikov function M t h , ( T ~ can be straightforwardly obtained from those associated with M* (to). 2.5.1 Suppression of chaos It was mentioned in Sec. 2.4 that Melnikov's method provides estimates in parameter space for the appearance of homoclinic (and heteroclinic) bifurcations, and hence for transient chaos. This means that only necessary conditions for steady chaos (strange chaotic attractor) are obtained from the method. Therefore, one may always get suficient conditions for the inhibition of even transient chaos (frustration of homoclinic/heteroclinic bifurcation) and, a fortiori, for the inhibition of the steady chaos that ultimately arises from such a homoclinic/heteroclinic bifurcation. This is the

)

41

The generic Melnikov function: Deterministic case

principal foundation of the utility of Melnikov's method in predicting the suppression of (steady) chaos when a homoclinic/heteroclinic bifurcation occurs prior to its emergence. According to the above discussion, the Melnikov function M*(to) (cf. Eq. (2.34)) will be used to illustrate how to obtain suppression theorems. Let us assume that for 77 = 0 the pendulum (2.31) presents chaotic behavior such that the associated Melnikov function M$ (to)= D* ACOS(wto) (2.40)

+

has simple zeros, i.e., A-(D*IEdkO,

(2.41)

where the equals sign corresponds to the case of tangency between the stable and unstable manifolds. If we now let the additional forcing act on the pendulum (77 # 0) such that C > d , i.e., (2.42) A< ,

c + I D *[

this relationship represents a necessary condition for Mi (to)to always have the same sign, which is q> ( 1 - T ) R . with

R=

y)

cosh ( cosh ( y ).

(2.43)

(2.44)

It is clear that, for general R and 0 (0 < 0 < 27r), this condition is not sufficient to assure that M* ( t o ) always has the same sign. In order to obtain such a sufficient condition, one will first need six lemmas. Preliminary lemmas

For the sake of clarity, the cases Mi ( t o ) will be treated separately.

A. Motion near the lower homoclinic orbit In this case we need three lemmas. Lemma I. Let R/w = p / q for some positive integers p and q . Then there exists t; satisfying cos (Rt; + 0 ) = - cos (wt;) = 1 if and only if f = for some integers 1 and m. Proof. If cos (Rt; + 0 ) = 1 then there exists some integer 1 such that Rtt + 0 = 217r. Also, if cos(wt6) = -1 then there exists some integer m such that wt; = (2m + 1 ) ~Solving . for t; in both relationships, one straightforwardly finds (2.45)

42

Theoretical Approach

Corollary 1. For a generic resonance ( p ,q arbitrary), the values of the initial phase satisfying (2.45) are all of the form r

Q=-

(2.46)

T,

S

where r and s are positive integers. Corollary 2. For a subharmonic resonance ( q = l), the values of the initial phase satisfying (2.45) are 7r,p odd (2.47) 0,peven).

Q={

Lemma 11. Let R/w be irrational. Then there is some tc* such that

+ 0 )- ACOS(wt:*) > A C. Proof. Using the change of variables 0 = 9 + 9 the above inequality reads -CCOS(Qt:*

-Csin(Rtr Let 9 =

-

+ 9)+ Asin

(2.48)

4 (1 - %) if w > R. Then taking t? = E one obtains

-Csin(Rtr

+ 9)+ Asin

< 1. Similarly, one takes 9 = since R and w are incommensurate, i.e., cos (2):. (1 - g ) if w < R. Let now 9 # 3 (1 - %) . Then taking t? = one obtains

2

4

-Csin (Rt:*

+ 9)+ Asin

4

since we have 8 # - 37r5 in this case. rn Lemma 111. Let f ( t ; p ,q ) = t real, p and q integers. Then f is finite if and only if q = 1. One also has that 0 5 f ( t ; p ,1) 5 p 2 ,t E (-m, co). Proof. Consider limt+2?rmf ( t ; p ,q ) . A necessary condition for such a limit to be finite is for the numerator and denominator of f to have their zeros at the same points. Let first q # 1. The numerator and denominator of f present their zeros at t = P and t = 27rs, respectively, with 1 and s integers. For arbitrary positive integers p , q , it is not always possible to find some integer r such that s = s being an arbitrary integer. Therefore, one needs q = 1 for f to be bounded. On the other

w,

F,

[a] sin t 2

hand, since f ( t ; p ,1) = , one straightforwardly obtains 0 5 f ( t ; p ,1) 5 p2 by using finite induction over p . rn

43

T h e generic Mehikov function: Deterministic case

B. Motion near the upper homoclinic orbit We need two additional lemmas for the case Df < 0. Lemma IV. Let R/w = p / q for some positive integers p and q. Then there exists t: satisfying - cos(Rt6 + 0 )= cos(wt;) = 1 if and only if = for some integers 1 and m. Proof. If cos(Rt: 0) = -1 then there exists some integer 1 such that Rt: + 0 = (21 1 ) ~ Also, . if cos(wtG) = 1 then there exists some integer m such that wt: = 2m7r. Solving for t: in both relationships one straightforwardly obtains

+

+

_p -- 21+1-0/7r 4

(2.49)

2m

w

Corollary 3. For a generic resonance (i.e., p , q arbitrary), the values of the initial phase satisfying (2.49) are all of the form

0 = %T,

(2.50)

S

with r, s being positive integers. Corollary 4. For a subharmonic resonance ( q = l), the unique value of the initial phase satisfying (2.49) is 0 = T , vp. (2.51) Lemma V. Let R/w be irrational. Then there is some tz* such that

Ccos(Rt7 + 0 )+ Acos(w~:*) > A - C. Proof. Let 0 = T . Then taking t;* =

C co~(RtG* + 0 )+ A cos (wtG*) = A

one obtains -

(

3

C cos 2 ~ - > A - C ,

since w and R are incommensurate, i.e., cos(27r;) < 1. Let now 0 # = 0 one obtains

tr

+ +

Ccos(RtG* 0) A cos(wtr) = A

T.

Then taking

+ C cos ( 0 )> A - C,

since we have cos(0) > -1 in this case. rn Suppression theorems The suppression theorems associated with the lower and upper homoclinic orbits can now be established as follows. Consider first the case of motion near the lower homoclinic orbit. Clearly, for Eq. (2.43) to also be a sufficient condition for M-(to) to be negative for all to, one must have A - C 2 -A C O S ( W ~~ )C cos(Rto 0). (2.52)

+

44

Theoretical Approach

One now looks for the values of w,R, and 0 permitting Eq. (2.52) to be satisfied for all to. From Lemma 11, a resonance condition is required: R = p w / q . In such a situation, Lemma I gives a condition for Eq. (2.52) to be fulfilled for an infinity of to values. Thus, let us assume that p , q, and 0 satisfy Eq. (2.45). We can then rewrite Eq. (2.52) in the form A 1 - cos ( p T / q ) (2.53) 1 - cos (7) '

c>'

+

with T = wto - (2m 1)7r. Finally, if q (2.53) to be fulfilled for all T :

=

1, Lemma I11 provides a condition for Eq.

R

(2.54)

V<-,

P2

where R is given by Eq. (2.44). In brief, one has the following theorem for subharmonic resonances. Theorem I. Let R = p w , p an integer, such that, for M - ( t o ) , the relationship p = (21 - O / n )/ (am + 1) is satisfied for some integers 1 and m. Then M-(to) always has the same sign, specifically M - ( t o ) < 0, if and only if the following condition is fulfilled:

qmin Vmax

= -

(I

+

5)

R,

R

(2.55)

= -

P2'

Consider now the slightly more complicated case of motion near the upper homoclinic orbit. Indeed, for Eq. (2.43) to be also a sufficient condition for M+(to) to always have the same sign, one must have A - C 3 -A

C O S ( W ~ ~)

C cos (Rto + 0 ),

A-C> A~os(w~~)+C~OS(R~~+O),

(2.56) (2.57)

for D+ > 0, D+ < 0, respectively. Note that Eqs. (2.52) and (2.56) are identical. Therefore, the Lemmas I, 11, and I11 apply to the case D+ > 0. Thus, we obtain a suppression theorem for subharmonic resonances which is formally identical to Theorem I: Theorem 11. Let R = p w , p an integer, such that for M+(to) and D+ > 0, p = (21 - 0/7r)/(2m 1) is satisfied for some integers 1 and m. Then M f ( t o )always has the same sign, specifically M+(to) > 0, if and only if the following condition is

+

The generic Mehikov function: Deterministic case

45

fulfilled:

(2.58) Finally, for the case D+ < 0, we similarly look for the values of w,R, and 0 permitting Eq. (2.57) to be satisfied for all to. From Lemma V, a resonance condition is again required: R = p w / q . In such a situation, Lemma IV gives a condition for Eq. (2.57) to be fulfilled for an infinity of t o values. Thus, let us assume that p , q and 0 satisfy Eq. (2.49). One can then rewrite Eq. (2.57) in the form given by Eq. (2.53) with 7 = wto - 2mn. Finally, for a subharmonic resonance ( q = l),Lemma I11 provides again a condition for Eq. (2.53) to be fulfilled for all 7 , which is given by Eq. (2.54) with R defined by Eq. (2.44). We have in brief the following suppression theorem for subharmonic resonances. Theorem 111. Let R = pw, p an integer, such that, for M+ ( t o ) and D+ < 0, p = (21 1 - 0/7r)/(2m) is satisfied for some integers 1 and m. Then M + ( t o ) always has the same sign, specifically M + (to) < 0, if and only if the following condition is fulfilled:

+

(2.59) Remarks. First, one can test the suppression theorems theoretically by considering the limiting Hamiltonian case (6 = 0) with no constant driving ( I = 0). Note that, in such a case, Eqs. (2.55), (2.58), and (2.59) must be rewritten as

(2.60) since qmincannot now be zero. Thus, by using the Lemmas I and IV one obtains = w , 0 = n, and 7 = 1 (cf. Eq. (2.44)) as a necessary and sufficient condition for suppressing stochasticity. (Observe that this result can be trivially obtained, to first perturbational order, from Eq. (2.32) with S = I = 0, i.e., having M*(to) = 0 for all

R

to.)

46

Theoretical Approach

Second, the lower threshold for the chaos-suppressing amplitude, qmin,takes into account the strength of the initial chaotic state through the factor ( 1 - ID*I / A ) , since one usually finds that the corresponding leading Lyapunov exponent A+ increases as the ratio lD*l / A decreases over a certain range of parameters. Therefore, for fixed chaos-inducing and chaos-suppressing frequencies (and hence R fixed), one would expect that qminwill increase as A+ is increased. Observe that the corresponding upper threshold, qmax,does not verify this important property, which is because qmaxarises from a necessary condition for the necessary condition yielding qminto be also a sufficient condition. This means that qm, slightly underestimates the upper threshold for the chaos-suppressing amplitude, as is numerically and experimentally observed in different instances. It is worth mentioning that this remark holds for any Melnikov function M&,(To) (cf. Eq. (2.37)). Third, the width of the allowed interval ]qmin,q,] for regularization is

(2.61) with R given by Eq. (2.44). Since R is a positive function, there always exists a mazimum resonance order p,, for suppression of homoclinic (and heteroclinic) chaos, for each fked initial chaotic state (i.e., ID*[ / A fixed), which is

(2.62) inwhere the brackets indicate integer part. From Eq. (2.62), one sees that p,, creases as the ratio ID* I / A is increased, which would be associated with the decrease of the corresponding leading Lyapunov exponent over a certain range of parameters. Observe that, for a given set of parameters satisfying the above theorems' hypothesis, as the resonance order p is increased, the allowed interval ]qminrqm,] shrinks quickly for low frequencies (cf. Eq. (2.61)). This means that initial chaotic states cannot necessarily be regularized to periodic attractors of arbitrarily long period, since numerical experiments indicate that the regularized responses are typically a period-1 attractor for p = 1 and a period-2 attractor for p = 2. On the other hand, the asymptotic behaviour Aq (w co) = 00 (the remaining parameters being held constant) means that chaotic motion is not possible in this limit, as expected. Fourth, the asymmetry between the upper and lower homoclinic orbits (cf. Eqs. (2.34) and (2.35)) gives rise to two distinct sets of optimal initial phases that are suitable for suppressing chaos. The optimal suppressory values of 0 (hereafter denoted as OTt) are those values allowing the widest amplitude ranges for the chaossuppressing excitation (the use of the adjective will be justified in the discussion of the suitable intervals of initial phase difference for taming chaos). Indeed, Theorems I and I1 require having 0 = OOpt= T , 0 for p = 2 n - 1 , 2 n ( n = 1 , 2 , 3 , ...), respectively, in order to inhibit chaos when one considers orbits initiated near the lower homoclinic ---f

47

The generic Melnikov function: Deterministic case

orbit or near the upper homoclinic orbit when D+ > 0. Also, Theorem 111requires having 0 = Oqt = n for all p , in order to tame chaos when one considers orbits initiated near the upper homoclinic orbit when Df < 0. The different values of the optimal initial phase (for p even) are those compatible with the surviving natural symmetry under the additional forcing. Indeed, the damped forced pendulum with no chaos-suppressing forcing (7 = 0) is invariant under the transformation

x

+

-x,

t

---t

t+(2n+1)z1

I

+

-I,

71

(2.63)

where n is an integer, i.e., if [x( t ),x ( t ) ]is a solution of Eq. (2.31) with 17 = 0 for a value I , then so is [-x (t (2n 1) 7 r / w ) , -x (t + (2n 1) 7r/w)] for -I. Observe that this pair of solutions may be essentially the same in the sense that they may differ by an integer number of (complete) cycles, i.e.,

+

+

+

+

+ 2x1,

x ( t ) = -x [t (2n + 1) ./W]

(2.64)

with 1 an integer, and they are termed symmetric. Otherwise, the time-shifting and sign reversal procedure yields a different solution, and the two solutions are termed broken-symmetric. When 7 # 0 and 0 is arbitrary the aforementioned natural symmetry is generally broken. It is straightforward to see that the reason for that breaking is - cos (Rt + 0 ) # cos [Rt (272 1)7rR/w 01 , (2.65)

+

+

+

for arbitrary w , R, and 0. Assuming a subharmonic resonance condition R = pw, the survival of the above symmetry implies cos (pwt

+ 0 ) = (-

l)P+'

cos (put

+ 0).

(2.66)

Obviously, this is only the case for p an odd integer. For p an even integer, one has the new transformation

x t

--+

I

+

0

-+

-x, 7r

--f

t+(2n+l);,

-I, OfT,

(2.67)

i.e., if [x( t ),Z ( t ) ]is a solution for values I and 0 , then so is

+

[-x (t + (2n + 1)n / w ) , -a: (t + (2n 1) 7r/w)] for -I and 0 k 7r. Thus, this explains the origin of the aforementioned differences between the corresponding (at the same resonance order) allowed Ooptvalues for the

48

Theoretical Approach

upper homoclinic orbit with D+ > 0 and the lower homoclinic orbit (recall that D- < 0 for any value of the parameters). As will be discussed in Chap. 3, this “maximum symmetry principle” appears to be the common background in the suppression of chaos by weak resonant excitations. Fifth, for a fixed constant driving I , the effectiveness of the chaos-suppressing excitation strongly depends upon a suitable choice for the dissipative force in the model of the system to be controlled. Note that, depending on the specific choice of the dissipative force, we can have D+ > 0 or D+ < 0 (cf. Eq. (2.36) and corresponding remark), and hence the suitable OTt can also change, according to the previous remark. Sixth, to establish the suppression theorem corresponding to any Melnikov function (2.37), it is enough to transform M,$(To) into the form given by Eq. (2.34), which implies, in particular, that the initial phase Q& (mod27r) must satisfy the following relationships

f Qcos,sin

= @

q&,,=

+

7l 2 1

R

@

+ ”-> W

(2.68)

Therefore, taking into account, for instance, that Qopt = 7r,Vp, for M+(to)with D+ < 0, and Eq. (2.68), one finds that in general there exist at most four suitable optimal values for the (suppressory) initial phase differences between the two (commensurate: 0 = pw) excitations: 0, ~ / 27 r , 3 ~ 1 2 The . optimal initial phases for the case D < 0, B > 0 (cf. Eq. (2.37)), and the respective values for B < 0 (in parentheses, cf. Eq. (2.38)), are given in Table 2.1 where n = 1 , 2 , ... .

49

The generic Melnikov function: Deterministic case

Table 2.1 p=4n-3

p=4n-2

p=4n-1

p=4n

Table 2.2 gives the optimal initial phases for the caseD>0,B>0(cf. Eqs. (2.37) and (2.47)), and the respective values for B<0 (in parentheses, of. Eq. (2.38)). Table 2.2

p=4n-3

p=4n-2

p=4n-1

p=4n

Suitable intervals of initial phase difference It has been stated above that the suitable values of the initial phase difference (between the two excitations involved) given by the Suppression Theorems are optimal, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation. One therefore could expect reliable control of the dynamics over certain suitable initial phase difference intervals, which would be centred on such optimal values, although this would imply a reduction of the respective amplitude ranges. To obtain the width of such suitable initial phase difference intervals for chaos suppression as a function of the system's parameters, it is sufficient to analyze the temporal behaviour of any Melnikov function M,&, ( T O ) (cf. Eq. (2.37)). Thus, the Melnikov function M + ( t o ) = D+ A sin (wto) C sin (Rto 0 ) (2.69)

+

+

+

50

Theoretical Approach

with D+ < 0 (cf. Eq. (2.35)), which corresponds to the damped driven pendulum (2.31) with forcing terms given by sine functions instead of cosine functions, will be used, for instance, to obtain the generic analytical expressions for the aforementioned intervals of the initial phase difference for which chaotic dynamics can be eliminated. By hypothesis, in the absence of any chaos-suppressing excitation (7= 0, i.e., C = 0, cf. Eq. (2.35)), the associated Melnikov function

(2.70)

M;(tO)=-ID+I+Asin(wto) changes sign at some to, i.e., ID+/ in Fig. 2.1.

< A.

A plot of M$ ( t o ) in this situation is shown

0

I to/ T

Figure 2.1: Functions M; ( t o ) and Cminsin (wto

+ 0 ) for two values of 0.

As was shown in Sec. 2.5, if one now lets the additional sinusoidal forcing act on the pendulum such that C < A - ID+l, this relationship represents a sufficient condition for M+ ( t o ) to change sign at some to. Thus, a necessary condition for M+ ( t o ) to always have the same sign is

C > A - ID+( Cmin.

(2.71)

+

Plots of Cminsin (wto 0 ) (i.e., for the main resonance) are shown in Fig. 2.1 for two limiting values of 0. First, we see that the value of 0 which corresponds to the situation in which the maxima of M; ( t o ) coincide with the minima of Cminsin (wto 0 ) (and vice versa), i.e., when such functions are an opposition, is T , corresponding to the respective value of 0,, for the main resonance (cf. Table 2.1). It must be emphasized that when the above two functions are in opposition for any resonance i2 = pw,

+

51

The generic Melnikov function: Deterministic case

one has that 0 matches the respective value OWt for that resonance, and vice versa. Thus, one has a geometrical characterization of the optimal (the use of the adjective will be justified below) values of 0 for chaos inhibition. Second, 0 = 0,t f A@,, (see Fig. 2.1), that is associated with the maximum deviation from Oqt such that there still exists a value of C (C > C,in) for which M+(to) < 0,Vto. This last inequality becomes impossible for 0 > OWt A@,,, or @ < OTt - A@,, so that chaos inhibition would not be expected. Therefore, (3,t fA@,, represent threshold values for inhibiting homoclinic/heteroclinic bifurcations. In order to obtain an analytical expression for A@,,,, consider the interval At0 = It; - tc;*l between (any two of) the nearest zeros of Cmjnsin (Rto 0,) and M$(t,), respectively (see Fig. 2.1). One straightforwardly obtains, for any resonance R=pw,

+

+

A0,,,

= Ipwt;

+ Oopt- wtg*l = arcsin (‘?+I) -

(2.72)

?

i.e., A@,,, does not depend on the additional excitation (unlike Oqt), which means that Eq. (2.72) in fact provides the generic expression of A@,, for any Melnikov function (2.37). Note that expression (2.72) takes into account the strength of the initial chaotic state through the ratio I D+I / A , as does qmin(cf. the second remark to the Suppression Theorems). For the Hamiltonian limiting case ( q = 0, and hence Df = 0) A@,, = 0, so that only particular values of 0 , namely OTt = {0,7r/2,7r, 3n/2}, are in general suitable for inhibition of homoclinic/heteroclinic bifurcations (OOpt= T for the pendulum (2.31), in complete agreement with the first remark to the Suppression Theorems). Indeed, an admissibility argument confirms that {0,7r/2, 7r, 3 ~ / 2 }are in fact the only possible values of Oqt for a generic subharmonic resonance R = pw (as expected from Tables 2.1 and 2.2). Consider first the Hamiltonian limiting case in Eq. (2.37). It is easy to see that R = w , A = B , and Q&, E (0, 7r/2, T , 3 ~ / 2 }(depending on the signs of A and B , and the functions har, har’) is a necessary and sufficient condition for suppressing stochasticity, i.e., having M& ( T O ) = 0, for all 70.For a general dissipative case ( D # 0), since &Aha?-( W T O ) and Bhar‘ (a70 Q,t) must be in opposition (by definition of Q W t ) , this interferencetype effect implies that, for any resonance R = pw, QOpt can only take some of the four aforementioned values. Consider now the dependence of the threshold values of C (cf. Eq. (2.69)) on the initial phase difference 0 , in particular on OTt fA@,,,. According to the above discussion, when 0 = 0,,kA0,, the frustration of a homoclinic/heteroclinic bifurcation will only be effective when the lower-threshold value of C (hereafter referred to as Cmin(0 = 0,t 4~A@,,)) is larger than Cmin= Cmin(0 = @,t) (cf. Eq. (2.71)). Indeed, Cmin (0 = Oqt k A@,,) can be easily obtained from Cmin(0 = 0,t) suitably weighted by a phase factor that depends on A@,,, (see Fig. 2.1):

+

Cmin(0 = Oqt & A@,,) with

= .

Cmin

sin (pwt;**

(0 = @opt) f A@,,)

+ 0,t

’

(2.73)

tr* = $ (the value of t o at which MZ ( t o ) presents its first maximum). Using

52

Theoretical Approach

Eq. (2.72) and taking into account that 0,, = T , 7r/2,0,3~/2for p = 4n - 3,4n 2,4n - 1,4n ( n = 1,2, ...), respectively (cf. Table 2.1), one can rewrite Eq. (2.73) in the form A - ID+[ Cmin (0 = @ o p t ) Cmjn(0 = 0,t f A 0 ) , (2.74) -

~1-cl.il/.,2= -J

i.e., Cmin(0 = 0, f Ao,,,) does not depend on the additional excitation, as expected. As Cmin(0 = Oqt f A@,,) only depends on the functions Df and A , Eq. (2.74) really provides its general expression for any Melnikov function (2.37). Once more, in the Hamiltonian limit (D+ = 0), from Eq. (2.74) we have

Cmin(0 = OoPtf Ao,,,)

= Cmin(0 = 0,t)

= A,

in agreement with the first remark to the Suppression Theorems. Observe that for an arbitrary deviation A 0 (0 6 A 0 6 A@,,) from Oqt one achieves the respective value Cmin(0 = 0,t f A@) similarly to the above lower-threshold case:

Cmin(0 = 0,t) - A - ID+ I (2.75) cos ( A 0 ) cos (A@) Consider now the upper-threshold value of C , as a function of 0 , for the damped driven pendulum whose associated Melnikov function is given by Eq. (2.69) with D+ < 0. From Theorem I11 one has A (2.76) c, = c,,, (0 = 0opt ) = p2 Cmin(0 = 0, f A@) =

'

plots of M: ( t o ) , in Fig. 2.2.

Cmax

sin (wto

+ O,,),

and C,,,

sin (wto

+ 0,,

- A@,,)

I

I

0

I

are shown

to/ T

Figure 2.2: Functions M: ( t o ) and C,,,

sin (wto

+ 0 ) for two values of 0.

53

The generic Mehikov function: Deterministic case

It is clear that for 0 = O, f Ao,, homoclinic chaos suppression will only be = possible when the upper-threshold value of C (hereafter referred to as C,,,(0 0,t f A@,,)) is lower than C,, (0 = O,,). If 0$

[@opt - A e m a x , @ q t

+ A@max]

homoclinic chaos inhibition is not guaranteed for any choice of C. Analogously to Cmin (0 = 0,, f Ao,,,), C,, (0 = Oqt f A@,,) can be directly obtained from C,, (0 = O,,) suitably weighted by a phase factor that depends on A@,, (see Fig. 2.2):

c,,, (0 = o,,

f Ao,,)

=

Cmax (0 = 0,) cosec (put;;** 0,, f A@,,,)

+

(2.77)

’

2

where tG**= corresponds to the value of t o at which M z (to)has its first maximum. Proceeding similarly to the case of the lower threshold] Eq. (2.77) can be put into the form

(2.78) i.e., C,,, (0 = 0, f A@,,,) does not depend on the additional excitation] as ex, f A@,,) only depends on p , A , and IDf/, Eq. (2.78) pected. Since C,, (0=, @ gives its general expression for any Melnikov function (2.37). Also, one recovers

C,,

(0 = 0,

f A@,,,)

=

C,,

(0 = 0,t)

=A

in the Hamiltonian limiting case, in agreement with the first remark to the Suppression Theorems. It is also clear that for an arbitrary deviation A 0 (0 A 0 Ao,,,) from O, one straightforwardly obtains C,, (0 = 0,t f A@) similarly to the above upper-threshold case:

<

c,,, (0 = o,,

f A@) = c , (0 =

cos (A@)=

A cos ( A 0 ) P2

<

.

(2.79)

Remarks. First, when the allowed suppressory values of 0 are Oopt,i.e., those indicated in Tables 2.1 and 2.2, the allowed range ]C,i, (0= O,t), C,, (0= 0,t)] (and hence the corresponding range for the associated chaos-suppressing amplitude, cf. Eq. (2.59)) for inhibition of homoclinic chaos is the widest possible. We have therefore denoted such initial phase difference values as optimal values: 0,, (cf. the fourth remark to the Suppression Theorems).

54

Theoretical Approach

Second, there always exists a maximum-range interval [@,t

- A@,,,

@qt

+ A@max]

7

A@,, given by Eq. (2.72) with the corresponding D instead of D f , of permitted initial phase differences for homoclinic chaos inhibition. A plot of this interval as a function of ID+I / A is shown in Fig. 2.3, where the grey region between the two functions, @ , & arcsin (ID+l /A) corresponds to the permitted values of 0 for regularization.

OOPI+ n/2

0,;

n/2 1

ID'I / A

f arcsin (ID+I /A) vs ID+ I /A.

Figure 2.3: Functions O,

For each value of 0 belonging to this interval there exists a reduced interval (with regard to the limiting case where the only suitable values of 0 are of amplitudes of the chaos-inhibiting excitation which is (cf. Eqs. (2.59), (2.75), and (2.79)) qmin(0 =, @ ,

f A@) < q < q,,

qmin(0 = O,

* A@)

qmax(0 =

o,,

f A@)

(0 = @opt f A@) >

( ) ' ' :I

E

R cos (A@) P2

where R is given by Eq. (2.44) and 0 6 A@ < A@,,,. for the chaos-inhibiting amplitude is Aq (0 = @,t

f A@) =

Rsec (A@),

1-

(2.80)

,

Thus, the width of the range

+

cos (A@)

(2.81)

The generic Melnikov function: Deterministic caSe

55

i.e., for fixed D f , A and A 0 there always exists a maximum resonance order p, homoclinic chaos suppression which is

for

(2.82)

where the brackets indicate integer part, and where Eq. (2.72) was used. Note that Eq. (2.82) generalizes Eq. (2.62) for initial phase differences apart from @opt, as expected. A density plot of p,, versus A 0 and A@,, is shown in Fig. 2.4, where homoclinic chaos inhibition is not possible within black regions and where increasing values of pmaXgo from darker (pmm= 1) to lighter (pmax= 7) regions. In particular, for the threshold values 0 = Oqt fAO,,,, the interval (2.80) is now written (cf. Eq. (2.72))

Vmax

(0 = 0, f Ao,,,)

=

4 J a ,

P

(2.83)

and the respective width of the interval for the chaos-inhibiting amplitude is

(2.84) i.e., homoclinic chaos suppression is only possible for the first resonance ( p = 1) between the two forcings because of the assumption of homoclinic chaos (10'1 < A ) in the absence of chaos-suppressing excitations.

56

Theoretical Approach

1.5

1.25

1 0.75 0.5 0.25

0 -1.5 -1 -0.5 0 Figure 2.4: Density plot of p,,

0.5

1

vs A@ and A@,,

1.5

no

(cf. Eq. (2.82)).

Third, observe that one can put (2.85) (cf. Eq. (2.72)). Thus, one can use a linear approximation for Ao,,, suitable for chaotic motions arising away from the limiting case of tangency between the stable and unstable manifolds (IP/A << 1). It is worth mentioning that the last inequality is usually associated with the observation of steady chaos (strange chaotic attractor). Fourth, according to the results presented above, one can extend the Suppression Theorems to the general situation where 0 E [O, - A@,, 0, A@,,,], A@,, given by Eq. (2.72), for which the permitted inhibitory interval Avo must satisfy A~Q,ptfA@m,, A%,pt, (2.86)

+

c

, Arlo, and A ~ e , pbeing t the permitted intervals corresponding to with A~e,pt~Ae,,, the respective values of 0 (cf. Eqs. (2.84), (2.81), and (2.59), respectively). 2.5.2 Enhancement of chaos We have seen above that the mechanism for suppressing (homoclinic) chaos is the frustration of a homoclinic/heteroclinic bifurcation, which prevents the appearance

The generic Melnikov function: Deterministic case

57

of horseshoes in the dynamics. In the following we shall see that the enhancement of the initial chaos is achieved by moving the system from the homoclinic tangency condition even more than in the initial situation with no second periodic excitation. Thus, enhancement of chaos can mean increasing the duration of a chaotic transient, passing from transient to steady chaos, or increasing the leading Lyapunov exponent. Consider again that the family of systems modeled by Eq. (1.1) satisfies the requirements of Melnikov’s method. Similarly to the discussion of the suppression of chaos, we can assume any particular form of the Melnikov function (2.37) to discuss the enhancement of chaos. Therefore, consider, for instance, the following nonlinearly damped, biharmonically driven, two-well Duffing oscillator: X -2

+ Pz3 = -6X (X(n-l + FCOS( ~ t-)7Pz3cos (at+ 0 ),

(2.87)

where 7 ,R, and 0 are the normalized amplitude factor, frequency, and initial phase, respectively, of the chaos-controlling parametric excitation (0 < 77 << l),and p, 6, n, F, and w are the normalized parameters of the potential coefficient, damping coefficient, damping exponent, chaos-inducing amplitude, and chaos-inducing frequency, respectively (0 < 6, F << 1,/3 > 0, n = 0,1,2, ...). The application of Melnikov’s method to Eq. (2.87) involves calculating the Melnikov functions (cf. Eq. (2.30)),

(2.88)

where the positive (negative) sign refers to the right (left) homoclinic orbit of the underlying integrable two-well Duffing oscillator (6 = F = 7 = 0):

zo,i(t) = + E s e c h t ,

Xo,+(t) = i f i s e c h t t a n h t .

(2.89)

After substituting Eq. (2.89) into Eq. (2.88) and computing the resulting integrals with the aid of standard integral tables, Eq. (2.88) can be written as

M*(to) = DfAsin(wt0) - C s i n ( R t o + @ ) ,

(2.90)

58

Theoretical Approach

with

(2.91) where B(m,n ) is the Euler beta function. Note that A and C are positive functions, while D is a negative function, over the complete range of their parameters. The Melnikov function M+(to) (cf. Eq. (2.90)) will be used to illustrate the theoretical approach to the enhancement of chaos by weak periodic excitations. In particular, it will be demonstrated that, in general, a second harmonic excitation can reliably play an enhancer or inhibitor role solely from adjusting its initial phase. Indeed, consider that, in the absence of any second parametric excitation (C = 0), the associated Melnikov function M$ ( t o ) = - (DJ Asin (wto) changes sign at some to, i.e., (Dl A. Figure 2.5 shows a plot of A4$ ( t o ) .

+

<

--. -

I 0

to/ T

Figure 2.5: Optimal enhancer and suppressor effects on MZ ( t o ) (see the text). If one now lets the second excitation act on the system such that C Q A - ID(,this relationship represents a sufficient condition for M+(to) to change sign at some to. Thus, a necessary condition for M+(to) to always have the same sign ( M + ( t o ) < 0) is C > A - ID( = Cmin.It was demonstrated in Sec. 2.5.1 that a sufficient condition for C > Cminto also be a sufficient condition for inhibiting chaos is R = pw (subharmonic resonance condition), C < C,, = A/p2,p an integer, and that M: ( t o )

59

The generic Melnikov function: Deterministic case

and -Cmin,max sin (Rto + 0 ) are in opposition (see Fig. 2.5). It was seen that this last condition yields the optimal suppressory values of the initial phase, E OTt, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation (cf. the fourth remark to the Suppression Theorems). Now one sees that imposing M$(to) to be in phase with -Cmin,maxsin(at0 0 )is a suficient condition for M+(to) to change sign at some to (see Fig. 2.6). This condition provides the optimal enhancer values of the initial phase, 0;!, in the sense that M+(to)presents its highest maximum at 0:;:; i.e., one obtains the maximum gap from the homoclinic tangency condition.

0z

+

1 1

I 0

t,/T

Figure 2.6: Optimal enhancer and suppressor effects on M,j+( t o ) (see the text). Remarks. First, for a given homoclinic orbit forming (part of) a separatrix, one has in general (i.e., for any Melnikov function M&, ( T O ) , cf. Eq. (2.37)) that

10 7

-

1

(2.92)

0:;: = T ,

for each resonance order. Second, it is straightforward to see that for C = Cminthere always exists a maximum-range interval

[O;; - A@ ::

(C = Cmin),0:;;

+ A0:k

(C = Cmin)]

(2.93)

of permitted initial phases for enhancement of chaos in the sense that, for values of 0 belonging to that interval, the maxima of M+(to)axe higher than those of M$(to). Similarly, for C = C,, there always exists a diflerent maximum-range interval

p ;;

-

n :o:

(C = Cma),0$

+ n:o:

(C = Cmax)]

(2.94)

60

Theoretical Approach

of allowed initial phases for enhancement of chaos, such as

n ;o:

(C = C,,,)

3a @ ;

(C = Cmin),

(2.95)

which is a consequence of the dissipation. It must be emphasized that the definition of 0;; is general, i.e., it refers to any resonance and any Melnikov function M,$, ( T O ) (cf. Eq. (2.37)). Third, for general separatrices, i.e., those formed by several homoclinic and (or) heteroclinic loops, the above scenario of control of chaos holds for each homoclinic (heteroclinic) orbit. However, it is common to find that the different homoclinic (heteroclinic) orbits of a given separatrix yield distinct 0;; values. This is a consequence of the survival of the symmetries existing in the absence of the second excitation (cf. the fourth remark to the Suppression Theorems). Thus, the actual , be the optimal scenario is usually more complicated. For instance, let O T , T 0%,[ values associated with the right and left homoclinic orbits, respectively, of a typical separatrix with a “figure-of-eight’’ loop, as in the two-well Duffing oscillator (2.87). It is straightforward to see that the best chance for enhancing chaos should now be (0:;&.- 0;$) /2 mod (27r). at 0;; 2.5.3 Case of non-subharmonic resonances The case of subharmonic resonance between the chaos-inducing and chaos-controlling frequencies has been discussed in previous sections. However, a number of theoretical, experimental, and numerical studies show that chaos can be reliably controlled by other non-subharmonic resonances. In this present subsection, a generic approach is discussed concerning the ultrasubharmonic case: R = p w / q , q > 1 ( p # q ) , p , q positive integers. Let us assume once more that the family of systems described by Eq. (1.1) satisfies the requirements of Melnikov’s method. Thus, one is concerned again with the generic Melnikov function given by Eq. (2.37). As in the case of a subharmonic resonance, one can consider any particular form of the Melnikov function to discuss the theoretical approach. Therefore, assume, for example, the nonlinearly damped, biharmonically driven, twewell Duffing oscillator given by Eq. (2.87). The Melnikov function M+ ( t o ) (cf. Eq. (2.90)) will be used to illustrate the approach to the case of an ultrasubharmonic resonance. Let us suppose that, in the absence of any chaos-suppressingexcitation (C = 0), the associated Melnikov function MZ(t0) = D Asin(wt0) changes sign at some t o , i.e., ID1 A . If we now let the chaos-suppressing excitation act on the D f i g oscillator such that C 6 A - ID[,this relationship represents a sufficient condition for M+ (to)to change sign at some to. Thus, a necessary condition for M + ( t o ) to always have the same sign is (2.96) C > A - ID1 G Cmin. N

<

+

It is obvious that, for Eq. (2.96) to also be a sufficient condition for M+ ( t o ) to be negative for all t o , one must have A - C 3 A sin (wto) - C sin (Stto

+ 0’) .

(2.97)

61

The generic Melnikov function: Deterministic case

Now we look for the values of w , 0, and O+ permitting Eq. (2.97) to be fulfilled for all to. We shall first need two preliminary lemmas: Lemma VI. Let R/w = p / q for some positive integers p and q. Then there exists tg satisfying sin(wt8) = sin (at; O+) = 1 if and only if = 4mf1-2ef 4n+l ?r for some integers m and n. Proof. If sin(Rt; O+) = 1 then there exists some integer m such that Rt; + O+ = (4m + 1)7r/2. Also, if sin(wt;) = 1 then there exists some integer n such that wtg = (471+ 1)7r/2. Solving for 1; in the two relationships, one readily obtains

+

+

p- - 4m + 1 - 20+/7r 4n

4

(2.98)

+1

Lemma VII. Let R/w be irrational. Then there is some tg* such that

A sin ( w t i * ) - C sin (at:) > A - C. Proof. Let O+ = ;(1 -

A sin (wt:)

-

2 ) if w > R. Then taking t:* =

C sin (at:*+ 0.)

one obtains

=A -

since R and w are incommensurable. Similarly, one takes O+ = f (1 - E) if w < R. Then taking tr = $ one obtains Let now O+ # (1 -

E).

Asin(wt;)

-

Csin (at:

+ 0')

=A

-

(5," ) >

Csin - + 0'

A - C,

since we have Of + # ;in this case. From Lemma VII, a resonance condition is required, R = p w / q , for Eq. (2.97) to be fulfilled for all to. In such a situation, Lemma VI provides a condition (Eq. (2.98)) for Eq. (2.97) to be satisfied for an infinity of t o values. Finally, for q = 1, (2.99) is a sufficient condition for Eq. (2.97) to be fulfilled for all t o (cf. Lemma 111). Now, although condition (2.98) with q > 1 ( p # q ) is a necessary condition but not a suficient one for Eq. (2.97) to be satisfied for all to (i.e., M+(to) may still present simple zeros), it provides the situation in which M + ( t o ) is as near as possible to the tangency condition for C = Cmin.This means that, although now chaotic transients cannot be inhibited (i.e., homoclinic bifurcations cannot be frustrated), one expects to have a fair chance of suppressing steady chaos. Note that an upper threshold for the amplitude of the chaos-suppressing excitation (i.e., Cmax) is obtained by imposing the condition that it m a y not enhance the initial chaos, i.e.,

C

< A + JDJE Cmax,

(2.100)

62

Theoretical Approach

which is a necessary condition for M+(to) to always have the same sign. For Eq. (2.100) to also be a sufficient condition for M+(to) to be negative for all t o , one must have Asin(wt0) - Csin(Rt0 O+) < C - A. (2.101)

+

Now we look for the values of w, R, and O+ permitting Eq. (2.101) to be fulfilled for all to. One shall first need two additional lemmas. Lemma VIII. Let R/w = p/q for some positive integers p and q. Then a t; exists such that sin(wt6) = sin(RtG Of) = -1 if and only if = 4m+3-2e+ 4n+3 for some positive integers m and n. Proof. If sin(Rt8 O+) = -1 then there exists some integer m such that s1t; Of = (4m 3) x/2. Also, if sin(wt6) = -1 then there exists some integer n such that wt; = (4n 3) 7r/2. Solving for t; in the two relationships, one obtains

+

+

+

+

+

p

- 4m+3--20+/7r

4

4n

(2.102)

+3

Lemma IX. Let R/w be irrational. Then there is some A sin (wt;*) - C sin(Rt;*

tr such that

+ Of) 3 C - A.

9 (1 - e) . Then taking t;*= one obtains A sin (wtr) - C sin (s1ti*+ 0') = C A 2 C - A. Let now Of # 9 (1 - e). Then taking tG* = (9 0.) one obtains Asin(wtr) - Csin(Ot(;*+ Of) = C + Asin [' (2- O')] > C - A, 0 2 since we have X (2 0') # 9 in this case. Proof. Let O+ =

-

-

-

From Lemma IX, a resonance condition is required, s1 = pw/q, for Eq. (2.101) to be fulfilled for all to. In such a situation, Lemma VIII provides a condition for Eq. (2.101) to be satisfied for an infinity of t o values. Observe that, although condition (2.102) with q > l ( p # q) is not a sufficient condition for Eq. (2.101) to be satisfied for all to, it makes (in analogy with Eq. (2.98) for the lower threshold C,in) A4+(to) as near as possible to the tangency condition for C = C,, and thus one again expects to have the best chance for eliminating chaotic attractors. By definition, OLin,max will denote the suitable (for each resonance) initial phases given by Eqs. (2.98) and (2.102), respectively. Remarks. First, for the Melnikov function associated with the left homoclinic orbit, M-(to) = D - A sin(wto)- C sin(Rt0 0 - ) ,one straightforwardly obtains

+

- Omin,,,

PT 4

= - (mod 27r)

(2.103)

63

The generic Melnikov function: Deterministic case

(cf. Eqs. (2.39), (2.98), and (2.102)), which is a consequence of the “maximum survival” of the symmetries under chaos-suppressing excitations (cf. fourth remark to the Suppression Theorems), i.e., while the two-well D a n g oscillator (2.87) with 7 = 0 has a symmetry with respect to the transformation

x t

--f

-2,

+

t+;,

-7T

(2.104)

the complete oscillator (2.87) (7 # 0) has a new symmetry with respect to the extended transformation

x

3

-x,

0

+

P-7T O+-((modZx) Q

(2.105)

Second, for each resonance note that (2.106) (cf. Eqs. (2.98), (2.102), and (2.103)), which is a general relationship (i.e., valid for any Melnikov function (2.37)) given the linear character of the relationships (2.39). In general, we have therefore two dierent sets of suitable (in the aforementioned sense) initial phases, associated with the upper and lower amplitude thresholds, respectively. For the two-well Duffing oscillator (2.87), these amplitude thresholds are qmax E (1

+

qmin

-

E

R E

(1

7) R,

q)

R,

6 m F w sinh (7rR/2)

(a4+ 4R2)cosh (xw/2)’

(2.107)

respectively (cf. Eq. (2.91)). Note that, in the limiting Hamiltonian case, Eq. (2.107) reduces to (2.108) v = Vmin = ~ m a x= R, i.e., A = C (cf. Eq. (2.90)), as expected.

Rational approach to the incommensurability case One can now use the above results to approach the case of incommensurate chaossuppressing excitations by means of a series of ever better rational approximations,

64

Theoretical Approach

which are the successive convergents of the infinite continued fraction associated with the irrational ratio R l w . This procedure has been much employed in characterizing strange non-chaotic attractors in quasiperiodically forced systems as well as in studying phaselocking phenomena in both Hamiltonian and dissipative systems. To illustrate the method one intentionally chooses the golden section n / w = @ = (& - 1) 12, since it is the irrational number which is the worst approximated by rational numbers (in the form of continued fractions). As is well-known, the golden section can be approximated by the sequence of rational numbers ( R l w ) , = Fi-l/F, where Fi = 1,1,2,3,5,... , are the Fibonacci numbers such that Jim 2-03

(5)

&-1

=2.

(2.109)

i

For each (Rlw),one replaces each quasiperiodically excited system (2.110)

(cf. Eq. ( l . l ) ) , where har(t) means indistinctly sin(t) or cos(t), by the respective periodically excited system (2.111)

giving a sequence of periodically excited systems whose associated frequencies satisfy an ultrasubharmonic resonance condition. Then one can apply the above theoretical predictions to each system (2.111) for increasing values of i. For the sake of clarity, the following nonlinearly damped, biharmonically driven, Helmholtz oscillator will be used to illustrate the analysis: (2.112) x - x + px2 = -6x liln-’ + F sin ( w t ) - qpx2sin (Rt + 0 ), where q,R, and 0 are the normalized amplitude factor, frequency, and initial phase, respectively, of the chaos-controlling parametric excitation (0 < q << l),and p, 6, n,F , and w are the normalized parameters of the potential coefficient, damping coefficient, damping exponent, chaos-inducing amplitude, and chaos-inducing frequency, respectively (0 < 6, F << 1,,B > 0, n = 0, 1,2,...). It is worth noting that the damped forced Helmholtz oscillator provides a simple model for a universal chaotic escape situation, i.e., when, before escape, chaotic transients of unpredictable duration are usually observed. The application of Melnikov’s method t o Eq. (2.112) involves calculating the Melnikov function (cf. Eq. (2.30))

+ [:io(t) -qP

+ t o ) ]dt io(t)x;(t)sin [n(t + t o ) + 01d t ,

1, m

sin [w (t

(2.113)

65

The generic Melnikov function: Deterministic case

where z o ( t ) = -sech

2($),

2P

i o (t)

=

-3 2P sech2

(i)

tanh

(s)

,

(2.114)

is the parametric representation of the homoclinic orbit of the underlying Helmholtz oscillator (6 = F = q = 0). After substituting Eq. (2.114) into Eq. (2.113) and computing the resulting integrals with the help of standard integral tables, Eq. (2.113) can be written as (2.115)

with

c

F

+

37v yf1(f12 2 1) (02+ 4) csch ( T O ) , 5P

(2.116)

where B(m,n) is the Euler beta function. Let us suppose that, in the absence of any chaos-inhibiting excitation ( q = 0, and hence C = O), the associated Melnikov function MO( t o ) = D - A cos(wt0) changes sign at some to, i.e., JDJ 6 A. If we now let the chaos-inhibiting excitation act on the system such that C 6 A - IDI, this relationship represents a sufficient condition for M(t0) to change sign at some to. Thus, a necessary condition for M(t0) to always have the same sign is C > A - ID1 F Cmin. (2.117)

It is straightforward to see that, for this relationship to also be a sufficient condition for &'(to) to be negative for all t o , we must have (2.118)

For the subharmonic case (0 = pw) one straightforwardly obtains (in analogy with the Suppression Theorems; see Table 2.1) the following theorem: Theorem IV. Let O = pw, p an integer, such that

p=

(2m + 1 - O / T ) 2n+1

(2.119)

66

Theoretical Approach

is satisfied for some integers m and n. Then M(t0) always has the same sign, i.e., M(t0) < 0, if and only if the following condition is fulfilled: Vmin

<

v

qmin

5

(1 -

Vmax,

y)

R, (2.120)

with

RE

10FPw2

R2

sinh (TO)

(R2 + 1) (R2 + 4) sinh ( T W )

(2.121)

'

For the ultrasubharmonic case ( q > 1 , p # q ) there exist distinct conditions (that given by Eq. (2.119) with p/q instead of p among them) which are sufficient for Eq. (2.118) to be satisfied for an infinity of t o values but not sufficient for Eq. (2.118) to be satisfied for all t o (i.e., M(t0) may still present simple zeros). Following the aforementioned criterion, one chooses among them that condition making M ( t o ) as neax as possible to the tangency condition for C = C m i n , in the sense that (at least) one of the local maxima of M (to)is the lowest:

(2.122) with m, n non-negative integers. This means again that, although now chaotic transients cannot be completely eliminated (i.e., homoclinic bifurcations cannot be s u p pressed), one would expect to have a fair chance of reducing chaotic escape.

0.4

MI 0.2 -0.2

10 7r"

Figure 2.7: Melnikov function M'

( T O ,0 ) for

p / q = 3/5 (see the text).

67

The generic Melnikov function: Deterministic case

Figure 2.7 depicts, as an example, the normalized Melnikov function M' ( T O , 0) = M ( t o ) / A versus T O = wto and 0 in the range [-0.2,0.4] for C = Cmin,CIA = 0.8 and the resonance p / q = 3/5. Observe that, in each period of M', its local maxima (with respect to T O ) are the lowest (with respect to 0)at the suitable initial phases, Osuit&Le, given by Eq. (2.122), i.e., { T , 97r/5,3~/5,77~/5,~/5}. Note that the variables M', T O , and 0 are dimensionless. Remarks. First, = 7r/q = 0) is the lowest suitable initial phase for all resonances p / q with q odd (even). Second, for a given resonance O/w = p / q , one has q suitable values of the initial phase which are uniformly distributed in the interval [0,27r],AOsuit&le= 27r/q being the gap between any two adjacent suitable initial phases. Observe that T is a suitable initial phase for all the resonances p / q . It should be emphasized that this remarkable property of OsUztable = 7r does not hold for any suitable initial phase of the two-well Duffing oscillator (cf. Eq. (2.87)). Note that for such a Duffing oscillator, an upper threshold for the amplitude is also deduced by imposing the condition that the chaos-suppressingexcitation may not enhance the initial chaos (cf. Eq. (2.100)). This upper threshold is pertinent since all the solutions of the two-well Duffing oscillator are bounded. In particular, one can observe chaos around the two wells or chaos confined within one of the wells depending on the values of the parameters. However, for the Helmholtz oscillator (2.112) (whose associated potential has a single well), one could expect that any value of the amplitude of the chaos-inhibiting excitation higher than its corresponding lower threshold would have an enhancing (or at least not suppressory) effect on the initial chaotic escape situation. Third, the lowest amplitude threshold, vmin,corresponding to Cmin is given by Eqs. (2.120) and (2.121) with R = p / q . Now one can discuss the possibility of reducing chaotic escape by incommensurate parametric excitations (i.e., R/w irrational) from the above results for ultrasubharmonic resonances. In particular, for the golden section one replaces the quasiperiodically excited Helmholtz oscillator (2.112) (with R = @w,@ = (& - 1) /2) by the periodically excited oscillator

x - IC + PIC'

= -62

121n-*+ F sin ( w t ) - q,Bz2sin ( % -wt l

+ 0) ,

(2.123)

giving a sequence of biharmonically excited Helmholtz oscillators whose associated driving frequencies satisfy an ultrasubharmonic resonance condition. Therefore, we can apply the predictions from Theorem IV to each oscillator (2.123) for increasing values of i. Thus, the respective values of the suitable amplitude and initial phase are (cf. Eqs. (2.120) and (2.122), respectively) (2.124)

68

Theoretical Approach

&=

lOFP ( F i ~ i / F i[w2 ) ~ (Fi-llFi)’

+

sinh ( 7 r ~ F , - ~ / F i ) , (2.125) 11 [w2 ( F $ - ~ / F ~41) ~ sinh (nu)

+

(2.126) where m, n are non-negative integers. Remarks. First, the successive convergents of @ (1/1,1/2,2/3,3/5,5/8,8/13,13/21, ...) exhibit one even denominator q for every two odd, and thuls whether 0 is or is not one of the associated suitable initial phases depends on the parity of q. Also 7r is a suitable initial phase for all the convergents. Second, for fixed w , the ratio qmin,i/qmin,oo converges very quickly to 1 as i -+ co. This means that the values of qmi+ corresponding to early convergents (3/5,5/8,8/13, ...) are really very close to the limiting value qmin,ooassociated with @. Third, in contrast to the mentioned asymptotic behaviour of the amplitudes qmin,i,the number of suitable initial phases Qsuitable,i tends to infinity as i -+ 00,and for each rational approximation to @, given by a certain convergent, the respective values of eszlitable,i are uniformly distributed in the interval [0,27r].Since two successive convergents Fi-l/Fi, Fi/Fi+l differ only in ever higher decimal places as i -+ 00,one should expect that the inhibitory effective (observed) values of the initial phase, Q e f f e c t i v e , associated with a certain convergent, would be related not only to its respective values @suitable,i but also to the suitable initial phases corresponding to its preceding convergents. Specifically, the effective values Q e f f e c t i v e should correspond to the points around which the suitable initial phases associated with the chosen convergent and its precedents concentrate. Moreover, one expects this prediction to gain in accuracy as i -+ 00. 2.5.4 The special case of the main resonance We have seen in previous sections that the theoretical approach based on the analysis of the simple zeros of generic Melnikov functions (2.37) yields analytical estimates for the intervals of initial phase difference 0 between the chaos-inducing and the chaossuppressing excitation, on the one hand, and for the intervals of the chaos-suppressing amplitude q, on the other, for which homoclinic/heteroclinic bifurcations can be inhibited. Regarding the predicted ranges of suppressory amplitudes, a weakness of the aforementioned theoretical approach is that the upper amplitude threshold qmax typically underestimates (unlike the lower amplitude threshold qmin)the corresponding numerically observed upper threshold (cf. second remark to the Suppression Theorems). Clearly, one would also wish to have analytical functions q = q (0)for the regularization (in the sense of frustration of homoclinic/heteroclinic bifurcations) boundaries in the 0 - q parameter plane, instead of separate sets of estimates for the (ranges of) suitable values of q and 0, respectively. Another point is that the Suppression Theorems deal indistinctly with parametric and external excitations (as

69

The generic Melnikov function: Deterministic case

chaos-inducing and -suppressing), although a general discussion of the relative effectiveness of any two of such chaos-suppressing excitations would be especially relevant in considering technological applications. In this section, generic functions 77 = ~ ( 0providing ) the regularization (in the above-mentioned sense) boundaries in the 0 - 77 parameter plane are derived on the basis of Melnikov’s method for the main resonance case R = w. Remarkably, such boundary functions yield more accurate upper amplitude thresholds than those predicted from the Suppression Theorems, and permit one to reliably determine the relative suppressory effectiveness (in the sense of the extension in the 0-77 parameter plane) of generic parametric and external excitations. The pertinency of the theoretical findings t o the elimination of chaotic attractors is illustrated with the example of a two-well Duffing oscillator. As previously noted, one can study the simple zeros of generic Melnikov functions M,&, ( T O ) (cf. Eq. (2.37)) by choosing quite freely the trigonometric functions in Eq. (2.37). Therefore, to illustrate the general procedure one can consider, for instance, the damped driven two-well Duffing oscillator, subjected to a parametric excitation of the cubic term, given by Eq. (2.87) with R = w and n = 1 (i.e., a linear damping). The corresponding Melnikov function associated with the left homoclinic orbit is M-(to) = -D - Asin(wt0) - Csin(wt0 + 0 ) (2.127) (cf. Eq.(2.90)) with D , A , C given by Eq. (2.91) with R = w,n = 1. The Melnikov function (2.127) will be used to illustrate the generic method. Note again that the Melnikov functions (2.37) and (2.127) are connected by linear relationships for each specific system (1.1) (cf. Eq. (2.39)). Let us suppose that, in the absence of any chaos-suppressing excitation (C = 0), the associated Melnikov function M;(to) = -D - Asin(wt0) changes sign at some t o , i.e., D < A. Clearly, Eq. (2.127) can be recast into the form

M- (to)

+ C cos 0 )sin (wto) C sin 0 cos (wto) < -D + [(A + c cos 0)’ + ~2 sin2 01’” .

=

-D - (A

-

(2.128)

If we now let the chaos-suppressing excitation act on the system such that

C2+ 2ACcosO + A 2 - D2 < 0,

(2.129)

this relationship represents a sufficient condition for M-(to) to be negative (or null) for all to. The equals sign in Eq. (2.129) provides the boundary of the region in the 0 - 77 plane where homoclinic chaos is suppressed: - cos

0 f Jcos2 0 - (1 - D2/A2)] R,

(2.130)

with (2.131)

70

Theoretical Approach

+

(cf. Eqs. (2.91) with n = 1,f 2 = w and (2.129)), and where the sign (-) before the square root corresponds to the upper (lower) branch of the boundary. Proceeding similarly, one finds that the boundary functions associated with the respective Melnikov functions (2.37) are

q,',,,,,,

= q:n,,in

*

= [Fcos 9 f dcos2 9 - (1

i

qcos,sin= -q,in,cos

E

[

-

D2/A2)]R,

sin 9 f dsin29 - (1 - D 2 / A 2 ) ]R,

(2.132) (2.133)

for C > 0 (cf. Eq. (2.37)), and f

qcos,cos = qc,,,sin i = -q:n,cos

5

+ cos 9 f dcos2 9 - (1

= [.sin

-

D 2 / A 2 ) ]R,

9 f dsin' 9 - (1 - D 2 / A 2 ) ]R,

(2.134) (2.135)

for C < 0 (cf. Eq. (2.37)). Note that the two signs before the square root apply to each of the sign superscripts of q&,, which, in its turn, is independent of the sign of

D. Remarks. First, the boundary functions (2.132)-(2.135)represent loops in the 9-7 plane which are symmetric with respect to the corresponding optimal suppressory value, i.e., that predicted from the Suppression Theorems (cf. Tables 2.1 and 2.2).

YwptAYnu y ' , ,

Ywt+AYmax

y

Figure 2.8: Properties of a generic boundary function (2.132)-(2.135).

71

The generic Melnikov function: Deterministic case

Second, the lower branch of each boundary function (2.132)-(2.135), qlozuer= qlouer(Q), exhibits a minimum at the corresponding Qt, value, which is q = qmin= (1 - ID1 / A )R, i.e., the lower amplitude threshold predicted by the Suppression Theorems (cf. Eq. (2.43)). Third, the maximum range of suppressory initial phase differences occurs when the upper and lower branches of the boundary functions coincide, i.e., when the square root cancels out (cf. Eqs. (2.132)-(2.135)). If AQmaxdenotes the maximum permitted deviation from Qopt, then, by substituting Q = Qt, ?C AQmaxinto the square root and taking into account the respective value Qqt (cf. Tables 2.1 and 2.2) for each Melnikov function M&, (cf. Eq. (2.37)), one readily obtains AQmax= arcsin,(ID1 / A ) for all the cases, i.e., one recovers the expression derived previously (cf. Eq. (2.72)). Fourth, the upper branch of each boundary function (2.132)-(2.135), qupper= qupper (Q), exhibits a maximum at the respective QTt value, which is (2.136) with qmm being the upper amplitude threshold predicted by the Suppression Theorems (cf. Eq. (2.54)). Computer simulations indicate that qkaxis clearly closer than qm, to the respective numerically obtained threshold. Fifth, the boundary functions (2.132)-(2.135) permit one to reliably compare the relative effectiveness of any two chaos-suppressing excitations, in particular, external and parametric excitations. Indeed, since the functions A, D are fixed for a given initial chaotic state (C = O), and the area enclosed by any boundary function is given by

1

*opt+A*max

AR

E

*opt

(qupper

- qlower) d~ = 4

-A*,,,

(T) R,

(2.137)

the relative effectiveness of any two chaos-suppressing excitations, denoted as I , 11, can be quantified by ARr (2.138) A R ~ &I' ~ Observe that one finds AR + 0 as D -+ 0, which corresponds to the limiting Hamiltonian case with no constant drivings, as expected. Figure 2.8 summarizes the aforementioned properties of a generic boundary function (2.132)-(2.135) encircling the region where homoclinic/heteroclinic bifurcations are frustrated in the suppressory Q - q parameter plane. As an illustrative example of the area criterion (2.138), consider the damped driven two-well Duffing oscillator, subjected now to an additional forcing term instead of a parametric excitation of the cubic term (cf. Eq. (2.87)):

5

z - 2 + p23 = -bi + F cos ( w t ) + qF cos (wt + 0 ).

(2.139)

The Melnikov function associated with the left homoclinic orbit can be straightforwardly calculated to be

M I- ( t o ) = -D

-

A sin(wt0) - C' sin (wto

+ 0 ),

(2.140)

72

Theoretical Approach

with (2.141) and D , A given by Eq. (2.91) with n = l , R = w. The corresponding boundary function is given by Eq. ( 2.130) (as in the parametric excitation case) now with

R

=RAF

(2.142)

1.

Thus, the area criterion (2.138) yields ~ 6~ f l F tanh A R ~4 w~+ w 3

A -R

(y ),

(2.143)

whose value is one-to-one determined for each initial chaotic state. Generally, this means that the choice of the most suitable chaos-suppressing excitation can strongly depend on the specific initial chaotic stated to be tamed. Next, one can compare the theoretical results obtained from Melnikov's method with Lyapunov exponent calculations of the two-well Duffing oscillator subjected to the two aforementioned types of chaos-suppressing excitations (cf. Eqs. (2.87) and (2.139)). As is well known, one cannot expect too good a quantitative agreement between the two kinds of results because Melnikov's method is a perturbation method generally related to transient chaos, while the Lyapunov exponents provide information concerning solely steady responses. Lyapunov exponents were computed using standard algorithms and the integration was typically up to 2000 drive cycles for the fked parameters p = 4,6 = 0.154, F = 0 . 0 9 5 , ~= 1.1. In the absence of the chaos-suppressing excitations (11 = 0), the Duffing oscillator presents a strange chaotic attractor with a leading Lyapunov exponent A+ (7 = 0) = 0.127 bits/s. The leading Lyapunov exponent was calculated and plotted for each point on a 100 x 100 grid, with (normalized) initial phase 0 and amplitude 11 along the horizontal and vertical axes, respectively, for both types of chaos-suppressing excitations. The results for the parametric and external chaos-suppressing excitations (cf. Eqs. (2.87) with R = w,n = 1 and (2.139), respectively) are shown in Figs. 2.9 and 2.10, respectively. The diagrams in these figures were constructed by only plotting points on the grid when the respective leading Lyapunov exponent was larger than lop3 (grey square) or than A f ( q = 0) (black square), and with solid black contours denoting the respective theoretical boundary functions (cf. Eqs. (2.132) and (2.134)). One sees that the complete regularization (A' (7 # 0) 6 0) mainly appears inside maximal islands which symmetrically contain the respective theoretically predicted areas where even chaotic transients are eliminated, the area of the former islands being notably larger than the theoretical prediction, as expected. The size of the main regularization islands is notably larger for the additional forcing case than for the parametric excitation case, which is in agreement with the area criterion: A R p E I A ~ A NF 0.264097 (cf. Eq.

73

The generic Melnikov function: Deterministic case

(2.143)). The structure of the secondary and minor islands of regularization is clearly more complex for the parametric than for the external chaos-suppressing excitation, as can be appreciated by comparison of Figs. 2.9 and 2.10. Another difference is that the whole diagram of Fig. 2.9 is periodic along the @-axis,with fundamental period equal to IT (note that there exist two optimal suppressory values eqt,1= 0, Oqt,2 = IT, which are associated with the right and left homoclinic orbits, respectively (cf. Table 2.1)), while the whole diagram corresponding to the external excitation (cf. Fig. 2.10) is symmetric with respect to the (unique) optimal suppressory value OOpt= IT (cf. Table 2.1). This is a consequence of the survival of the symmetries existing in the absence of chaos-suppressing excitations (cf. Chap. 3). It is worth mentioning that the criterion based on the area in the suppressory amplitude-initial phase parameter plane, where frustration of homoclinic/heteroclinic bifurcations is guaranteed, is a valuable criterion in choosing the optimal chaos-suppressingexcitation since this choice exhibits sensitivity to the particular initial chaotic state to be tamed. This sensitivity to the initial chaotic state is closely related to the sensitivity to initial conditions. For real-world systems there is always intrinsic noise so that the initial conditions are never precisely the same. If the parameters are such that one is in a chaotic state, one does not know what chaotic solution will emerge.

0

1

o/z Figure 2.9: The controllable region of the twewell Duf€ing oscillator (2.87).

74

Theoretical Approach

0

1

2

0 / 7 r Figure 2.10: The controllable region of the twewell DufFing oscillator (2.139). The theoretical approach discussed in the present chapter has focused on the case of a single chaos-inducing excitation together with a single chaos-controlling excitation. A desirable generalization of the results would be to consider the action of several harmonic chaos-inducing excitations (as is indeed the general case in real-world chaotic systems) together with the action of multiple harmonic chaoscontrolling excitations, which could extend the experimental applicability, flexibility, and effectiveness of the control technique. The rest of the chapter is devoted to presenting an extension of the previous approach to the case of multiple harmonic chaos-inducing and chaos-controlling excitations for the main case of a single common driving frequency. Let us consider the important family of dissipative non-autonomous systems, described by the differential equations (2.144)

where U ( x ) is a general potential, Ef, g:h(x, k ) F f h ( t )is a general multiple chaosinducing excitation, -d(z, k ) is a generic friction force, and C,”=, g y ( x ,k ) F y ( t ) is an as yet undetermined suitable multiple chaos-controlling excitation, with F f h ( t ) , F J ! ( t )being harmonic functions of common frequency w and (without loss of generality) initial phases 0 (i = 1, ...,N ) ,qj ( j = 1,..., M ) . Once more, it is assumed

75

The generic Melnikov function: Deterministic case

that the complete system (2.144) satisfies the Melnikov’s method requirements. The application of Melnikov’s method to Eq. (2.144) yields the Melnikov function Mh,,h:(to) = D

N

M

z= 1

3=1

+ x A z h u r z(wto) +- x B ~ h u r : ( w t o+ G 3 ) ,

(2.145)

where the notation bar(.) means indistinctly sin (.) or cos (.), and D , A,, B3 arc diffcrent functions of the corresponding parameters for each particular system. Specifically, D contains the effect of the constant driving and dissipative forces. In the absence of any constant driving, D < 0, while one has the three cases D 5 0 when (positive) constant driving acts on the system besides the dissipative forces. Also, the non-null functions B3 and A, contain the effect of the respective chaos-controlling excitations and chaos-inducing excitations, respectively. It is straightforward to see that the Melnikov function (2.145) can be put into the form Mh,h‘(TO) = D

+ A h U T ( @ T o )+ BhUr’

+ Q),

(@TO

(2.146)

with (2.147)

(2.148)

(2.149)

(2.150) (2.151) (2.152) where ( z j ) indicates that the summation is over all pairs of (respective) excitations, (2.153) and 8i is defined by the condition A i h ~ r (i d o )

[Ailh ~ (wto r + 6’i) ,i = 1, ...,N .

(2.154)

Specifically, 0; = {0,7r/2,7r, 3r/2} depending on the functions huri, har, sign (Ai). Since one assumes that the chaos-inducing excitations are known, the functions Ai, Bi

76

Theoretical Approach

are o n e b o n e determined, and hence so are A, 8 from Eqs. (2.147) and (2.149), respectively. Now the previously discussed approach applies directly to the equivalent Melnikov function (2.146), and thus an equivalent control theorem can be deduced for each particular system (2.144). In particular, these theorems provide analytical estimates for the intervals of initial phase Q (and hence of [, cf. Eq. (2.151)) and for the ranges of the chaos-controlling function B (which are closely correlated, cf. Eq. (2.81)) for which homoclinic/heteroclinic chaos can be suppressed/enhanced. Thus, in view of Eqs. (2.148), (2.150), this implies that we have a wider choice of suitable chaos-controlling functions Bj (and hence chaos-controlling amplitudes) and suitable initial phases $j (corresponding to a particular chaos-controlling excitation gy(z,Z)F,CO(t))than for the case when it is the only chaos-controlling excitation -provided that the complete choice { B , @, Bj, $ j ; j = 1, ..., M } satisfies Eqs. (2.147)-(2.152) and the predictions of the equivalent control theorem. To provide an illustrative instance, the above multiple-control scenario will be discussed in the following in the simplest case: one chaos-inducing excitation and two chaos-controlling excitations. To this end, consider, for example, the two-well Duffig oscillator 5--2+p53

= -SZ

~~l"-l+ycos (wt)-pq1z3 cos (wt

+ 7~~~)+y77~ cos (wt + $,I,

(2.155)

whose associated Melnikov function is

M ( t o )= -D f Asin (do) - B1sin (wto + $1)

+ B2 sin (wto + q2),

(2.156)

where the positive (negative) sign refers to the right (left) homoclinic orbit of the underlying integrable two-well Duffing oscillator (cf. Eq. (2.89)),

+

D = - S ( i ) ' "+l)I2 B (n + T 2 , n~ 1 ) )

A= B1

B2 =

(a) (a)

112

xyw sech

7rq1 (4w2

(2.158)

csch

(2.159)

xq2yw sech

(2.160)

6P

112

+

(7), (y ), (y ),

(2.157)

w4)

where B(m,n ) is the Euler beta function, and y cos ( w t ) is the single chaos-inducing excitation while -pq1x3 cos (wt + g1),yq2 cos (wt $,) are chaos-controlling excitations. From Eq. (2.148), it is straightforward to obtain

+

(2.161)

77

The generic Melnikov function: Deterministic case

cos ($, - q2)f Jcos2

1

( G -~ G ~+) B ~ / B -; 1 R ~ ~ ,

with

(2.162)

(2.163) In the case where B E ]B,i,, B,,,] (the predicted suppressory range, cf. Sec. 2.5) and B2 3 B , B1 3 B is satisfied in Eqs. (2.161) and (2.162), respectively, these equations represent the boundaries of the regions in the planes (& - &) - ql, (G1 - $ J ~-) q2, respectively, where the Melnikov function (2.156) is negative (or null) for all to, i.e., where homoclinic chaos is suppressed.

Area = 4BB,"RI2 (I+BB,-') R,,

(~-B~B,-~ R,,) ' "

...............

( ~ - B B ~R,, -')

Figure 2.11: Generic boundary function (Eq. (2.162)). Figure 2.11 summarizes the main properties of a generic boundary function using the case associated with Eq. (2.162). One sees that the area enclosed by the boundary function, 4BB;'RlZ = 4q2BBT1,depends on the initial chaotic state (through B )

78

Theoretical Approach

and on the corresponding chaos-controlling excitation (through q2BC1),but not on the second chaos-controlling excitation, unlike the maximum deviation,

(2.164)

~ ~ is , the phase from the optimal suppressory initial phase difference ($1 - $ J ~ )which difference allowing the widest suppressory range of qz. Observe that (2.165)

with $20pt being the optimal suppressory initial phases when q2 = 0, ql respectively.

=

0,

Computer simulations clearly indicate that the suppressory effectiveness of a single chaos-controlling excitation is sharply increased when there is a second chaoscontrolling excitation chosen from the above theoretical predictions.

0.5

0.0 0

1

Figure 2.12: The controllable region of (2.155) for q2 = 0.

Figures 2.12-2.15 show the results corresponding to the twewell Duf€ing oscillator (2.155) for the fixed parameters n = 1,b = 0.154, = 4, y = 0.095, w = 1.1.

79

The generic Melnikov function: Deterministic case

0.0

!

0

I

1

w, In2

Figure 2.13: The controllable region of (2.155) for v2 = 0.1, $2 = 7r.

1 '72

0

0 0

1

Figure 2.14: The controllable region of (2.155) for q1 = 0.

80

Theoretical Approach

1.o

772

0.5

I >0 0.0

1

w,

2

Figure 2.15: The controllable region of (2.155) for v1 = 0.1, $J1

= T.

In the absence of any chaos-controlling excitation (771,2 = 0) the oscillator presents a strange chaotic attractor with a maximal Lyapunov exponent X (771,2 = 0) = 0.127 bits/s. The diagrams in Figs. 2.12-2.15 were constructed by plotting points on the grid (of 100 x 100) when the respective maximal Lyapunov exponent was larger than lop3 (grey squares) or than X (91,2= 0) (black squares). The results for single chaoscontrolling excitations, parametric (v2= 0) and external (vl = 0), are shown in Figs. 2.12 and 2.14, respectively, for which $Jlqt = ( 0 , ~ ,$20pt ) = 7r. As expected, one finds a marked enlargement of the regularization areas towards lower values of the amplitude factor when the respective second chaos-controlling excitation acts on the oscillator at the optimal value(s) of its initial phase, as can be appreciated in Figs. 2.13 and 2.15. 2.6

The generic Melnikov function: The noise effect

In any real physical system, external noise is always present and will eventually cause transitions between previously stable attractors as well as diverse noise-induced phenomena such as multistability, stochastic resonance, multimodality, and stochastic ratchets, to quote a few. There exists a great diversity of internal noise sources: quantum fluctuations in lasers, random switching of ion channels and quasi-random release of neurotransmitter by the synapses in neurons, and finite-size effects in chemical reactions are well-known examples. Also, it is common practice nowadays to apply noise externally in experiments as well as to include external noise sources in theoretical models so that the role of noise can be systematically investigated by controlling its features.

81

The generic Melnikov function: The noise effect

The present section is dedicated to the question of how bounded stochastic (random) excitations affect the above (deterministic) control scenario. In particular, the effect of noise on the onset of homoclinic/heteroclinic chaos is discussed. 2.6.1 Additive noise Consider fhstly the broad and important class of dissipative non-autonomous systems described by the differential equations (2.166)

where U ( x )is a general potential, -d (2, i)is a generic dissipative force, p , (x,i)fc ( t ) is a general chaos-inducing excitation, p , (x,i)f, ( t ) is an as yet undetermined suitable chaos-suppressing excitation (with f c ( t )f,(t) , being harmonic functions of frequencies w , 0, and initial phases O,$, respectively), and(2.167)

[-el, el],

is an additive noise term, with zero mean, oscillating within and Fourier spectrum A(w) which, for simplicity, is taken as even, i.e., A (-w) = A ( w ) . Instances of this rather generic type of noise are Gaussian noise (2.168) fractal noise

A(u)

1

-,0

iwiC3

3

< c3 < 2,

(2.169)

,

(2.170)

and band-limited noise

ci

where are parameters. It is also assumed that the complete system (2.166) satisfies Melnikov’s method requirements, i.e., the excitation, dissipation, and noise terms are small-amplitude perturbations of the underlying conservative system x dU(x)/dz = 0 which has a separatrix. The application of Melnikov’s method to Eq. (2.166) yields the Melnikov function

+

I’d,$, (to)= D k Ahar (wto)

+ Char’ (Oto +

+

Radd

ci),

(to;

(2.171)

where the notation har (x)means indistinctly sin(x) or cos (x),and A is a non-negative function, while D , C can be negative or non-negative functions depending upon the respective parameters for each specific system. In particular, D contains the effect of the damping and constant drivings. In the absence of any constant driving, D < 0,

82

Theoretical Approach

while one has three cases D 5 0 when a (positive) constant driving acts on the system besides the dissipative force. Also, A and C contain the effect of the chaos-inducing and chaos-suppressing excitations, respectively, while the function (2.172)

represents the effect of the additive noise, with xo ( t ) being the velocity associated with the homoclinic (or heteroclinic) orbit which is being considered. Note that changing the sign of C is equivalent to having a fixed shift of the initial phase: C -+ -C p&, -+ p&, f T where the two signs before T apply to each of the sign superscripts of p. Before discussing the effect of additive noise on the control scenario, consider first its effect on the onset of homoclinic chaos when the (deterministic) chaossuppressing excitation is absent (C = 0). It is well known that noise-induced transitions are characterized by a qualitative change of the system’s state as the intensity of noise acting upon it increases. This change can manifest itself in diverse forms or mechanisms such as either stabilization or destabilization of system equilibrium states (noise-induced multistability) and excitation of oscillations. Concerning the present case, it seems plausible that the former mechanism underlies the effect of noise on the onset of homoclinic chaos, since it (i.e., the change of stability properties of equilibria) is essentially the same as proposed for the purely deterministic case. A paradigmatic example of this basic mechanism is provided by a simple pendulum with a harmonically excited axis of suspension, for which the unstable upper equilibrium position can be transformed to a stable one if the frequency of the excitation is sufficiently high. The same effect is achieved in the case of random, but sufficiently high-frequency, excitation of the suspension axis. As mentioned above, the simple zeros of a deterministic Melnikov function imply transversal intersection of stable and unstable manifolds, giving rise to Smale horseshoes and hence hyperbolic invariant sets. However, the Melnikov function in Eq. (2.171) is a random process rather than a deterministic function and it can be treated only in some statistical sense. Thus, one has to deduce an effective (deterministic) Melnikov function (and hence to discuss the existence of its simple zeros) to obtain approximate predictions concerning the random Melnikov process. For the sake of concreteness, consider in the following the bounded noise term

G,(t)

+ aB (t)+ I’] ,

= Xsin [Clt

(2.173)

where X and 0’ are the amplitude and averaged frequency, respectively, B ( t ) is a unit Wiener process, a represents the intensity of random frequency, and r is a random initial phase uniformly distributed in [ O , ~ T ) . In this case, the Melnikov function

83

The generic Melnikov function: The noise effect

(2.171) can be written

(2.174) J -09

where R a d d (to; A, a', a, r) is the component of the random Melnikov process due to additive noise. Next, one aSsumes that in the absence of periodic excitations (fc,, (t)= 0), the threshold amplitude of bounded noise excitation for the onset of chaos occurs when the random Melnikov process has a simple zero in the mean-square sense, i.e., (2.175)

where (2.176)

are the spectral density of 6 (t)and the frequency response function of the system, respectively. Now, in the presence of periodic excitations (fc,, ( t )# 0), one defines an effective Melnikov function (2.177)

(2.178)

so that

*

Mh,hl,X>O

(to> 6

M:hl,ef

f,ad$

(to> Vto'

(2.179)

7

It is worth noting that Eq. (2.179) connects the effective Melnikov function with the random Melnikov process. Thus, one can apply the above deterministic theory to Mzh,:eff,aa( t o ) . In particular, for the main resonance case, one obtains new boundary functions %&,cos

=dn,sin

cos

dcos2 ( -

- D:f f , a d d / A 2 ) ]

R,

(2.180) (2.181)

84

Theoretical Approach

for C > 0 (cf. Eq. (2.37)), and

k cos 9 71cfos,cos= rlk,n,sin

[

*

.\/cos2 9 - (1 - D$l,ndd/A2)]R,

= fsin 9 f qcos,sin f = -qsin,cos f

(2.182)

J V R,

(2.183)

for C < 0 (cf. Eq. (2.37)), instead of Eqs. (2.132)-(2.135),respectively. Recall that Eqs. (2.180)-(2.183) represent sufficient conditions for M,$,,eff,add( t o ) 0, Vo, and hence for M&,,x>o ( t o ) 6 0,Vo (cf. Eq. (2.179)). Remarks. First, the theoretical boundaries of the regularization regions associated with the random and deterministic cases have identical form and are symmetric with respect to the same optimal suppressory values, while the respective enclosed areas are smaller for the former than for the latter case. Second, there exists a critical amount of noise, aR,c = D , beyond which regularization is no longer possible, and that this critical value depends upon the damping strength, as expected.

<

2.6.2 Multiplicative noise Consider now the general system dU(x) +7 = -d

! i

i)+ Pc (z, i)f c ( t )+ P,

(2,

(z,).

fs

(t)+ Pn (x,).

<,

(t)

1

(2.184)

where p , (z, i) ( t ) is a multiplicative noise term and the remaining functions are as in Eq. (2.166). The corresponding Melnikov function is given by Eq. (2.171) but with k Z , l( t o ;

Ci) = C1 Re

SrnSa -w

ZO ( t )P, [zo(t),ZO ( t ) ]A ( w ) exp [iw (t + t o ) ]b d t

-w

(2.185) instead of Eq. (2.172). For the bounded noise term (2.173), one obtains the Melnikov function

(to) M&t,x=o ( t o )

M&{,X>O

=

E

( t o ) + R,d (to; A a', a, r) D h Ahar (wto) Char' (Rto p&) ,

M&,x=o

L W

R,ui

(to; An', 0 ,r)

I

+

+

<

i o ( t ) ~[zo(t), , i o (t)] (t + t o ) d t ,

(2.186)

where LZ,~ (to; A, R', a, I') is the component of the random Melnikov process due to multiplicative noise. Next, one assumes as in the additive case that in the absence of periodic excitations (fc,, ( t )= 0), the threshold amplitude of bounded noise excitation

85

Notes and references

for the onset of chaos occurs when the random Melnikov process has a simple zero in the mean-square sense, i.e.,

(2.187) where Hmul (w)=

/

W

XO

( t )pn Izo(t),io (t)]exp ( i w t ) dt

(2.188)

-00

is the corresponding frequency response function of the system, and Sc (w)is given by Eq. (2.176). Now, in the presence of periodic excitations (fc,, ( t )# 0), one again defines an effective Melnikov function (2.189)

(2.190) so that M&,X>O

(to)

G M&f,ef f,mul ( t o ) vto. 1

(2.191)

From Eq. (2.191) the same implications hold as for the case of additive noise. 2.7

Notes and references

The original work by Melnikov appeared in [l],which was generalized by Arnold [2] to a particular instance of a time-periodic Hamiltonian perturbation of a two-degreeof-freedom integrable Hamiltonian system. Fifteen years later, Holmes [3] was the first to apply Melnikov’s method (to a damped forced two-well Duffing oscillator) in the west. From then on the method began to be popular. Chow, Hale & MalletParet [4] rediscovered Melnikov’s results using alternative methods and emphasized that homoclinic and subharmonic bifurcations are closely related. Through the 1980s a great diversity of extensions and generalizations of Melnikov’s approach were developed [5,6,9-131. The interested reader is referred to the books by Lichtenberg & Lieberman [7], Guckenheimer & Holmes [8],Wiggins [14], and Arrowsmith & Place [15] for more details and references. The simplest extension of Melnikov’s method to include perturbational time-delay terms is presented in [16] The application of Melnikov’s method t o controlling chaos in low-dimensional systems by weak periodic perturbations began in about 1990. Indeed, Lima and Pettini provided a heuristic argument to extend the idea that parametric perturbations can modify the stability of hyperbolic or elliptic fixed points, in the phase space of linear systems, to the case of nonlinear systems, and hence that “parametric perturbations could also provide a means t o reduce or suppress chaos in nonlinear systems”

86

Theoretical Approach

([17]). They used for the first time Melnikov’s method to demonstrate this conjecture analytically in the case of a damped driven two-well Duffing oscillator subjected to a chaos-suppressing parametric excitation. However, their insufficient analysis of the corresponding Melnikov function led them into gross errors in their final results and conclusions. Specifically,they failed both to demonstrate theoretically the sensitivity of the suppression scenario to the initial phase of the chaos-suppressing excitation and to find it numerically: “Finally, it is worth mentioning that all the observed phenomenology is independent of the initial phase shift between the two cosines in Eq. (3)” ([17]). They also failed theoretically to predict the suppression of chaos in the case of subharmonic resonances (between the chaos-inducing and chaos-suppressing excitations) higher than the main one: “Another thing that is not theoretically ex= 0” plained is the existence of higher resonances, i.e., X @) = 0 and X a;’

0

0

(A being the maximal Lyapunov exponent, [17]). Although a part of their erroneous analysis of the Melnikov function originated from a mistake in its calculation [20,21], its main weakness was in not providing a correct necessary and suficient condition for the Melnikov function to always have the same sign (i.e., for the frustration of homoclinic bifurcations). For the two-well Duffing oscillator that they considered, such a correct necessary and sufficient condition was first deduced for the general case of subharmonic resonances in [23], where the extremely important role of the initial phase (of the chaos-suppressing excitation) on the suppression scenario was demonstrated theoretically. Cicogna and Fronzoni [18]studied the suppression of chaos in the Josephson-junction model = - 11 + ( cos (at + 0)] sin 4 - 6 4 + y cos ( w t ) , where the parametric excitation -[ cos (fit+ 0) sin4 is the chaos-suppressingexcitation, for the single case of the main resonance R = w by using Melnikov’s method. Their insufficient analysis of the Melnikov function (in particular, that of the role played by the initial phase 0) led them also into gross mistakes in their final conclusions: “We see that the global effect of the modulation is that of lowering the threshold, i.e., of favoring chaos. ... The opposite effects of the modulation in these two cases [in reference to the two-well Duffing oscillator studied by Lima and Pettini [17]]actually depends on the different form of the potential functions and the homoclinic orbits involved, which produce different types of contributions in the Melnikov function.” On the contrary, it was demonstrated in [24] that the effect of the above parametric excitation (in the aforementioned Josephson-junction model and for the general case of subharmonic resonances R = pw, p an integer) is to suppress the chaotic behaviour when a suitable initial phase is used and only for certain ranges of its amplitude. It was also shown in [24] for the first time that such suitable initial phases are compatible with the surviving natural symmetry under the parametric excitation. In [25] it was conjectured that such maximum survival of the symmetries of solutions from a broad and relevant class of systems, subjected both to primary chaos-inducing and chaos-suppressing excitations, corresponds to the optimal choice of the suppressory parameters, specifically, to particular values of the initial phase differences between

4

87

Notes and references

the two types of excitations for which the amplitude range of the suppressory excitation is maximum. The result was supported by analysis of a damped, harmonically driven one-well Duffing oscillator subjected to two qualitatively different kinds of chaos-suppressing excitations: additional forcing and parametric excitation terms. Rajasekar applied Melnikov’s method to study the suppression of chaos in the D&g-van der Pol oscillator x = a2z-pz3-p (1 - z2).+f cos (wt)+qcos (at R4), where the additional forcing q cos (Rt 04) is the chaos-suppressing excitation, for the single case of the main resonance 0 = w. He pointed out the important role of the initial phase (of the chaos-suppressing excitation) on the suppression scenario for the first time: “The system dynamics is sensitive to the initial phase shift” ([22]). He also deduced the analytical expression of the boundaries of the regions in the q - 4 phase plane where homoclinic chaos is inhibited. A generalization of Rajasekar’s approach concerning the relative effectiveness of any two weak excitations in suppressing homoclinic/heteroclinic chaos of an important and wide class of dissipative, low-dimensional and non-autonomous systems, for the main resonance between the chaos inducing and chaos-suppressing excitations, is discussed in [30]. There, general analytical expressions are derived from the analysis of generic Melnikov functions providing the boundaries of the regions as well as the enclosed area in the amplitude/initial phase plane of the chaos-suppressing excitation where homcclinic/heteroclinic chaos is inhibited. Also, a criterion based on the aforementioned area was deduced and shown to be useful in choosing the most suitable of the possible chaos-suppressing excitations ([30]). Cicogna and F‘ronzoni analyzed the Melnikov functions associated with the familyof systems x = f(z)-Sk+ycos (wt)+cg(z) cos (Rt 6’), where Eg(x)cos (Rt 6’) is the chaos-suppressing excitation, for the single case of the main resonance R = w. They deduced both the suitable suppressory values of the initial phase 6’ and the associated chaotic threshold function (y/b)threshold when the chaos-suppressingexcitation acts on the system ([19]). In [26]general results concerning suppression of homoclinic/heteroclinic chaos are derived on the basis of Melnikov’s method for the broad family of dissipative systems x + d V ( z ) / d x = -d(x,k)+pc(x,k)F,(t)+p,,(x, k)Fnc(t),where Fc(t),Fnc(t) are harmonic functions with frequencies w,R, and initial phases 0 , @ , respectively, and where pnc(z,i)F,,(t) is the chaos-suppressing excitation, for the general case of subharmonic resonance R = p w , p an integer. A generic analytical expression was deduced for the maximum width of the intervals of the initial phase iP for which homoclinic/heteroclinic bifurcations can be frustrated. It was also demonstrated that {0,7r/2, 7r, 37r/2} are, in general, the only optimal values of such initial phase, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation. Generic results concerning enhancement or maintenance of chaos for the aforementioned family of systems ([26])were presented in [27],where the connection with the results on chaos suppression ([as])was discussed in a general setting. It was also demonstrated that, in general, a second harmonic excitation can reliably play an

+

+

+

+

88

Theoretical Approach

enhancer or inhibitor role by solely adjusting its initial phase. A preliminary Melnikov-method-based theoretical approach concerning s u p pression of chaos by a cham-suppressing excitation which satisfies an ultrasubharmonic resonance condition with the chaos-inducing excitation was discussed in [28]. This approach was further applied to the problem of the inhibition of chaotic escape from a potential well by incommensurate escape-suppressing excitations in [29]. A recent review on the Melnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations is given in [31]. Finally, the discussion of many of the results of this chapter originated in work by the present author and coworkers [26-321. [l] Melnikov, V. K., (1963) “On the stability of the center for time periodic perturbations,” Trans. Moscow Math. Soc. 12, pp. 1-57.

[2] Arnold, V. I., (1964) “Instability of dynamical systems with many degrees of freedom,” Sov. Math. Dokl. 5 , pp. 581-585.

[3] Holmes, P. J., (1979) “A nonlinear oscillator with a strange attractor,” Phil. Trans. ROY. SOC.A 292, pp. 419-448. [4] Chow, S. N., Hale, J. K., and Mallet-Paret, J., (1980) “An example of bifurcation to homoclinic orbits,” J. Diff. Eqns. 37, pp. 351-373.

[5] Greenspan, B. D., (1981) Bifurcations in Periodically Forced Oscillations: Subharmonic and Homoclinic Orbits (Ph.D. thesis, Cornell University). [6] Holmes, P. J . and Marsden, J. E., (1982) “Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom,” Comm. Math. Phys. 82, pp. 523-544. [7] Lichtenberg, A. J. and Lieberman, M. A., (1983) Regular and Stochastic Motion, Springer-Verlag, New York, Heidelberg, Berlin.

[8] Guckenheimer, J. and Holmes, P., (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York. [9] Lei-man, L. M. and Umanski, Ia. L., (1984) “On the existence of separatrix loops in four dimensional systems similar to integrable Hamiltonian systems,” PMM U.S.S.R. 47, pp. 335-340. [lo] Greundler, J., (1985) “The existence of homoclinic orbits and the method of Melnikov for systems in R”,”SIAM J. Math. Anal. 16, pp. 907-931. [ll] Salam, F. M. A., (1987) “The Melnikov technique for highly dissipative systems,” SIAM J. App. Math. 47, pp. 232-243. [12] Schecter, S., (1987) “Melnikov’s method at a saddle-node and the dynamics of the forced Josephson junction,” SIAM J. Math. Anal. 18 (6), pp. 1699-1715. [13] Wiggins, S., (1987) “Chaos in the quasiperiodically forced Duffing oscillator,” Phys. Lett. A 124, pp. 138-142. [14] Wiggins, S., (1988) Global Bifurcations and Chaos, Springer-Verlag, New York. [15] Arrowsmith, D. K. and Place, C. M., (1990) An Introduction to Dynamical Systems, Cambridge University Press, Glasgow.

Notes and references

89

[16] Cai, C., Xu, Z., and Xu, W., (2002) “Melnikov’s analysis of time-delayed feedback control in chaotic dynamics,” ZEEE Puns. Circuits Syst. Z 49 (12), pp.1724-1728. (171 Lima, R. and Pettini, M., (1990) “Suppression of chaos by resonant parametric perturbations,” Phys. Rev. A 41 (2), pp. 726-733. [18] Cicogna, G. and Fronzoni, L., (1990) “Effects of parametric perturbations on the onset of chaos in the Josephson-junction model: Theory and analog experiments,” Phys. Rev. A 42 (4), pp. 1901-1906. [19] Cicogna, G. and Fronzoni, L., (1993) “Modifying the onset of homoclinic chaos: application to a bistable potential,” Phys. Rev. E 47 (6), pp. 4585-4588. I201 Cuadros, F. and Chacbn, R., (1993) “Comment on ‘Suppression of chaos by resonant parametric perturbations’,’’ Phys. Rev. E 47 (6), pp. 4628-4629. [21] Lima, R. and Pettini, M., (1993) “Reply to ‘Comment on “Suppression of chaos by resonant parametric perturbations” ’,” Phys. Rev. E 47 (6), pp. 4630-4631. 1221 Rajasekar, S., (1993) “Controlling of chaos by weak periodic perturbations in Duffingvan der Pol oscillator,” Prurnana J . Phys. 41 (4), pp. 295-309. [23] Chacbn, R., (1995) “Suppression of chaos by selective resonant parametric perturbations,” Phys. Rev. E 51 (l),pp. 761-764. 1241 Chacbn, R., (1995) “Natural symmetries and regularization by means of weak parametric modulations in the forced pendulum,” Phys. Rev. E 52 (3), pp. 2330-2337. [25] Chacbn, R., (1998) “Comparison between parametric excitation and additional forcing terms as chaos-suppressing perturbations,” Phys. Lett. A 249,pp. 431-436. (261 Chacbn, R., (1999) “General results on chaos suppression for biharmonically driven dissipative systems,” Phys. Lett. A 257,pp. 293-300. [27] Chacbn, R., (2001) “Maintenance and suppression of chaos by weak harmonic perturbations: a unified view,” Phys. Rev. Lett. 86 (9), pp. 1737-1740. [28] Chacbn, R., (2001) “Role of ultrasubharmonic resonances in taming chaos by weak harmonic perturbations,” Europhys. Lett. 54 (2), pp. 148-153. [29] Chacbn, R. and Martinez, J. A., (2002) “Inhibition of chaotic escape from a potential well by incommensurate escape-suppressing excitations,” Phys. Rev. E 65,pp. 036213/1-7. [30] Chacbn, R., (2002) “Relative effectiveness of weak periodic excitations in suppressing homoclinic/heteroclinic chaos,” Eur. Phys. J. B 30 (2), pp. 207-210. [31] Chacbn, R., (2005) “Melnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations,” Phil. Puns. R. SOC. Lond. A (to be published). [32] Martinez, P. J. and Chacbn, R., (2004) “Taming chaotic solitons in FrenkelKontorova chains by weak periodic excitations,” Phys. Rev. Lett. 93,pp. 237006/14.

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Chapter 3 PHYSICAL MECHANISMS 3.1

Energy-based approach

We have seen in the previous chapter that the frustration of homoclinic/heteroclinic bifurcations is the basic mechanism underlying the suppression of chaos by secondary excitations in systems capable of being studied by Melnikov’s method. However, there is no doubt that such a control procedure may also be reliable in more general systems. It is thus necessary to identify a more general physical mechanism to understand the basic (universal) features of the control scenario by periodic excitations. In the present chapter, a recent approach is discussed which is based on a nonlinear generalization of the standard concept of (frequency) resonance, which in its turn is based on energy considerations. 3.1.1 Motivation

Apparently, Galileo Galilei mentioned and discussed for the first time the notion of resonance (“risonanza”) in his book Discorsi e dimostrazioni matematiche, intorno B due nuoue scienze (1638). He provided illustration of such a notion through comments on three examples: resonance of strings (of citherns and spinets), bells (of a belfry), and pendula (church lamps). In all cases, Galileo pointed out, as a distinctive feature of resonance, a temporal increase of the oscillations’amplitude when the forcing frequency matched a natural frequency of the system. Clearly, such a resonant effect originates in the fact that, under the experimental conditions in which Galileo observed the aforementioned systems, they reasonably behaved as linear oscillators. Remarkably, this linear-system-based concept has survived up to now: resonance (nonlinear resonance) is identified with how well the driving period fits (a rational fraction of) a natural period of the underlying conservative system. However, the genuine effect of the Galilean resonance (i.e., the secular growth of the oscillations’ amplitude) can no longer be observed in a nonlinear system. The reason is simple: while a linear oscillator has a unique period which is energy-independent, nonlinear oscillators generically present an infinity of energy-dependent periods. This means that, although a Galilean resonance can still be momentarily induced in a nonlinear system by exciting it with a driving period that exactly matches the intrinsic period of the current motion, the subsequent growth of the nonlinear oscillations changes the intrinsic period of the motion, which no longer matches the excitation period, 91

92

Physical Mechanisms

and thus takes the system out of Galilean resonance. From this point of view, the notion of exact Galilean resonance has no physical sense for nonlinear systems. Since linear oscillations represent a limiting degenerate (energy) case with respect to nonlinear oscillations, it seems that any truly nonlinear generalization of the notion of resonance, in its etymological sense of resonare, i.e., awaken an echo (of some underlying nonlinear oscillation), must be based on energy (or action) considerations. Also, one would no longer expect any notion of nonlinear resonance to involve the secular growth associated with the Galilean resonance, when the system is excited by a strictly periodic excitation. In this regard, the phenomenon of autoresonance provides a mechanism of oscillation growth. Up to now, this phenomenon has been mainly studied for certain classes of nonlinear systems subjected to a swept period excitation, such as when the system automatically adjusts its amplitude so that its instantaneous nonlinear period matches the driving period. The notion of geometrical resonance aims to extend the aforementioned sense of “resonare” to general nonlinear systems. An important point, which will be shown below, is that the notions of autoresonance and geometrical resonance naturally arise when the power of dissipative and non-autonomous systems is discussed in a general setting. In particular, this energy-based theory explains the aforementioned adiabaticity requirement of the previous approach. 3.1.2 Geometrical resonance Consider, for the sake of simplicity, the case of a general one-dimensional Hamiltonian system subjected to general, dissipative and non-autonomous, perturbations

(3.1)

where HO( q , p ) is the Hamiltonian of the conservative system. It is convenient to express the system (3.1) in terms of the canonically conjugate variables action I and angle $:

(3.2)

where E = Ho(q,p) is the energy of the (unperturbed) conservative system while w = w ( E ) = dHo/dI is the frequency of the conservative system at a given energy E , with Eminbeing the minimal (over all q , p ) energy E = Ho(q,p) and qmin = qmin(E)

93

Energy-based approach

being the minimal (for a given E ) coordinate q = q ( E , p ) . Thus, the variations corresponding to system (3.1) can be shown to be

4,E

(3.3)

Now, for each pair of initial conditions ( q ( t = O),p(t = 0)), the system (3.1) is at geometrical resonance, b y definition, when the non-autonomous terms of the perturbation are such as to preserve the unique natural solution from the underlying conservative system associated with such initial conditions, i.e., when a solution of the conservative system resounds or echoes in the sense that it is e x a c t l y the solution of the perturbed system for the same initial conditions. Note that the definition of geometrical resonance is necessarily local, unlike the Galilean resonance, to take into account the nonlinearity property. The denomination geometrical resonance originates in the fact that the shape of the non-autonomous perturbative terms is just as meaningful as other characteristics, such as the amplitude and the period, for the general (nonlinear) case. In general, if (qGR, p G R ) is a geometrical resonance solution of (3.1), then N

i= 1 M

(3.4) j=1

must be satisfied, which is equivalent to the local energy conservation requirement E = Ho(qGR,PGR) = const. (cf. Eq. (3.3)). In other words, one can obtain the particular expression(s) for the geometrical resonance non-autonomous terms by requiring that they satisfy Eq. (3.4). To illustrate this point with an example, consider the (general) case described by the unperturbed Hamiltonian Ho(q,p) = p 2 / 2 U ( q ) and the f q . j ( Q , P ,t ) = - 4 q , P ) + 9 ( 4 , P ) F ( t ) , perturbations f*,i(q,P, t ) = 0, = 1, '.'IN, where U ( q ) is an arbitrary timeindependent potential, - d ( q , p ) is a general dissipative force, and g ( q , p ) F ( t ) is a general temporal excitation. In this case, the geometrical resonance condition (3.4) is

,:c

-d(qGR,PGR)

+ g(qGR,PGR)FGR(t) = 0

+

(3.5)

or, equivalently, p& (t)/2 + U [qGR (t)]= const. When the geometrical resonance solution is periodic, this means that the amplitude, period, and waveform of & ~ ( t ) will be oneto-one determined by those of ( q G R ( t ) , p G R ( t ) ) via Eq. (3.5).

94

Physical Mechanisms

Remarks. First, given a conservative integrable system Ho(q,p ) , a necessary condition for the system to be able to undergo a geometrical resonance is that at least two perturbative terms (at least one of them being non-autonomous) to act on it. Second, when there exist many non-autonomous perturbative terms, one has, in general, several choices to drive the system to a given geometrical resonance solution. Third, in the fully linear limit one recovers the Galilean resonance requirement: the non-autonomous perturbative terms are harmonic having the same period as the unperturbed solution. Fourth, since the notion of geometrical resonance implies a procedure to locally “Hamiltonianize” an otherwise dissipative system, the question of the structural stability of the system in passing through a geometrical resonance acquires especial relevance. Fifth, the notion of geometrical resonance is quite general. It can be applied to multidimensional non-autonomous systems described by sets of ordinary differential equations as well as to spatio-temporal systems described by partial differential equations (such as perturbed sine-Gordon, nonlinear Schrodinger, and complex Ginzburg-Landau equations).

3.1.3 Autoresonance As mentioned above, autoresonance phenomena occur when a system continuously adjusts its amplitude so that its instantaneous nonlinear period matches the driving period, the effect being a growth of the system’s energy. Autoresonant effects were first observed in particle accelerators, and have since been noted in nonlinear waves, fluid dynamics, nonlinear oscillators, atomic and molecular physics, and planetary dynamics. A previous theoretical approach to autoresonance phenomena provided an early explanation of the mechanism inducing the growth of the oscillation for particular classes of resonantly driven nonlinear systems which stay locked with an adiabatically varying perturbing oscillation (the drive). The adiabatic excitation yields the autoresonant effect by automatically adjusting the system’s amplitude so that the instantaneous nonlinear period matches the driving period. However, a fundamental part (hereafter referred to as adiabatic autoresonance (AAR) theory) of the aforementioned previous theoretical approach to autoresonance phenomena presents severe limitations of applicability and insight: essentially, it was developed for nonlinear oscillators that reduce to a Duffing oscillator

x + w; (x + b 2 3 )

= -sx

+

& cos

(wot

+ &2)

(3.6)

for small amplitudes, where (Y is the linear sweep rate and S > 0. In the context of AAR theory, it has been found numerically that autoresonance solutions only occur if (i) the damping coefficient 6 is not too large, and (ii) the amplitude of the autoresonance oscillations grows on the average, but also oscillates around the

95

Energy- based approach

average growth. Also, AAR theory predicts that (iii) there exists a threshold for autoresonance, in particular, if the normalized excitation amplitude ~ w i exceeds ~ / ~a threshold proportional to a3I4,the system will follow the excitation to high amplitude, while the amplitude will stay very low otherwise, (iv) that the threshold sweep rate f f t h scales as b2, (v) that the autoresonance effect is solely expected for the case with initial conditions near some equilibrium of the (unperturbed) nonlinear system, and (vi) that there exists a breaking time for autoresonance, t b . Properties (ii), (iii), (v), (vi) also hold in (vii) the case with no dissipation, but AAR theory provides no theoretical explanation of that fact. In the following, a general and energy-based theory for autoresonance phenomena in non-autonomous systems is discussed and applied to the above D f f i g oscillators to explain conjointly points (i)-(vii) as well as to deduce new properties concerning autoresonance phenomena in generic systems (including Duffmg-like systems). The theory arises from the question as to whether there exists an upper limit for the growth rate of the system’s amplitude when a small-amplitude force acts on the system. Consider the general family of systems

2

= 9 (x)- d ( 2 ,i)+ p ( x , k ) F(t),

(3.7)

where g (x) = - d V ( z ) / d x [V(x) being an arbitrary time-independent potential] -d (2,i) is a general damping force, and p (x,i)F ( t ) is an as yet undetermined suitable autoresonance-inducing force. Clearly, the corresponding equation for the energy is

E

= X [-d

(z,i)+ p (x,k ) F (t)] P (2,X,t ) ,

(3.8)

+

where E ( t ) = (1/2) x2 (t) V [x(t)]and P (x,k, t ) are the energy and power, respectively. In the spirit of the aforementioned energy-based approach to resonance phenomena, the autoresonance solutions are defined by requiring that the energy variation

AE

=

6’

P ( 2 ,x,t) dt

(3.9)

is a maximum (with t l , t 2 arbitrary but fixed instants), where the power is considered as a functional. This implies a necessary condition (hereafter referred to as autoresonance condition) to be fulfilled by autoresonance solutions and excitations, which is the Euler equation (3.10) From Eq. (3.10), a relationship between x,x,and F can be deduced such that the solutions of the system given by Eqs. (3.7) and (3.10) together provide the autoresonance excitations, FAR ( t ) ,and the autoresonance solutions, XAR (t). It is worth noting that the autoresonance condition (3.10) represents a feedback autoresonancecontrolling mechanism, which is absent in the aforementioned previous approach to

96

Physical Mechanisms

autoresonance phenomena where an explicit, coordinate-independent, and adiabatic force is used from the beginning. In this regard, autoresonant control has previously been discussed in the context of vibro-impact systems on the basis of the analysis of nearly sinusoidal self-oscillations. Also, the corresponding autoresonance equations for the multidimensional case can be straightforwardly obtained from the same principle. To compare the present approach with the previous one, consider the power functional P (x,2, t ) = 2 [-62 F ( t ) ]. (3.11) For the particular case of Duffing oscillators, the system (3.7), (3.10) reduces to

+

zAR

+

2 WO (zAR

+ b&R)

= sXAR,

FAR= 262AR.

(3.12) (3.13)

Note that Eq. (3.13) gives the autoresonance condition, i.e., the autoresonance excitations and the corresponding autoresonance solutions have the same instantaneous nonlinear period, at all instants, but without the adiabaticity requirement of the AAR theory. Generally, the autoresonance condition means that the instantaneous period of the autoresonance solution fits a rational fraction of the instantaneous period of the autoresonance excitation. This property generalizes (and contains as a particular case) the persistent phaselocking condition of the AAR theory. To obtain autoresonance solutions (and hence autoresonance excitations, cf. Eq. (3.13)) consider the ansatz (3.14) XAR ( t )= 7.f ( t )cn IPS ( t )+ 4;ml where cn is the Jacobian elliptic function of parameter m, and where the constants

P, m, and the functions f ( t ),g (t)have to be determined for the ansatz to satisfy Eq. (3.12), while y,4 are arbitrary constants. After substituting this ansatz into Eq. (3.12), one finds the exact general autoresonance solution zAR(t) = y0ebt/3cn “P ( t ); 1/21 , p (t)

=

3 y O w 0 h(e6t/3- 1) / 6

+ po,

(3.15)

+

with the constraint w i = 2S2/9 and where po = 4 3yoWo&/6, yo = y. Clearly, the exact autoresonance excitation corresponding to solution (3.15) is ~ dn / ~[p (t); 1/21 , jyo6 2 2 eb t / 3 cn [‘p ( t );1/21 - 2 y ~ S ~ ~ dsni[pe (~t );~1/21 (3.16) where sn and dn are Jacobian elliptic functions. Observe that the particular time dependence of the autoresonance solution (3.15) directly explains the above point (ii). In comparing the present predictions with those from AAR theory, recall that the latter solely exist for the case with z (0) N 0, 2 (0) N 0, for b > 0 (point(v)). Thus, for this case yo 1: 0 and hence Eq. (3.16) can be approximated by F A R (t)=

FAR( t )N ty06’ (1

+ ,a1t + ...) cn [yo&

(wot

+ ;w06t2 1 + ...) ; 1/21

(3.17)

97

Energy-based approach

and, using the Fourier expansion of cn, one finally obtains (3.18)

where K K'

= =

7r&'csch(7r/2)/K(1/2) 7r/(2K(l/2)) N 1.

1,

+

Now, one sees that to consider the excitation ECOS (wot at2/2)(cf. Eq. (3.6)) as a reliable approximation to FAR(t) (cf. Eq. (3.18)) implies that the damping coefficient has to be sufficiently small (point (i)) so as to have a sufficiently large breaking time, tb 6-' (point (vi)). Thus, for t 5 t b , one obtains E t h d2, ath W o 6 (Cf. Eqs. (3.6) and (3.18)). When wo 6 (recall that w i = 2b2/9 for the exact autoresonance solution (3.15)),one finds (Yth 6' (point (iv)), which explains the adiabaticity requirement of AAR theory for dissipative systems, &thIW;" N c'th/aip = a:p (point (iii)), and the cosine's argument in Eq. (3.18) can be reliably approximated by the first two terms, as in AAR theory (cf. Eq. (3.6)). N

N

N

N

N

I

*

,

,

6 = 0.4

6 = 0.8 coo = 0.4

0.8

-

W"

= 0.2

a = 0.0s 6 = 0.4

0

10

20

30

40

50

60

Time

Figure 3.1: Autoresonant responses (energy vs time, see the text).

98

Physical Mechanisms

o'20

8 = 0.4

6 = 0.4

5;;

2

0.15

-

0.10

-

6 = 0.2

E = 0.22

W

a = 0.04

0.05 -

6 = 0.2

0.00

1 0

20

40

80

60

100

Time

Figure 3.1: (Continued) Figure 3.1 shows an illustrative comparison between the autoresonance responses yielded by autoresonance excitations given by E cos (wot (rt2/2), where in all cases b = 5, z (0) = j: (0) = 0, yo = lop3, (po = 0, and & > &th, and FAR(^) (cf. Eq. (3.16)), respectively, for the cases wo b ( E = 0.5, Fig. 3.l(a)) and wo >> b ( W O = 27r, Fig. 3.1(b)). Point (vii) is rather striking in view of the very different properties of Hamiltonian and dissipative systems, and its explanation is a little more subtle. Firstly, note that current autoresonance theory provides an unsatisfactory result for the limiting Hamiltonian case. For example, Eq. (3.10) yields r ( z ) F ( t )= 0 for the family given by Eq. (3.7) with d (z,i) = 0 , p (z, i) = T (z), i.e., including the cases of external and parametric (of a potential term) excitations. Clearly, the two possible types of corresponding particular solutions, equilibria and those yielded by a constant excitation (cf. Eqs. (3.7) and (3.10)), can no longer be autoresonance solutions. Secondly, for the above Duffing oscillators one has

+

N

(3.19) (cf. Eqs. (3.12) and (3.13)). Therefore, it is natural to assume the ansatz F ( t ) =

Xi ( t ), X > 0, for the case with no dissipation, where now the autoresonance rate, A, is a free parameter which controls the initial excitation strength. Thus, the corresponding autoresonance solutions are given by Eq. (3.15) while autoresonance excitations are given by the expression in Eq. (3.16) multiplied by 1/2, both with X instead of 6, which explains point (vii) and hence the adiabaticity requirement of AAR theory

99

Energy- based approach

for Hamiltonian systems (recall point (iv)). It is worth mentioning that this valuable

result holds for the broad family of dissipative systems

(3.20)

where V ( x )is a generic time-independent potential and -6x Iiln-'is a general dissipative force (b > 0, n = 1,2,3, ...). The corresponding autoresonance equations (cf. Eqs. (3.7) and (3.10)) are

(3.21)

For the limiting Hamiltonian case (6 = O), it is therefore natural to assume the ansatz F ( t ) = nXx Iiln-l , X > 0. Thus, autoresonance solutions are the same for the dissipative and Hamiltonian cases, while the autoresonance excitations associated with the Hamiltonian case are the corresponding autoresonance excitations associated with the dissipative case multiplied by n/ ( n + l), with X instead of 6 for the Hamiltonian case. In the light of the exact autoresonance excitation (cf. Eq. (3.16)), one can readily obtain a reliable approximation for arbitrary initial conditions, i.e., not just those near the equilibrium of the unperturbed D f f i g oscillator:

(3.22)

+

where 6" = .rr2fisech ( ~ / 2 /) K 2(1/2) 11 1.61819 N (1 &) /2 (i.e., the golden ratio). Thus, for t 5 t b 6-' ( A p 1 ) one obtains the general (i.e., valid for any initial condition) 1st-order adiabatic excitation N

FA,^ ( t )= ECOS (wot + at2/2)- €'sin (wot + at2/2) , with the above scaling for Eth, a t h , and E&

N

EthYoblf2.

(3.23)

100

Physical Mechanisms

4

3

$

w2

2

1

0 0

5

10

15

20

25

30

Time Figure 3.2: Autoresonant responses (energy vs time, see the text). Figure 3.2 shows illustrative examples for several initial conditions far from z (0) = 0, x (0) = 0. Black and grey lines represent the autoresonant responses to a general 1st-order adiabatic excitation (cf. Eq. (3.23)) and to a harmonic and linearly swept excitation (cf. Eq. (3.6)), for the parameters b = 5, 6 = 0.4, wo = 0.2, E = 0.5 &th, a = 0.08 a t h , and initial conditions II: (0) = 0.8, x (0) = 0.107 (yo = 0.8, &' = 0.9 &ih, thick lines) and z(0) = 0.6, x (0) = 0.08 (yo = 0.6, E' = 0.67 &ih, thin lines). Another fundamental consequence of the present approach is the derivation of the scaling laws for the thresholds corresponding to higher-order chirps. Indeed, consider the general nth-order adiabatic excitation N

N

N

N

FA,^

=

E

cos [w ( t )t] - &'sin [w ( t )t] , (3.24) n=l

+

instead of ECOS (wot a t 2 / 2 ) in Eq. (3.6), where an is the nth-order sweep rate (a1= a/2). For this general case, the above analysis straightforwardly yields the scaling law Eth [3n + 1 ) ! 1 3 / ( 2 n + 2 )%?th 3/(2n+2) (3.25) - N

6

for t 5 t b 6-' (Ap1), where an,$,is the threshold nth-order sweep rate and N (m) = [3" ( n+ l)!]3/(2n+2) is a monotonous increasing function. Thus, the 3/4 scaling law N

101

Energy- based approach

is a particular law which solely applies to a linear chirp. For the case of a single chirp term (w ( t )= wo ant*),the dependence of the above general scaling law on n indicates that one can expect a similar autoresonance effect for ever s m a l l e r values of a, as n increases.

+

0.8 0.1 0.6

0.5

B

0.4

w

0.3

6 c

0.2 0.1

0.0 -0.1

0

5

10

15

20

25

30

Time Figure 3.3: Autoresonant responses (energy vs time, see the text). Computer simulations confirm this point: an illustrative example is shown in Fig. 3.3. Grey and black lines represent the autoresonant responses t o a harmonic excitation with a linear chirp (w ( t )= wo + alt) and with a quadratic chirp (w ( t )= wo a&, respectively, for the parameters b = 5, b = 0.4, wo = 0.2, & = 0.5 &th, a1 = 0.04 al,th, a2 = 0.003 a!2,thI and the initial conditions x (0) = lop3, x (0) = 0 (thick lines) and x (0) = 0 , x (0) = 1 (thin lines). A further question remains to be discussed: We have seen why AAR theory requires autoresonance excitations to be adiabatically varying perturbing oscillations, but which are the underlying adiabatic invariants? To answer this question, note that Eq. (3.12) (with X instead of S for the case with no dissipation) can be derived from a Lagrangian, which one defines as

-

-

+

-

(3.26) p

= 2 , and whose associated Hamiltonian is (3.27)

102

Physical Mechanisms

The form of this Hamiltonian suggests the following simplifying canonical transformation:

(3.28) It is straightforward to see that the generating function of the canonical transformation is F2 ( x ,P, t ) = xPe-atf2.The new Hamiltonian therefore reads:

K(X,P,t)

=

H ( z , p , t ) - -aF2 at P2

(3.29)

In the limiting linear case ( b = 0), one sees that K is conserved, i.e., the autorescnance solutions corresponding to the linear system are associated (in terms of the old canonical variables) with the invariant p2eat

+ wix2e-6t+ bpx,

(3.30)

while for the nonlinear case ( b # 0) one obtains (after expanding eat) that the respective autoresonance solutions are associated with the adiabatic invariant (3.31)

< <

over the time interval 0 t tAI 6-’ (i.e., the same scaling as for the breaking time, tb, deduced above), where E is the energy of the underlying integrable D f f i g oscillator. Observe that the adiabatic invariant reduces to E provided that 6 (A) is sufficiently small (as required in AAR theory) and that the same result is obtained for a general potential V ( x )instead of Duffing’s potential. The broad scope of current autoresonance theory suggests its application to diverse approaches to chaos control. One example could be the control of chaos by altering the system’s energy. 3.1.4 Stochastic resonance Over the last quarter of a century, the phenomenon of stochastic resonance has been the subject of intense study in the context of multistable (frequently bistable) model systems of low dimension. The notion refers to a phenomenon that occurs when generally feeble input information (such as a weak signal) can be amplified and optimized with the assistance of noise. The basic models used to determine the stochastic resonance features are generally described by a “particle-in-a-potential” equation, X =

N

dU

ax + F ( t )+ 71 ( t ),

--

(3.32)

103

Energy-based approach

where F ( t ) and 7 ( t ) represent a T-periodic excitation and noise, respectively. In particular, one has the simplest one-dimensional model for a symmetric stochastic bistable system when U ( x ) = -ax2/2 bx4/4 and 7 ( t ) is a Gaussian white noise with zero mean and (7 ( t )7 (t s ) ) = 2a6 (s). In the following, geometrical resonance analysis is applied to this case in the context of the standard adiabatic assumption (i.e., the frequency 1/T has to be small compared to the local relaxation rate). Let us consider firstly the purely deterministic case (q = 0). Geometrical resonance analysis permits us to describe equivalently the dynamics arising from (3.32), for T large enough depending upon the excitation waveform, by means of an equivalent autonomous system

+

+

(3.33) where U,, ( x ) is the equivalent potential which retains, in a static way, all the information concerning the seesawing behavior of the original potential (alternate rising and falling of the wells) caused by periodic excitation. Although the notion of geometrical resonance is presented above in the context of second-order non-autonomous differential equations, it is essential for the present case to mention that, by definition, the precise characteristic of a geometrical resonance (for a generic driven dynamical system) is that of preserving a given solution, whether periodic or not, of the underlying unperturbed system. Now, with that in mind, and replacing F ( t )by Ff ( t )in (3.32), with f ( t ) and F being an a priori indefinite T-periodic function with unit amplitude, and an amplitude parameter, respectively, let us consider the particular (steady) solutions z, (t)= Af ( t ) ,i.e., when the temporal excitation and the response have both the same period (if any) and the same waveform. With this assumption, those solutions (hereafter referred to as geometrical resonance solutions) satisfy (3.34) where aeg= a+ F / A is the equivalent intrinsic rate, and U,, (2,) is the equivalent potential associated with U ( x ) = -ax2/2 bx4/4. Since geometrical resonance solutions are equilibrium states (cf. Eq. (3.34)),one would suppose that a rectangular wave would satisfy, intermittentlv over time, the requirements of a geometrical resonance whenever its alternating constant steps corresponded to two of those equilibrium states. Indeed, let us assume (for instance) that Asn (wt @; m) is an intermittentgeometrical resonance solution, i.e., satisfying (3.34), where sn (wt @; m) is the Jacobian elliptic function of parameter m and period T (w = w ( m ) z 4 K ( m ) / T with K ( m ) the complete elliptic integral of the first kind) and @ is an initial arbitrarg phase. One straightforwardly obtains that a necessary and sufficient condition for that is F = bA3 - a A (3.35)

+

+

+

together with m = 1 (a square-wave excitation). Really, Eq. (3.35) provides the necessary and sufficient condition for A sn (wt @; m = 1) to be a solution of Eq.

+

104

Physical Mechanisms

(3.34) except on a set of instants that has Lebesgue measure zero. Observe that, using aeq = a F / A , Eq. (3.35) can be written bA3 - aeqA= 0 , i.e., aeq is the key parameter allowing one to deal with the non-autonomous problem as an equivalent autonomous one. The relationship (3.35) does not depend on the initial phase a, which is relevant in the complete stochastic problem. The aforementioned scenario is exact for a rectangular excitation ( m = 1) for any period T . However, one would like to determine if it is also pertinent for other excitation waveforms (for instance, increasing m from 0 to 1 the function sn ( w t a;m) undergoes a smooth transition from a sine function ( m = 0) to a square wave (m = 1)). Contrary to the case m = 1, sn ( w t a;m < 1) does not satisfy (3.34) during temporal intervals which have finite measure, even if the relationship (3.35) holds. However, for any waveform 0 6 m < 1, the discrepancies (relative to a period) decrease as T -+ co,i.e., for T large enough one expects the aforementioned scenario to hold with (approximate) validity. As will be shown in the following, this result is fundamental for a complete stochastic problem under an adiabatic assumption. Consider now the effect of additive Gaussian noise. Under geometrical resonance and T large enough, one can replace system (3.32) by an equivalent purely stochastic system x. = -~ aueq ( X I + rl ( t ) , (3.36)

+

+

+

ax

where U,, ( x )is given by Eq. (3.34) and the relationship (3.35) is assumed. The corresponding Fokker-Planck equation for the equivalent probability distribution Peq( x ,t ) reads (3.37) It has the stationary solution

Peq,.(x)= C,, exp

,

(3.38)

where C,, is a normalized constant. Notice that the effect of the deterministic excitation is now concentrated on the parameter aeq and one can study the evolution of the stationary probability distribution as this parameter varies. As expected, the extrema of the distribution Peq,s(z)satisfy the equation for the equilibrium states of the equivalent autonomous system (Eq. (3.34)). Let us calculate the Lyapunov exponent X for the system (3.36). Because of the ergodicity of the process z ( t ) ,the Lyapunov exponent may be calculated as an ensemble average: (3.39) One readily finds X = aeq - 3b (z2),, ,

(3.40)

105

Energy- based approach

where the notation (3.41) ( n integer) is used. ,..Jw, given the special relevance of aeq for stoc--astic resonance phenomena, it is natural to consider X as a function of ueq. Its maxima must satisfy the condition (3.42)

which can also be considered as a transcendental equation for 0. Thus, the noise intensity urnax, for which the Lyupunov eqonent presents u maximum and (x2)eq presents a minimum (as functions of u) is non-zero. Therefore, with fixed a, b, F (and hence A , cf Eq. (3.35)), one sees that 0 = urnaxmight be closely related to the respective noise strength USR at which the maximum noiseinduced enhancement of the response from the original system (3.32) takes place, i.e., when a stochastic resonance condition is met. In this regard, it is worth mentioning that the main quantifiers of stochastic resonance (i.e., the response amplitude at the frequency of the periodic signal, the signal-to-noise ratio, the time-scale matching condition, and the measure based on the residence-time distribution function) yield distinct maximizing noise strengths. 25

20

%lax=4%t

5

0 0.00

0.09

0.18

0.27

0.36

F Figure 3.4: Quantity

u r n a x / u S Rvs

excitation amplitude F for a = b = 1.

Therefore, it would be interesting to quantitatively relate the Lyapunov exponent characterization of stochastic resonance to (some of) the above-mentioned quantifiers. In order to obtain a preliminary relationship between the respective values of urn=and USR the signal power amplification factor (which is proportional to the

106

Physical Mechanisms

quotient between the first Fourier coefficient and the excitation amplitude) corresponding to a rectangular excitation F ( t ) in the dimensionless Eq. (3.32) (i.e., U(x) = -ax2/2 bx4/4 with a = b = 1) was calculated for an adiabatic situation (T = 27r. lo3) and several values of the amplitude F . The rectangular excitation was generated by a Fourier series with 20 odd harmonics. Numerically, one finds that for excitation amplitudes sufficiently lower than the static bistability threshold Fth = 2 4 1 9 , the relationship u,,, = const . USR is approximately satisfied for const = 4, while ffmm/(TSR strongly increases near Fth, as is shown in Fig. 3.4. Since the above proportionality relation is satisfied over the excitation amplitude interval where the greatest effect of stochastic resonance occurs, such a relationship represents a new genuine characteristic of the stochastic resonance phenomenon in the simple symmetric bistable system (3.32).

+

3.2

Geometrical resonance analysis: Chaos, stability and control

Diverse applications of the concept of geometrical resonance are discussed below, including the reshaping-induced modification of the onset of chaos of a perturbed pendulum, the stability of the responses of an overdamped bistable system, an improvement of a classical result (due to Cartwright and Littlewood) on the driven van der Pol oscillator, the suppression of chaos in damped harmonically driven systems by resonant harmonic excitations, and the suppression of spatio-temporal chaos and the stabilization of localized solutions in general spatio-temporal systems.

3.2.1 Geometrical resonance in a damped pendulum subjected t o periodic pulses We have seen above that the denomination geometrical resonance arises because shape driving is just as meaningful as period driving for a completely nonlinear problem. One can then conjecture that the corresponding threshold (in parameter space) for the chaosttorder transition under solely shape-driving changes could be explained in terms of geometrical resonance. In other words, one expects that regularization of the dynamics will be guaranteed when the system response sufficiently approximates a geometrical resonance solution by means of such changes. To test such a supposition analytically in a simple system, let us consider the example of a simple pendulum subjected to weak damping and driven by a small-amplitude periodic string of pulses:

x + sin x = -6x

+ A dn ( w t ;m) ,

(3.43)

where 6 , A << 1, dn is the Jacobian elliptic function of parameter m, and time is regarded as dimensionless. Making the frequency w (m)3 2 K ( m ) / T (where K ( m ) is the elliptic integral of the first kind), for fixed amplitude A and period T , one can vary the shape of the pulses by changing m between 0 and 1. The application of

107

Geometrical resonance analysis: Chaos, stability and control

Melnikov’s method to Eq. (3.43) gives the Melnikov function

c -03

M* ( t o )

=

-86 f An2

an(m)bn(T) cos

(F) ,

(3.44)

n=-m

(3.45)

bn(T)

=

sech

($)

,

(3.46)

where 2K(m)/T has been substituted for w , and where the positive (negative) sign refers to the top (bottom) homoclinic orbit of the underlying integrable pendulum. It is straightforward to demonstrate that a homoclinic bifurcation is guaranteed for orbits whose initial conditions are sufficiently near the unperturbed separatrix if

(3.47) (3.48)

(3.49)

It is worth noting that the existence of a lower threshold (i.e., one needs 6 > 0 for the onset of chaos) is a consequence of the positive character of the dn function. Now, one defines a chaos window in parameter space ( m ,T):

(3.50)

m ) ,(T) given by Eqs. (3.45) and (3.46), respectively. Let us now with a ~ ~ + ~ (b2n+l consider the case in which the periodic string of pulses is modeled by a periodic 6 function:

x + sinz

=

-6x

+ Ad1 (t;T), (3.51)

108

Physical Mechanisms

After applying Melnikov's method to Eq. ( 3.51) one readily obtains the corresponding threshold-chaos condition, 6

2 < uk,(m,T),

ur!nin(miT) <

(3.52)

w

(3.53) n=-w M

Ug,,(m,T)

=

4

bn(T),

(3.54)

n=-w

and bn(T)given by Eq. (3.46). The associated chaos window is now

A' ( T )

UL,(T) - ULin(T) (3.55) n=O

As mentioned above, geometrical resonance is concerned with the survival of solutions of the underlying integrable system. It is well known that for the conservative pendulum there is a family of periodic orbits inside the separatrix ZSep(t) =

f (4arctand

ZSep(t)

~t2secht

=

-T),

(3.56)

given by

z g ( t ) = 2arcsin [Jmsn(t;m)] , iE(t) = aJmcn(t;m),

(3.57)

where sn, cn are Jacobian elliptic functions with periods

T,,(m)

= 4K(m).

(3.58)

Also, there exist two families of periodic orbits outside the separatrix (3.56), given bY

(3.59) where the positive (negative) sign refers to counterclockwise (clockwise) pendulum rotations. The periods of these orbits are

T,t(m)

= 2JmK(m).

(3.60)

109

Geometrical resonance analysis: Chaos, stability and control

The two families of inner and outer periodic orbits converge, in the limit m + 1, to the homoclinic orbits forming the separatrix (3.56). From Eq. (3.59) it is clear that when the periodic pulses are given by the dn function, the whole system (3.43) presents the exact geometrical resonance solutions

(3.61) with period given by (3.60), and corresponding, respectively, to the initial conditions if and only if

z(t = 0) = 0, i ( t = 0) = &2/+,

6 J m (3.62) A - 2 ' Observe that such geometrical resonance solutions can exist if A 26, while for the limiting value 6/A = 1/2 they converge to the separatrix solutions (3.56). Also, it is obvious from Eq. (3.57) that geometrical resonance is not possible with inner orbits. Since Melnikov's method is only valid for motions based at points sufficiently near the separatrix of the unperturbed system, the geometrical resonance concerning the separatrix (3.56) (i.e., the special orbit with period Tsep= m) will be considered in the following. The forcing corresponding to a geometrical resonance for the separatrix (i.e., the forcing permitting the survival of this separatrix) is written as F G R , ~= ~~ f26sech ( ~ ) t,

(3.63)

which can be recovered (for some m value) from A dn [2K(m)t/T;m],but not from 2 f i c n ( t ;m). However, one can require the dn forcing to be (period) resonant with either the inner or the outer orbit of the unperturbed pendulum, then take the limit T -+ m to see how well the geometrical resonance forcing (3.63) is approximated, depending upon the shape of the resulting functions. In this way, one can test the exclusive characteristic of the geometrical resonance (shape) by calculating the corresponding threshold functions (cf. Eqs. (3.47)-(3.49)). From Eqs. (3.58) and (3.60), it is obvious that T;n,out(m-+ 1) -+ 00. Therefore, one obtains lim dn [2K(m)t/T;,; m] = sech (t/2) ,

m-+1

lim dn [2K(m)t/Tmt;m]

rn-1

=

sech t .

(3.64)

It is straightforward to calculate the associated threshold functions for the limiting cases (3.64):

U z n ( m +l , T = T , , )

=

Ukax( m+ l , T = T%,) = nU::,!, ( m -+ 1,T = Tmt) + 1, T = Tout)

Urk ( m

=

=

(3.65)

0,

di 0

0, 1/2.

-

1 /2,

(3.66) (3.67) (3.68)

110

Physical Mechanisms

In terms of chaos windows (cf. Eq. (3.50)), one finds kn> Aout,i.e., the range of b/A for the onset of chaos is broader for the inner orbit (period) resonance than for the outer orbit case. As all these results were obtained for T = 00, they should be explicable in terms of how near or far the shape (of the limiting forcings) is from the geometrical resonance shape. Indeed, for the outer (inner) orbit case, the separatrix is (not) a geometrical resonance solution of the whole system for T + 00, which explains the different values for the upper threshold functions (3.66), (3.68). 3.2.2 Geometrical resonance in a n overdamped bistable system Without regard for any specific scientific context at the outset, let us begin with the overdamped model

dx dt

- = -U'(x)

+Ff(t),

(3.69)

where f ( t )is an a priori arbitrary T-periodic function with unit amplitude and the prime refers to the derivative of the arbitrary potential U ( x ) . Observe that the precise characteristic of the geometrical resonance, for a general system to be driven, is that of preserving a given natural response (whether periodic or not) of the unperturbed system. Now, with that in mind, let us consider the special (steady) solutions z s ( t ) , with amplitude A, such that x s ( t )= A f ( t ) ,i.e., when the external excitation and the response have both the same period (usual resonance) and the same shape, including when the natural responses are not periodic. Under this assumption, the dynamics is equivalent to that of a particle having total energy E given by

E

1. z

= -5, 2

+ Ueq(xS)= const,

(3.70)

with (3.71) the equivalent potential. Note that the energy conservation requirement can also be . away from the linear satisfied for a more general choice: f ( t ) = (1/A) g [ z s ( t ) ]But case, the resonance (period) requirement generally is not verified. With E = 0 (the system is overdamped) one directly obtains the solutions corresponding to the general model (3.69), rz

2-

lLL

jx0F x / A

-

U'(x)=

(3.72)

For the sake of concreteness, consider in the following the two-well potential U ( z ) = -z2/2 + x4/4. Then, the corresponding particular responses (3.72) are 3.73)

xs(t)

=

zs(t)

=

k x o [l + 2xgt1

kp[%] 2

71--0

1/4

,

(3.74)

1 e q t / ' s e ~ h 1 / ' { 7 1 t +71~-1 2n0[ ~ ] } ,

(3.75)

Geometrical resonance analysis: Chaos, stability and control

for q < 0 , q = 0 , q > 0, respectively, with x ( t = 0) = ~

111 X O and ,

F A

q=1+-.

(3.76)

Observe that the asymptotic behaviour is that of equilibrium states lim x ( t ) F x s ( t ) =

(3.77)

t+oJ

which do not depend on the initial condition 50. Note that q = q, = 0 is the critical value for the topological change (symmetry breaking) in the shape of the corresponding equivalent potential (cf. Eq. (3.71)). Therefore, one would suppose that a rectangular forcing would satisfy, intermittently over tame, the requirements of a geometrical resonance if F/A 1 > 0. Indeed, let us assume that x s ( t ) = A sn (wt; m) is an intermittent geometrical resonance solution, i.e., that it should verify dxs - = qx, - X,” . (3.78) dt One straightforwardly obtains

+

dXS

- =

dt qz, - x,”

=

[4AK(m)/T] cn (wt;m)dn ( w t ; m) ,

(3.79)

A [q - A2sn2(wt;m)]sn ( w t ; m) ,

(3.80)

where cn and dn are Jacobian elliptic function of parameter m, and w = w ( m ) = 4 K ( m ) / T . Now, it is possible to rewrite (3.79) in the form D*(t;T,m)dn(wt;m)

dt

(3.81)

D(t;T/2,m)cn(wt;m), with

D*(t;T,m)

[-;m,]

2 K ( m )cn 4 K ( m ) t 7T

(3.82) (3.83)

These functions have the remarkable properties

D* ( t ;T ,m = 0) D* (t;T ,m = I ) D(t;T,m=O) D ( t ;T ,m = 1)

= cos ( 2 r t / T ) , = 61+ (t;T / 2 ), = 1, =

61,,(t; T ) ,

(3.84) (3.85) (3.86) (3.87)

112

Physical Mechanisms

where d1,, (t;T/2) [61,a(t;T)] is the symmetrical (asymmetrical) periodic 6 function of period T/2 (T),i.e., they provide non-ideal representations of periodic sequences of pulses. Now, taking the limit m --+ 1, one sees that the right-hand side of (3.81) vanishes on a set of points that has Lebesgue measure zero. This is also the case for the right-hand side of (3.80) if one sets q = A2.

(3.88)

Therefore, a squarewave function of a certain amplitude A* (see below) is an intermittent geometrical resonance response to

dx - x3 + Fsn(wt;m) dt if the following cubic equation is satisfied (cf. Eqs. (3.76) and (3.88)) -=x

A3 - A - F

= 0.

(3.89)

(3.90)

Its solution provides the amplitude-response curve for the a priori possible intermittent geometrical resonance responses. The expected solutions z s ( t )= A* sn (wt;m = 1) will be observed only if they are stable, i.e., if any small perturbation 6x of 2, is damped. Writing x = 2, 62, for 6x << 1 one gets (to first order)

+

3qsn2 (wt; m = I)] 62,

(3.91)

with solutions

hX(t)= 6 x ( ~ ) e ( 1 - 3 7 ) ) t .

(3.92)

Therefore, the above intermittent geometrical resonance responses will be observed if and only if A+3F/2 > 0 (cf. Eq. (3.76)). That sgn(A) # sgn(F) means that the forcing and the rectangular response are phase shifted by 2K(m) (i.e., T/2 in time). It is worth mentioning that over the range F E (-2&/9,24/9) one can expect the amplitude A* of the rectangular responses to be given by A* = (IA,,1I - IAs,zI)/2, where As,I, As,2 (lAs,ll > 1As,21)are the predicted (stable) solutions from the amplituderesponse relation (Eq. (3.90)). This can be understood by noting that such solutions would represent stable pure geometrical resonance responses (cf. Eqs. (3.77) and (3.88)). Hence the expected response under rectangular driving will visit periodically those two equilibrium states. On the other hand, for JFI > 2&/9, A* will be given by the single solution of (3.90), which may be understood as the orbit exploring both wells, describing large amplitude oscillations around the origin. The amplitude-response curve (3.90) indicates that the bistable system (3.89) subjected to a rectangular forcing ( m= 1) should (at geometrical resonance) exhibit a discontinuous transition between the two aforementioned periodic behaviours when F is varied slowly. This discontinuous jumping between the two stable (interior and exterior) orbits is a consequence of the nonlinear amplituderesponse relation (Eq. (3.90)).

113

Geometrical resonance analysis: Chaos, stability and control

3.2.3 Geometrical resonance approach to control of chaos by weak periodic perturbations To illustrate the usefulness of the concept of geometrical resonance in the problem of controlling chaos by weak periodic excitations, consider the family of dissipative svstems

x

+ g ( x ) = -yx + F, har

(7)+

Fnchar

(F+

4) ,

(3.93)

where the notation har(x) means indistinctly sin(x) or cos ( x ) , and p, q are relatively prime integers. When the suppressory driving term is absent (F, = 0), it is assumed that the system is in a chaotic state for a certain damping y and forcing F,, and for a given initial condition. Now, the necessary and sufficient condition to be verified by the total driving force in order for the system (3.93) to be found in a geometrical resonance is written (cf. Eq. (3.4)) (3.94)

where xGR(t) is a TI-periodic response (based on the same aforementioned initial condition) of the underlying conservative system. Generally, xGR(t) will be a nonlinear periodic response, and so one can write m

Canhar

i ~ ~= ( t )

(F+

y:) .

(3.95)

n= 1

Clearly there cannot exist an added harmonic suppressory driving force exactly satisfying the geometrical resonance condition (3.94). However, one can find the optimal values of Fnc,4,and p/q which most closely preserve the energy in the following sense. Let us assume that for the optimal choice (and the same initial condition) the corresponding actual solution x ( t ) remains (after the transient) close to the geometrical resonance solution: x ( t ) = xGR(t) bx(t), where bx(t) is a small deviation with d (62)l d t << bx/T'. Then, one conjectures that the energy is a local almost adiabatic invariant to lowest order in bx, i.e.,

+

(3.96) where local means that each almost adiabatic invariant is only postulated for trajectories based on initial conditions on a given TI-periodic solution from the associated conservative system. For the sake of concreteness, consider in the following the cubic oscillator

x

+ x3 = -yx + F, cos

(7)+

cuF,cos

(7+ 4 )

,

(3.97)

114

Physical Mechanisms

where a = F,/F,.

The TI-periodic solutions of the conservative system are

[

z ( t ) = Acn ““(m; 1/2)t

x(t)

=

-A2sn

[

+ 4’; 1/21

4K(m; 1/2)t

+ 4’;

(3.98)

9

dn

[

4 K ( m = 1/2)t

T/

+ 4;1/21 ,

where cn, sn, and dn are Jacobian elliptic functions, and A = 4K(m = 1/2)/T’. From the assumption z ( t ) = Z G R ( ~ ) bz(t),the energy is given (to first order in bz) by = EGR ( ~ G R z$R) dz, and thus

+

+

+

(3.99)

Finally, from the almost adiabatic invariant condition (3.96), one obtains

(3.100)

with T‘ = qT/p. From Eq. (3.100), it is then obvious that the phase difference 4 between the two forces plays a fundamental role for general systems, in agreement with the results obtained in Chap. 2 for systems capable of being studied by Melnikov’s method. For a given T, the relationship T‘ = q T / p gives the period T‘, depending on the approximation ( p , q ) we are using, and hence the energy (i.e., A = 4 K ( 1 / 2 ) / T t of ) the underlying geometrical resonance. With fixed F,, T ,p , q, Eq. (3.100) provides the curve in the 4 - a parameter plane where the energy of the initial state is expected to be conserved as an almost adiabatic invariant. Next, one can compare the theoretical predictions obtained from the geometrical resonance analysis and Lyapunov exponent calculations of the cubic oscillator (3.97). Lyapunov exponents were computed using the fixed parameters F, = 11.52, y = 0.3, T = 27r, q = 1. In the absence of the chaos-suppressing excitation ( a = 0), the cubic oscillator presents a strange chaotic attractor with a leading Lyapunov exponent A’ ( a = 0) = 0.135 bits/s. The leading Lyapunov exponent was calculated and plotted for each point on a 70 x 70 grid, with (normalized) initial phase 4 and amplitude a along the horizontal and vertical axes, respectively, for p = 1,3,5 (Figs. 3.5, 3.6, 3.7, respectively).

115

Geometrical resonance analysis: Chaos, stability and control

a

0.0

0.5

1 .o

1.5

2.0

Figure 3.5: The controllable region of the cubic oscillator (3.97) for p

=

1.

a

0.0

I

I

I

I

0.5

1.o

1.5

2.0

O/n Figure 3.6: The controllable region of the cubic oscillator (3.97) for p = 3.

116

Physical Mechanisms

0.0

0.5

1.o

1.5

2.0

+l.n Figure 3.7: The controllable region of the cubic oscillator (3.97) for p = 5 . The diagrams in these figures were constructed by only plotting points on the grid when the respective leading Lyapunov exponent was larger than 0 (grey circles) or than A+ ( a = 0) (black circles), and with solid contours denoting the curves associated with the respective almost adiabatic invariants (cf. Eq. (3.100)). One sees that the geometrical resonance approach is reliable for small a values and that the boundaries of the chaotic regions exhibit the same shape as the curves associated with the respective almost adiabatic invariants.

3.2.4

Geometrical resonance and globally stable limit cycle in the uan der Pol oscillator Consider now the problem of chaos suppression for the van der Pol oscillator subjected to two harmonic forcing terms

i +d

(2- 1) z + z = F,cos(wt) + aF,cos

(Rt + @),

(3.101)

where d , F, > 0 , 0 < a < 1, and time is regarded as dimensionless. The forcing F, cos ( w t ) is considered as a chaos-inducing excitation, for given parameters and initial conditions, and the second excitation aF,cos ( R t a) as a chaos-suppressing excitation added a posteriori. Then one looks for the parameter-space regions ( a ,R,@) in which the energy is an almost adiabatic invariant, i.e., the regions in which one expects regularized dynamics. It is demonstrated below that the geometrical resonance analysis of Eq. (3.101) gives an improvement of a classical result due to Cartwright and Littlewood. To obtain the geometrical resonance excitation FGB(t) associated

+

117

Geometrical resonance analysis: Chaos, stability and control

with the autonomous part of Eq. (3.101), let us note that the corresponding unperturbed conservative system is just the simple harmonic oscillator:

+ +

A cos (t @') , iGR(t) = -Asin (t (a')

ZGR(~) =

(3.102)

Therefore, from Eq. (3.4) one easily obtains

sin (t

dA3 + a') - -sin 4

(3t

+ 3@),

(3.103)

which depends on the initial conditions through A and a'. For A -+ 0, F G R ( t ) -+ dAsin (t (a') because the nonlinear damping term approximates to -dx in such is consistent with the wella limit. Notice that the above expression for FGR(~) known result that the autonomous van der Pol oscillator has a stable limit cycle, for small d, close to the circle x 2 ( t ) Z2(t) = 4, i.e., A = 2 (cf. Eqs. (3.102) and (3.103)). Let us now consider the system (3.101), assuming that the dynamics is chaotic for a = 0 (for given values of d, F,,w). It is clear that no trio ( a , Q @ ) provides an exact geometrical resonance forcing (cf. Eq. (3.103)). However, when dissipation and external modulation are represented by small amplitude terms, it is natural to suppose that the suitable values (&,a,@) for regularization will be those providing the best approximation (in the energy-conservation sense) to the corresponding geometrical resonance solution, i.e., those values derived from the local almost adiabatic conservation of the energy

+

+

(3.104)

where X G R ( ~ ) ,iGR(t) are given by Eq. (3.102). It is worth noting that condition (3.104) provides a chaos threshold. Also, condition (3.104) holds for any relationship between w and R, i.e., it is valid for both commensurate and incommensurate cases. For the sake of concreteness, consider in the following the commensurate case q w l p = 0 = 1, i.e., 2r/R = Tn = TGR = 2 r and assume that there exist some integers q , p verifying Tn = ( p / q ) T , (T, = Z.rr/w). Then, substituting Eqs. (3.102) and qwlp = R = 1 into Eq. (3.104), the resulting integrals can be easily evaluated from

118

Physical Mechanisms

standard integral tables. The final result can be written

a

M

R’ (” 7r”

[$(l-$)+

”’1

[sin (a’ - a)]-’ ,

R’ ( p ,q, (a’) = -7r sin a’, R’(P,q,@’) = 0, R’b,q,@’) =

-2 sin

(7)[sin (7)cos -

-

2 cos Q

(7) sin 0’1

(3.105) (3.106) (3.107) (3.108)

(:)2

for p = q, p = nq ( n = 2,3,4, ...), and p # nq ( n = 1,2,3,...), respectively. As in the problem of chaos suppression one is interested first in the situation where the amplitude of the inhibitory excitation is much smaller than the amplitude of the main driving excitation, the optimal values of must verify sin ((a’ - a) = f l , i.e., (a = (a’ f ~ / 2 respectively. , One has then instead of Eqs. (3.105)-(3.108) (3.109) (3.110) (3.111) R(p,q,@)

=

2sin(7) [sin(7)sina+:cos(7)cosa] (3.112) -

(:)2

for p = q , p = nq (n = 2,3,4, ...), and p # nq ( n = 1,2,3,...), respectively. After substituting a from Eqs. (3.109)-(3.112)into F, [cos( w t ) acos (02t a)]and taking , straightforwardly obtains the whole forcing into consideration = a‘ f ~ / 2 one

+

sin (t

+ (a’) + F, cos (t + a’)cos a’,

(3.113)

sin(t+@)+F,cos

(3.114)

+ a’) + F,cos

(3.115)

sin (t

+

+

):( [:cos (7) sin a’ - sin (7)cos a’] sin (t + a’)

2Fcsin

[I-

(3.116)

(:)2]

for p = q, p = nq ( n = 2,3,4, ...), and p # nq ( n = 1,2,3, ...), respectively. The first of the two terms forming F G R ( ~thus ) appears in all the approximations to the

119

Geometrical resonance analysis: Chaos, stability and control

geometrical resonance forcing. Moreover, for p = 3q and the initial condition x ( 0 ) = 0, X(0) = - (4Fc/d)’I3, obtained for A = (4Fc/d)lI3, @‘ = ~ / 2 ,the bichromatic excitation (3.114) exactly coincides with FGR(t), i.e., for

[(5)

-213

a: = f41/3

- 2-213] ,

@ = 0,7r,

(3.117)

respectively. For the particular case F, = 2 d , Eq. (3.117) yields LY = 0 and also A = 2; in other words, the monochromatically (w = 3) forced van der Pol oscillator has an .2 exact limit cycle given by xgR + xGR = 4. Thus this agrees with and explains the aforementioned result concerning the autonomous van der Pol oscillator, namely, that for small d it has a stable limit cycle close to the solution x2+x2 = 4. Indeed, because of the relationship F, = 2d, F, + 0 is equivalent to d + 0. Another point concerning this particular case refers to a classical result due to Cartwright and Littlewood: if F,/d > 2w/3 and d > do (F,, w), Eq. (3.101) with a: = 0 has a stable periodic solution of period 27r/w to which all trajectories converge as t co (i.e., a globally stable limit cycle). Note that the result is stated without restrictions as to how large d can be, i.e., it is only required to surpass a lower threshold. However, in the above discussion for the case w = 3, it was deduced that d = dGR = Fc/2 so that the condition F,/d > 2w/3 is now written dGR > d. In other words, the result of Cartwright and Littlewood requires also, for the specific case w = 3, an upper threshold for d, derived from a geometrical resonance analysis of the problem. similarly, the case d > dGR, d large enough, corresponds to the situation for which Cartwright and Littlewood noted the possibility of “strange” behaviour. 3.2.5 Geometrical resonance in spatio-temporal systems As mentioned above, the concept of geometrical resonance can be applied to any system irrespective of its dimensionality. A relevant case is that of spatic-temporal systems described by partial differential equations: ---$

where F0[4] and 92 [4] are functions of the field 4 and derivatives &, &, &z, &, ..., while Fl[$, x ] includes dissipative terms and g ( x ,t ) R [4]represents a general driving term. It is also assumed that the equation FO[4]= 0 represents an integrable Hamiltonian system, such as those described by the sineGordon equation, the nonlinear Schrodinger equation, the Toda lattice, or the Boussinesq equation. In general, if 4GR is a geometrical resonance solution of (3.118), then (3.119

must be satisfied, where gGR(x,t) is the geometrical resonance driving term. The usefulness of the geometrical resonance analysis in this context is shown by the discussion of some particular examples as follows. Consider first the damped driven

120

Physical Mechanisms

sineGordon equation 4tt

- 4m

+ sin4 = -74t + 9

( 2 7

t)

(3.120)

1

where g ( 2 ,t ) is a generic spatic-temporal driving term. As is well known, the integrable sineGordon equation exhibits the exact breather solution

4 ( 2 ,t) = 4 arctan

4-

sin (wt)

w cosh (4-x)

I

(3.121)

’

for arbitrary w provided that w2 < 1. For this breather solution, the corresponding geometrical resonance driving term (cf. Eq. (3.119)) is written gGR ( 2 ,t )

cosh (d-2)

4 7 4 W c o s (wt) - 1)sin2 (wt) / cosh (4-2)

+(w-~

(3.122) ’

It is worth mentioning that when g ( 2 ,t ) = gGR ( 2 ,t ) , the breather solution (3.121) is asymptotically stable, i.e., it is a spatic-temporal attractor of (3.120). This is an important property which permits one to explain previous stability results concerning driving terms of the form g ( x , t ) = s ( z ) f ( t )with s ( 2 ) and f ( t ) being a bell-shaped function and a time-periodic function, respectively. As a second example, consider the following damped nonlinear Schrodinger equation subjected to a generic spatic-temporal driving term: (3.123)

The integrable nonlinear Schrodinger equation presents the one-soliton solution

4 ( 2 ,t ) = \/;JeiWtsech (&x)

,

(3.124)

for arbitrary w > 0. In this case, the corresponding geometrical resonance driving term (cf. Eq. (3.119)) is written (3.125) It is well known that when the driving term is purely temporal, ~ e x p ( i w t )Eq. , (3.123) exhibits chaotic dynamics for certain regions of the parameter space ( E , a,w). Comparison of the driving terms eexp (zwt) and gGR ( 2 ,t ) (cf. Eq. (3.125)) tell us that a temporally periodic and spatially localized driving yields regularization of the chaotic dynamics into the region € / aM &. In the case of a two-soliton solution, (2’

=

+

4 eit cosh (32) 12 egitcosh ( 2 ) cosh (42) 4 cosh (22) 3 cos (8t)’

+

+

(3.126)

121

Notes and references

the corresponding geometrical resonance driving term is now (Y

gGR (z’ t , =

[4 ei(t+?r/2) cosh (3z)+ 12 ei(9t+x/2) cosh (x)] cosh (45) 4 cosh (22) 3 cos (8t)

+

+

(3.127)

Note that this function can be rewritten as E ~ S (z) I ei(t+K/2) + E Z S Z ( I C )ei(9t4-”/2) where s1,2(z) are sharply localized functions. This result explains that the chaos induced by the purely driving term €1 exp (iwlt) €2 exp (iwzt) at certain €1 > 0 when €2 = 0, w1 = 1 is regularized at certain €2 > 0 when w2 = 9.

+

3.3

Notes and references

In his original work [l],Galileo states that “...the fact that a vibrating string will set another string in motion and cause it to sound not only when the latter is in unison but even when it differs from the former by an octave or a fifth. A string which has been struck begins to vibrate and continues the motion as long as one hears the sound [risonanza];these vibrations cause the immediately surrounding air to vibrate and quiver; then these ripples in the air expand far into space and strike not only all the strings of the same instrument but even those of neighbouring instruments. Since that string which is tuned to unison with the one plucked is capable of vibrating with the same frequency, it acquires, at the first impulse, a slight oscillation; after receiving two, three, twenty, or more impulses, delivered at proper intervals, it finally accumulates a vibratory motion equal to that of the plucked string, as is clearly shown by equality of amplitude in their vibrations. This undulation expands through the air and sets into vibration not only strings, but also any other body which happens to have the same period as that of the plucked string,” which is believed to be the first time the term resonance (risonanza in the Tuscan original) is used in the west. Concerning resonantly excited pendula, Galileo writes “Thousands of times I have observed vibrations especially in churches where lamps, suspended by long cords, had been inadvertently set into motion; but the most which I could infer from these observations was that the view of those who think that such vibrations are maintained by the medium is highly improbable: for, in that case, the air must needs have considerable judgment and little else to do but kill time by pushing to and fro a pendent weight with perfect regularity... First of all one must observe that each pendulum has its own time of vibration so definite and determinate that it is not possible to make it move with any other period [altro period01 than that which nature has given it. For let any one take in his hand the cord to which the weight is attached and try, as much as he pleases, to increase or diminish the frequency Ifre4uenzal of its vibrations; it will be time wasted. On the other hand, one can confer motion upon even a heavy pendulum which is at rest by simply blowing against it; by repeating these blasts with a frequency which is the same as that of the pendulum one can impart considerable motion. Suppose that by the first puff we have displaced the pendulum from the vertical by, say, half an inch; then if, after the

122

Physical Mechanisms

pendulum has returned and is about to begin the second vibration, we add a second puff, we shall impart additional motion; and so on with other blasts provided they are applied at the right instant, and not when the pendulum is coming toward us since in this case the blast would impede rather than aid the motion. Continuing thus with many impulses [impulsi] we impart to the pendulum such momentum [impeto] that a greater impulse Iforza] than that of a single blast will be needed t o stop it.” The linear-system-based concept of Galilean or frequency resonance was latter rather blindly extended to nonlinear systems to indicate the case when the driving period fits a rational fraction of a natural period of the underlying conservative system. This extension is the so-called nonlinear resonance, which was introduced by Chirikov [2] in the context of periodically driven Hamiltonian systems. The energy-based concept of geometrical resonance was introduced by Chac6n in the context of controlling chaos [3]. Geometrical resonance analysis has been applied to diverse nonlinear problems [4-111. In particular, a geometrical-resonance-based non-feedback control was introduced in [8], where a weak excitation drives the trajectory to follow an a priori chosen natural solution of the unperturbed Riissler system. Also, a feedback approach to controlling chaotic oscillators by altering their energy is discussed in [12]. The present discussion of the application of geometrical resonance analysis to spatiotemporal systems follows works by Gonztilez and coworkers [7,10,11]. The basics and diverse applications of the adiabatic autoresonance theory are given in [13-161, while the energy-based approach t o autoresonance phenomena appears in [17]. It is worth noting that the autoresonance condition (3.10) represents a feedback autoresonancecontrolling mechanism, which is absent in the aforementioned previous approach to autoresonance phenomena [13-161 where an explicit, coordinate-independent, and adiabatic force is used from the beginning. In this regard, autoresonant control has been previously discussed in the context of vibro-impact systems [18] on the basis of the analysis of nearly sinusoidal self-oscillations [19], where the term self-resonance was introduced to indicate “resonance under the action of a force generated by the motion of the system itself’ (cf. [19], p. 166). Regarding noise effects on nonlinear dynamics, a review of the particular topic of stochastic resonance is given in [20], while a more general perspective, including stochastic ratchets, noise-induced multistability, multimodality, and noise-induced oscillations, is found in the reviews [21,22]. [l] Galilei, G., (1954) Dialogues Concerning Two New Sciences, translated by Henry Crew & Alfonso de Salvio. Dover, New York, pp. 97-99.

[2] Chirikov, B. V., (1959) “Resonance processes in magnetic traps,” At. Energiya 6, pp. 630-638 (in Russian).

[3] Chacbn, R., (1996) “Geometricalresonance as a chaos eliminating mechanism,” Phys. Rev. Lett. 77,pp. 482-485. [4] Chacbn, R., (1996) “Geometrical resonance in a driven symmetric-bistable system subjected to strong-weak damping,” Phys. Rev. E 54, pp. 6153-6159. [5] Chach, R., (1997) “Chaos and geometrical resonance in the damped pendulum sub-

Notes and references

123

jected to periodic pulses,” J . Math. Phys. 38, pp. 1477-1483. [6] Chacbn, R., Shchez, M., and Martinez, J. A., (1997) “Geometrical resonance analysis of chaos suppression in the bichromatically driven van der Pol oscillator,” Phys. Rev. E 56, pp. 1541-1549. [7] Gonziilez, J . A. et al., (1998) “Resonance phenomena of a solitonlike extended object in a bistable potential,” Phys. Rev. Lett. 80, pp. 1361-1364. [8] Tereshko, V. and Shchekinova, E., (1998) “Resonant control of the Rijssler system,” Phys. Rev. E 58, pp. 423-426. [9] Chacbn, R., (2003) “Resonance phenomena in bistable systems,” Int. J. Bifurcation and Chaos 13, pp. 1823-1829. [lo] Gonziilez, J. A. et al., (2003) “Geometrical resonance in spatiotemporal systems,” Europhys. Lett. 64, pp. 743-749. [ll] GonzAlez, J. A. et al., (2004) “Pattern control and suppression of spatiotemporal chaos using geometrical resonance,” Chaos, Solitons and Fractals 22, pp. 693-703. [12] Tereshko, V., Chacbn, R., and Preciado, V., (2004) “Controlling chaotic oscillators by altering their energy,” Phys. Lett. A 320, pp. 408-416. [13] Fajans, J., Gilson, E., and Friedland, L., (1999) “Autoresonant (nonstationary) excitation of the dicotron mode in non-neutral plasmas,” Phys. Rev. Lett. 82, pp. 4444-4447. [14] Nakar, E. and Friedland, L., (1999) “Passage through resonance and autoresonance in xZn-typepotentials,” Phys. Rev. E 60, pp. 5479-5485. [15] Fajans, J., Gilson, E., and Friedland, L., (2001) “The effect of damping on autoresonance (nonstationary) excitation,” Phys. Plasmas 8, pp. 423-427. [16] Fajans, J. and Friedland, L., (2001) “Autoresonant (nonstationary) excitation of pendulums, Plutinos, plasmas, and other nonlinear oscillators,” Am. J. Phys. 69, pp. 1096-1102. [17] Chacbn, R., (2005) “Energy-based theory of autoresonance phenomena: Application to Duffing-like systems,” Europhys. Lett. 70,pp. 56-62. [18] Babitsky, V. I., Astashev, V. K., and Kalashnikov, A. N., (2004) “Autoresonant control of nonlinear mode in ultrasonic transducer for machining applications,” U1trasonic 42, pp. 29-35 [19] Andronov, A. A., Vitt, A. A., and Khaikin, S. E., (1966) Theory of Oscillators, Dover, New York. [20] Gammaitoni, L. et al., (1998) “Stochastic resonance,” Rev. Mod. Phys. 70,pp. 223-287. [21] Landa, P. S. and McClintock, P. V. E., (2000) “Changes in the dynamical behavior of nonlinear systems induced by noise,” Phys. Rep. 323, pp. 1-80. [22] Lindner, B. et al., (2004) “Effects of noise in excitable systems,” Phys. Rep. 392, pp. 321-424.

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Chapter 4 APPLICATIONS: LOW-DIMENSIONAL SYSTEMS The present chapter is devoted to discussing the application of the results demonstrated in Chap. 2 to two physically relevant problems: the control of chaotic escape from a potential well and the suppression of chaos in a driven Josephson junction. Also, the taming of chaotic charged particles in the field of a non-ideal electrostatic wave packet is studied. 4.1

Control of chaotic escape from a potential well

Escape from a potential well is a general and ubiquitous phenomenon in science. One finds it in very distinct contexts: the orbits of a photon near a Schwarzschild black hole, the capsizing of a boat subjected to trains of regular waves, the escape of stars from a stellar system, and the stochastic escape of a trapped ion induced by a resonant laser field, are some notable examples. Remarkably, such complex escape phenomena can often be well described by a low-dimensional system of differential equations (as indeed seems to generally be the case for any set of phenomena arising from complex real-world systems). Here we are concerned with the situations where escape is induced by an external periodic excitation added to the model system, so that, before escape, chaotic transients of unpredictable duration (due to the fractal character of the basin boundary) are usually observed for orbits starting from chaotic generic phase space regions (such as those surrounding separatrices), in both dissipative and Hamiltonian systems. It is worth mentioning that chaotic escape is often undesirable from an engineering standpoint, because erosion of the basin of attraction (of bounded states) would limit the engineering integrity of large amplitude states which could easily be destroyed in a noisy environment. In other words, the performance of a specific nonlinear system with a potential well subjected to a periodic excitation would generally be considered optimal if it operates in a periodic mode (i.e., inside the well). Here, we are concerned with those situations where one cannot make a system operate in a safe domain of parameter space and chaotic escape will be unavoidable. The main properties of (one-way) escape phenomena can be suitably described by a simple oscillator model with a quadratic nonlinearity:

2

+

2 - x2 =

-d(2,i)

+ Fcos ( w t ) ,

(4.1) 125

126

Applications: Low-dimensional systems

where -d(x, i)is a general damping force, and w, F are the normalized parameters of frequency and forcing amplitude, respectively. It is worth noting that the cubic potential corresponding to the driven oscillator (4.1) typically models a metastable system close to a fold. Also, the effects of phase space damping and external harmonic excitation on the synchrotron equation of motion in quasi-isochronous storage rings are reliably modeled by Eq. (4.1), which is also a useful model for the capsizing of a boat subjected to trains of regular waves. It is well known that the erosion of the non-escaping basin generally increases as the forcing amplitude is increased, the remaining parameters being held constant. 4.1.1 Model equations The effect of two different parametric excitations on suppressing chaotic escape will be studied separately and the corresponding results will be compared.

Escape-suppressing parametric excitation of the linear potential term Consider first the effect of a weak parametric excitation of the linear potential term on the escape suppression scenario: . n-1 x + x - x2 = -62 1x1

+ Fcos ( w t ) - qxcos (Rt + 0 ),

(4.2)

where q , R , and 0 are the normalized amplitude, frequency, and initial phase, respectively, of the parametric excitation (0 < q << l), which will have a suppressory effect on the chaotic escape of the remaining system (0 < 6, F << l),and w, F, 6, and n are the normalized parameters of the forcing frequency, forcing amplitude, damping coefficient, and damping exponent, respectively. In this case, the corresponding Melnikov function (2.30) is

M(t0)

=

-6 -q

1: 1:

i: ( t )(io(t)l"-' d t + F

io ( t )cos [w (t + t o ) ]dt

+ + 01dt,

io ( t )20 ( t )cos [R (t t o )

(4.3)

where

3 1 cosht' 3 sinh t (1 cosh t)2'

1O'

=

+

(4.4)

+

is the parametric representation of the separatrix of the underlying conservative system (6 = F = q = 0). After substituting Eq. (4.4) into Eq. (4.3) and computing the resulting integrals with the help of standard tables, Eq. (4.3) can be put into the form M(t0) = D - Asin(wt0) Csin (Rto 0 ), (4.5)

+

+

127

Control of chaotic escape from a potential well

with

D

=

A =

c =

-26

(i)n+l

B

(F, n + 1) ,

67rFw' sinh (T W ) 2T7)R' (1 - 0') sinh (TO) '

(4.6)

where B(ml n ) is the Euler beta function. It is worth mentioning that the criterion for a homoclinic tangency (accurately predicted by Melnikov's method) is coincident with the change from a smooth to an irregular, fractal-like basin boundary. These results connect Melnikov's method predictions with those concerning the erosion of the basin boundary. Escape-suppressing parametric excitation of the quadratic potential term Consider now the effect of a weak parametric excitation of the quadratic potential term on the escape suppression scenario: X

+x

-' X

=

-6X

lXln-l

+ F cos (wt)+ VX'

COS

(Rt

+0)

(4.7) where 0,0 and 7 are the normalized frequency, initial phase, and amplitude, respectively] of the parametric excitation (0 < 7 << l), which will yield an inhibitory effect on the erosion patterns (chaotic escape) of the non-escaping basin of the remaining system (0 < 6, F << l),and w, F, 6, and n have the same meaning as in the Eq. (4.2). The application of Melnikov's method to Eq. (4.7) yields the following Melnikov function (cf. Eq. (2.30)): J-w

J -w

(4.8)

where (zo(t),Xo(t)) is the separatrix (4.4). Substituting Eq. (4.4) into Eq. (4.8), after some algebraic manipulation one obtains

M(to) = D

-

A sin(to)- C sin (Rto

+ 0 ),

(4.9)

with

D

=

A =

c =

-26 (')^+'B

(lln n+2

6~Fw' sinh(7rw)' -3T7 R' (R2 - 1) (522 - 4) 5 sinh(7rR) '

+ 1) ,

(4.10)

128

Applications: Low-dimensional systems

where B(m,n) is the Euler beta function. 4.1.2 Escape suppression theorems Once the respective Melnikov functions have been calculated, one can straightforwardly establish the corresponding suppression theorems by using the results from Chap. 2.

Escape-suppressing parametric excitation of the linear potential term From Eq. (4.6)one sees that C vanishes at R = 1, i.e., for this frequency (the 1 : 1 parametric resonance of the underlying Hamiltonian system ( b = F = = 0 ) ) suppression of chaotic escape is not possible for any set of the remaining parameters. Also C > 0 (C < 0) for 0 < 1 (R > 1). Therefore, the two cases will be studied separately. Theorem V. Let R = p w , p an integer, be such that, for R < 1,

2 m - 1/2 - Q / n = 2n - 1/2

(4.11)

is satisfied for some integers m and n. Then M(t0) (cf. Eq. (4.5))always has the same sign, i.e., M(t0) < O,Vto, if and only if the following condition is fulfilled: Vmin

< rl 6 Vmax,

rlmin

iE

(1-9) JRJ,

(4.12) with

RE

3Fw2 sinh (TR) R2 (1 - R2) sinh ( T W ) ’

(4.13)

and D, A given by Eq. (4.6). Theorem VI. Let R = p w , p an integer, be such that, for R > 1, =

2 m + 1/2 - Q / T 2n - 1/2

(4.14)

is satisfied for some integers m and n. Then M(t0) (cf. Eq. (4.5))always has the same sign, i.e., M ( t o ) < 0, Vto, if and only if the following condition is fulfilled: Vmin

<

v 6 Vmax,

qmin

=

(1

-

rlmax

R

= -

P2 ’

-

7) R,

(4.15)

129

Control of chaotic escape from a potential well

with R and D, A given by Eqs. (4.13) and (4.6), respectively. Remarks. First, note that, for a given set of parameters satisfying the theorems’ hypotheses, as the resonance order p is increased, the allowed interval ]vmin,vmax]for escape suppression shrinks rapidly. The width of this interval is given by

AV

Vmax - ~ m i n

1- p2 p 2 (1- p2w21 p 2 sinh (TW)

- 3 F sinh (p7rw)

[

Figure 4.1 shows A7 versus w for n, 6, F

B (y, n + 1) + 6 ($)”+’3nFw2

1

= const.

.

(4.16)

0.6 0.3 0.0 0

2

0.6

AT

03 0.0 0

2

06 0.3

0.0 0.2

04

08

0.6

1.0

12

0

Figure 4.1: AT/vs w (cf. Eq. (4.16)) for qmaX= 0.8. For fixed v,,, there exists a forbadden frequency range for each resonance that is centred at the frequency w = l/p. Clearly, this is a consequence of the 1 : 1 parametric resonance of the underlying Hamiltonian system. As is well known, its effect becomes more important when dissipation is taken into account in the model equation, i.e., the amplitude of oscillations becomes larger, which should favour incidental escape. From Fig. 4.1 one sees that the width of the forbidden frequency range (gray regions), h f , h d d e n , decreases as the resonance order is increased, as expected from the decreased efficacy of the escape-inducing forcing as w -+ 0 (cf. Eq. (4.6)). Note that there also exists a minimum-range frequency for each resonance. Second, for R < 1 (R > 1) Theorem V (VI) imposes having 0 = 0, = 0,7r/2,7r, 37r/2 (0 ii OOpt= T , 3 ~ / 2 , 0~, / 2 )for p = 4n - 3,471 - 2,4n - 1,4n (n = 1 , 2, 3,...), respectively (cf. Eqs. (4.11) and (4.14)), respectively, and Table 2.1).

130

Applications: Low-dimensional systems

Escape-suppressing parametric excitation of the quadratic potential term Similarly to the preceding case, from Eq. (4.10) one sees that C vanishes at R = 1 , 2 , i.e., for these frequencies (the 1 : 1 and 1 : 2 parametric resonances of the underlying Hamiltonian system) inhibition of chaotic escape is not possible for any set of the remaining parameters. Moreover, C > 0 (C < 0) for R < 1 or 52 > 2 (1 < R < 2). Therefore, the two cases will be considered separately. Theorem VII. Let 0 = p , p an integer, be such that, for R E ]0,1[U]2, m[,

(4.17) is satisfied for some integers m and n. Then M(t0) (cf. Eq. (4.9)) always has the same sign, i.e., M(t0) < O,&, if and only if the following condition is fulfilled:

qmin

=

(1 -

y)

R, (4.18)

with

R=

10Fw2 sinh(7rR) 1)(a2- 4) sinh (TW) ’

R2 ( 0 2 -

(4.19)

and D , A given by Eq. (4.10). Theorem VIII. Let R = p w , p an integer, be such that, for R E ]1,2[,

(4.20) is satisfied for some integers rn and n. Then M ( t 0 ) (cf. Eq. (4.9)) always has the same sign (specifically, M(t0) < 0, Vto) if and only if the following condition is fulfilled:

(4.21) with R and D , A given by Eqs. (4.19) and (4.10), respectively.

Remarks. First, for R E ]0,1[ U 12, cm[ (R E ]I,2[) Theorem VII (VIII) imposes having 0 = OTt = T, 3~/2,0,7r/2 (0 = OTt = 017r/2,7r,37r/2) for p = 4h - 3.4h - 2,4h -

131

Control of chaotic escape from a potential well

1,4h ( h = 1,2,3,...), respectively (cf. Eqs. (4.17)and (4.20),respectively, and Table 2.1). Second, similarly to the former case, for a given set of parameters satisfying the theorems' hypotheses, as the resonance order p is increased, the allowed interval ]qmin,qmaX]for escape suppression shrinks quickly. The width of this interval is given bY

Ad

qmax -

-7min

~ O F Psinh(prw) -~

1 - p2 l(p2w2- 1) (p2w2- 4)1 p2sinh(7rw)

[

(y,

B n + 1) (4.22) 37rFw2

+6

1

Figure 4.2 shows Aq' versus w for F,d = const and several values of the damping exponent.

P=l 0.00

I

I

I

0.5

1.o

I .5

-

0.75

0.50

Aql 0.25 ..

0.0

2.0

0

Figure 4.2: Aq' vs w (cf. Eq. (4.22))for n = 1 (black line) and n = 2. Observe that, for fixed qmax,there exist two forbidden frequency ranges for each resonance that are centered at frequencies w = l/p,2/p,respectively. Classical perturbation theory indicates that this is a consequence of the 1 : 1 , 1 : 2 parametric resonances of the underlying Hamiltonian system. For fixed n, one observes (cf. Fig. 4.2) that the width of the respective forbidden frequency ranges Aw forbiddenJ, b f o r b i d d e n , 2 (SSOCiated with w = l/p,2/p,respectively) decreases as the resonance order p is increased, as expected from the reduced efficacy of the escape-inducing excitation as w --+ 0 (cf. Eq. (4.10)). This last behaviour also explains why Alc'f&&&n,2> AWf&idden,l

132

Applications: Low-dimensional systems

for each resonance and for each value of the damping exponent. There also exist minimum-range frequencies for each resonance, as in the case of the escapesuppressing parametric excitation of the linear potential term. Third, in order to compare the quadratic and linear cases, one takes the same values of the parameters in both cases. One then obtains

(4.23) (cf. Eqs. (4.16) and (4.22)). Figure 4.3 shows the relative width (Aq’/Aq) ( w ) versus w for several values of p.

0.0

1.0

0.5

1.5

2.0

2.5

3.0

0

Figure 4.3: Relative width Aq’/Aq vs w (cf. Eq. (4.23)). With fixed p , the asymptotic behavior [Aq’/Aq](w 0 ) = 5/6 means that a linear parametric excitation inhibits chaotic escape more easily (i.e., for a larger interval of amplitudes) than a quadratic parametric excitation for larger driving periods, i.e., for resonances with orbits (of the underlying conservative system 6 = 77 = F = 0, cf. Eqs. (4.2) and (4.7)) very close to the separatrix (for which T = m). 4.1.3 Inhibition of the erosion of non-escaping basins In order to test numerically the theoretical predictions of the previous section, consider, for instance, the case of an escapesuppressing parametric excitation of the quadratic potential term (4.7). For such a universal escape model, the initial conditions will determine, for a fixed set of its parameters, whether the system escapes to an attractor at infinity (with x m as t ---t co),or settles into a bounded oscillation. It is well known that there can exist a rapid and dramatic erosion of the safe basin (union of the basins of the bounded states) due to encroachment by the basin of the attractor at infinity (escaping basin). It will be shown in the following how the safe basin is restored under the conditions established by Theorem VIII. To generate numerically the basins of attraction, one selects a grid of (uniformly distributed) 300 x 300 starting points in the region of phase space { x ( t = 0) E [-0.7,1.3] ,i ( t = 0 ) E [-0.8,0.7]}. From this grid of initial conditions, each integration is continued until x exceeds 20, ---f

--f

133

Control of chaotic escape from a potential well

at which point the system is deemed to have escaped (i.e., to the attractor at idinity), or the maximum allowable number of cycles (of the chaos-inducing excitation), here 20, is reached. The colour white represents the non-escaping basin, and black the escaping basin. In the absence of an escapesuppressing excitation ( q = 0), one assumes that the system presents a dramatic erosion and stratification of the basin (as in the instance shown in Fig. 4.4 for b = 0.1, n = 2, F = 0.05, and w = 0.85).

X

Figure 4.4: Erosion of the safe basin (see the text).

For these parameters one obtains, for the main resonance R = w , qmin= 0.223323, qmax= 0.549748, 0 = = 7r from Eqs. (4.17) and (4.18), and 0 = 0;: = 0 from Eq. (2.92). Figure 4.5 shows a basin restoration/deterioration sequence for three values of 0 increasing from 7r/2 to 3x12 and three values of q increasing from 0.223323 to 0.549748 containing the above control interval. One sees that the restoration of the safe boundary is optimal just when the theoretical predictions are met (note, however, that a thin finger-like projection still protrudes into the basin at the lower threshold, qmin,cf. version ( b ) ) . For 0 = 3n/2 the erosion of the basin is greater for any amplitude q than in the initial situation q = 0, in agreement with the theoretical prediction. The initial phase 0 therefore plays a fundamental role in the control (inhibition and enhancement) of chaotic escape.

@z

4.1.4

Role of nonlinear dissipation

The purpose of the present section is to compare the escape control results for a linear damping term with those associated with a nonlinear damping term. Two different types of nonlinear damping term will be considered separately.

134

Applications: Low-dimensional systems

(a) 0 = n / 2, q = 0.223323

(b) 0 = n, I-= 0.223323

(c) 0 = 3n /2,q = 0.223323

(d) 0 = R / 2, q = 0.3

(e)0 = n , q = 0.3

(0 0 = 3rr / 2 , q = 0.3

(9)0 = n / 2, q = 0.549748

(h) 0 = n, 11= 0.549748

(i) 0 = 3r / 2, q = 0.549748

Figure 4.5: Basin restoration sequence (see the text).

Lienard nonlinear damping term

Let us consider, for instance, the following universal escape equation with an escapesuppressing parametric excitation of the linear potential term: X

+ z - X’ = - ( ~ 1 , ’ + ~

2 5 X ~-) vzcos

+

(Ot + 0 ) F cos ( w t )

(4.24)

in comparison with the model (4.2) with a linear damping term (n= 1). Proceeding similarly to that case, the Melnikov function corresponding to Eq. (4.24) is Y

(4.25)

with N

(4.26) and A , C given by Eq. (4.6). As Eqs. (4.5) and (4.25) become identical with the N

substitution D -+ D , one obtains identical escape suppression theorems t o Theorems

135

Control of chaotic escape from a potential well

V and VI but with (4.27) instead of qmin (cf. Eqs. (4.12) and (4.15)) and R given by Eq. (4.13). In order to compare the linear (n = 1, cf. Eq. (4.2)) and nonlinear cases, one sets 6 = v1 = vp and defines Aq (F,vl,v2,w,p) = qmax- qmin.One then obtains (cf. Eqs. (4.12) and (4.27)) N

N

A77 (46,w, P, n = 1) A< (F,v1 = 6, u2 = 6 , w, p )

;+%(F,W,P,b)

$, + i$+ & 92 (F,W , p , 6) '

(4.28)

with % (F,w, P, 6)=

67rFw2 (p2 - 1) 6p2sinh(7rw)

(4.29)

'

Clearly, Eq. (4.28) indicates that Aq > A./ for any set of the parameters, i.e., the effectiveness of the parametric excitation in the presence of the nonlinear damping term -6 (z2 z4)x is less than that for a linear damping term -62.

+

Nonlinear damping term proportional to the nth power of the velocity Consider now the universal escape equation with an escapesuppressing parametric excitation of the quadratic potential term given by Eq. (4.7) and compare the cases of nonlinear ( n > 1) and linear ( n = 1) damping. One obtains (cf. Eq. (4.22))

Aq(n = 1) 1/5 - P (F, w, P, 6) Q(n) Aq ( n > 1) (3n/2n+1)B ( ( n 2) /2, n 1) - p ( F ,w,p, 6)

+

+

(4.30)

with 7rFw2 (p2 -

1)

P (F,w, P , 6 ) = 6p2sinh(7rw)

(4.31)

For any set of parameters F,w , p , 6 such that Aq ( n = 1) > 0 one obtains Q(n) 2 l , h ,i.e., the effectivenessof the parametric excitation in the presence of a nonlinear strictly dissipative force -6x Iiln-lis less than for a linear dissipative force -hi. As the nth power of the velocity is increased, the allowed interval A77 (n)shrinks quickly, as can be appreciated in Fig. 4.6 where the function Q(n) (cf. Eq. (4.30)) is plotted versus n for p = 1 , 6 = 0.06, F = 0.096, and w = 0.61.

136

Applications: Low-dimensional systems

1

2

3

4

5

n Figure 4.6: Function Q(n) (cf. Eq. (4.30)) for p = 1. Note that the solid line represents the eponential fit 0.70969 exp (n/1.60194) 0.33251.

4.1.5 Robustness of chaotic escape control In order to illustrate the robustness of the control (reduction and enhancement) of chaotic escape by weak parametric excitations against external noisy perturbations, consider, for instance, the following universal escape equation with a Newtonian drag force: (4.32) X 2 - z2 = -6X 1 51 ~ / Z ~ C O(Rt S 0 ) F f c ( t ) fnoise(t),

+

+

+ +

+

is a random force, and 6 = 0.1,w = where fc(t) is a chaos-inducing excitation, fnoise(t) 0.85. Consider first the case of a harmonic chaos-inducing excitation f c ( t )= cos ( w t ) ( F = 0.05, R = w ) in the absence of any random force (fnoise(t)= 0). Figure 4.7 shows the corresponding escape probability normalized to that of the respective case with no escapesuppressing excitation, P(q)/P(r/= 0), versus the initial phase 0 for several values of q : (m) q = 0.07, (0) 77 = qmin= 0.223323, (0)77 = 0.3, (A) q = vmax= 0.549748, (*) q = 0.7. The theoretical predictions are q E ]0.223323,0.549748],02 = 7r (0%;= 0) for the reduction (enhancement) of chaotic escape (cf. Eqs. (4.18), (4.17), and (2.92), respectively). The asymptotic behavior of the normalized escape probability as 0 + 0;; indicates that the non-escaping basin has been (almost) completely destroyed. Observe that the maximum reduction of chaotic escape occurs just over the theoretically predicted amplitude range, and that this reduction is uniform for every amplitude 77 belonging to the theoretical interval. Let us assume now an aperiodic chaos-inducing excitation f c ( t )= sin y ( t )with

Control of chaotic escape from a potential well

137

y(t) a chaotic response from the system

i + siny = -0.ly

+ 2cos ( 2 w t ) ,

no random force (fn0ise = 0) , and the main resonance case R

(4.33) = w.

1.2

--.

h

F 0.8

v

nl

0.6 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

O/n Figure 4.7 Normalized escape probability vs initial phase (see the text).

1.2 h

0

. F

v

0.8

e,

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Oln

Figure 4.8: Normalized escape probability vs initial phase (see the text). Here F

=

0.1 to have an initial escape situation similar to that corresponding to

138

Applications: Low-dimensional systems

Fig. 4.7. It must be stressed that the chaotic signal y ( t ) exhibits a power spectrum with a strong peak at frequency w. Naively, one would expect that P ( q ) / P ( q= 0) is insensitive to 0 , provided that f c ( t )has no definite phase in this case. However, the existence of a sharp Fourier component, with a sufficiently high power, seems to be enough to permit the resonant control excitation to reliably act as an enhancer or inhibitor excitation. It should be emphasized that this represents a new aspect of the robustness in the suppression and mainteIiance by weak harmonic excitations, which extends the robustness against external noise. Figure 4.8 shows the corresponding normalized probability versus the initial phase for the same values of q as in Fig. 4.7. Clearly, comparison of Figs. 4.7 and 4.8 indicates that the suppressory effectiveness of the second excitation is less for an aperiodic escapeinducing excitation than for a periodic one, as expected. To study the robustness against external noise, consider now the effect of a harmonic excitation, f c ( t ) = cos(wt), in the presence of (additive) noise fnoise(t)= sin@( t ) with ( t ) a Gaussian random phase, and the remaining parameters being F = 0.05, R = w. Figure 4.9 shows the respective findings for the same values of the amplitude q as in Figs. 4.7 and 4.8.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0l.n Figure 4.9: Normalized escape probability vs initial phase (see the text). Comparison of Figs. 4.9 and 4.7, 4.8 clearly indicates that the suppressory effectiveness of the second excitation is greater for a periodic escape-inducing excitation in the presence of random noise than for such excitation with no random noise or for an aperiodic escapeinducing excitation in the absence of random noise, respectively, which is in agreement with the results of Chap. 2. This noise-induced effect is especially evident in the almost null enhancement of chaotic escape around 0;;; = 0. Note that such a suppressory effect on chaotic escape, which is induced by adding a certain amount of noise to the system, is restrictive for the range of the initial phase in which it occurs in comparison with the case with no random noise, as can be appreciated by comparison of Figs. 4.7 and 4.9. However, one sees that the reduction

Control of chaotic escape from a potential well

139

of chaotic escape over a narrow range of the initial phase centred around 0% = T , and for certain values of the amplitude q (viz. q = qmax= 0.549748 and q = 0.3), is greater in the presence of noise than in its absence. This reveals in general a new aspect of the hyper-sensitivity of the control scenario t o the initial phase of the chaos-controlling excitation. It is worth mentioning that there exist other general phenomena where noise preserves or enhances certain types of ordered motion. Indeed, the noise-induced synchronization in an extended array of nonlinear elements, the noiseimproved ability of some nonlinear systems to transfer information reliably via the stochastic resonance phenomenon, the noise-induced stabilization of bumps (position of a localized region of active neurons) in systems with long-range spatial coupling, and the emergence of regular spatio-temporal patterns in arrays of chaotic coupled oscillators by adding a certain amount of noise to each of the oscillators, are some illustrative examples. 4.1.6 Case of incommensurate escape-suppressing excitations The reduction of the erosion of non-escaping basins by incommensurate excitations applied to (the escape model given by) the Helmholtz oscillator (2.112) will be discussed in the present section. To generate the basins of attraction numerically in this case, a grid of uniformly distributed 300 x 300 initial conditions in the region of the phase space { z ( t= 0) E [0,1.8],k ( t = 0) E [-0.8,0.8]} is selected. From this grid of starting points, each integration is continued until either x exceeds 20, at which point the system is deemed to have escaped (i.e., to the attractor at infinity), or the maximum allowable number of cycles, here 20, is reached. In the absence of an escapesuppressing excitation ( q = 0), one has that the system presents a dramatic erosion and stratification of the basin for the parameters p = 1 , d = 0.1, F = 0.08, w = 0.85,n = 1. For these parameters, one calculates, for each resonance p / q , the normalized escape probability, P ( q ) / P ( q = 0), versus the initial phase 0 for several values of q. Typically one finds that no reduction of initial escape is attained for q > qminand arbitrary initial phase, with qmingiven by Eqs. (2.120) and (2.121). For small amplitudes ( q 5 qmin),the probability P ( q > 0) becomes lower than P ( q = 0) over short ranges of 0 which are typically centred on (some of) the predicted suitable initial phases (in the sense of the first remark to Theorem IV) for each resonance p / q . As an instance, Fig. 4.10 shows the normalized escape probability versus the initial phase for the resonance p / q = 314 and the amplitudes q = (0.025 ( A ) ,0.015 (A),0.010 (*) ,0.008 (U)}, and where dotted vertical lines indicate the predicted suitable initial phases. The theoretical predictions are qmin M 0.0242 and @suitable = { 0 , ~ / 2 , ~ , 3 ~ /(cf. 2 } Eqs. (2.116) and (2.122)). One sees that the normalized probability presents minima, as a function of the initial phase, that come progressively closer to the predicted suitable values as the amplitude decreases. The reduction of chaotic escape is achieved for q = 0.008 over a range centred on the initial phase 0 = @&table = 3 ~ 1 2 . A further instance is shown in Fig. 4.11 for the resonance p/q = 819 and q = (0.08 ( A ) ,0.03 (A),0.022 (*)}. In this case, the theoretical predictions are qminM

140

Applications: Low-dimensional systems

0.0217 and = { ~ / 9~, / 3 , 5 ~ / 9 , 7 ~ 11~/9,13~/9,15~/9,17~/9}. /9,~, For 71 = 0.08 the normalized probability reaches its maximum value (-J 1.34) for any initial phase, which represents the situation where the non-escaping basin has been completely destroyed (for the resolution considered here). For 71 < 0.08 the normalized probability solely presents minima at the values predicted above. One sees that for 7 = 0.022 ( s qmin)the reduction of chaotic escape is achieved over two ranges of the initial phase centred on each of the suitable initial phases { 7 ~ / 9T, } .

0.9 I 0

I

I

I

1

2

3

I 4

2$/x Figure 4.10: Normalized escape probability vs initial phase.

0

2

4

6

8

10

12

14

16

18

941 z Figure 4.11: Normalized escape probability vs initial phase.

141

Control of chaotic escape from a potential well

Consider now the case of incommensurate escape-suppressingexcitations. The only "off-resonance" excitations that can be numerically considered are those with irrational frequencies to the limits of computational precision. As in the previous instances of arbitrary resonances p / q , one finds (for sufficiently small amplitudes ( q 5 qmin))that the overall behaviour of the normalized escape probability presents a minimum near 0 = Osvatdle= 7r (see, e.g., Fig. 4.11), as expected (cf. the second remark to Eq. (2.122)). 1.035

I

0

:

,

2

(a) p/q = 315

. ,

I

4

,

6

.

,

.

8

,

10

5+/x

. . .

I *

(c) p/q = 0.618033988749894

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4/2n

Figure 4.12: Normalized escape probability vs initial phase.

142

Applications: Low-dimensional systems

Figure 4.12 shows the normalized escape probability versus the initial phase for the amplitude q = 0.010 and three values of p / q : the two convergents 315,8113, and 0.618033988749, this last being @ (golden section) to the limit of computational precision considered in the simulations. The theoretical predictions for the threshold amplitude are qminM 0.02857,0.02800, and 0.02791, respectively (cf. Eqs. (2.124) and (2.125)), which are very close to the predictions (cf. the second remark to Eq. (2.124)).

558 I R

558 I R

1.029 1.022 1.016

(c) q = 0.010

It

1.009 1.003 .5 0.996 a 0.989 v

2

h

0

55 5541I x

110

Figure 4.13: Normalized escape probability vs initial phase.

143

Control of chaotic escape from a potential well

In Fig. 4.12 (a), there are some clear-cut additional minima (which are not associated with any of the suitable initial phases = {n/5,37r/5,7r, 77r/5,97r/5}) that are absent from the two lower plots in Fig. 4.11. Typically, the number of such additional “off-prediction” minima increases with increasing convergent order, as can be seen for p / q = 8/13 in Fig. 4.12 (b). This confirms (see also Fig. 4.13 for additional instances) the prediction in the third remark to Eq. (2.126). It is also observed that, in general, the plot of the normalized escape probability is asymmetric with respect to the particular suitable initial phase @suitable = 7r: the ranges where reduction of chaotic escape occurs are larger for 0 < 7r than for 0 > 7r (see Figs. 4.12 and 4.13). Figure 4.13 gives plots of the normalized escape probability versus the initial phase for the convergent p / q = 34/55 and the amplitudes q = {0.020,0.015,0.010}in versions (a), (b), (c), respectively. One typically finds that the overall behaviour of P ( q ) / P ( q= 0) versus 0 for a fixed amplitude q (and also the remaining parameters fixed) is quite similar for all the successive convergents beginning with the third convergent or so, as can be appreciated by comparison of Figs. 4.12(b), 4.12(c), and 4.13(c). This is coherent with the insensitivity of vminwith respect to the convergent order (cf. the second remark to Eq. (2.124)). Figure 4.13(b) (and to a lesser degree Fig. 4.13(c)) shows that the overall behaviour of the normalized escape probability presents a few dominant minima at certain effective initial phases effective (cf. the third remark to Eq. (2.126)), which cannot be explained solely on the basis of the predicted suitable initial phases @suitable = ((2n - 1)7r/55, n = 1,..., 55).

p/q = { I h , 112,213,

7 6 2 5 .g

2

13/21,21/34, 34/55}

Figure 4.14: Density of suitable initial phases (see the text). In order to test the conjecture about the origin of the effective initial phases (cf. the third remark to Eq. (2.126)), one calculates the histogram shown in Fig. 4.14, which is a plot of the density of suitable initial phases associated with the complete series of convergents of 9 up to the ninth order p / q = f, f , $, $, versus

(i,

3,5,%, g}

144

Applications: Low-dimensional systems

the initial phase. One observes three remarkable properties. First, the histogram is asymmetric with respect to the central value 0 = = K : the number of maxima of the density function is larger for 0 < K than for 0 > 7r. This is coherent with the aforementioned asymmetry of the distribution of ranges where a reduction of chaotic escape occurs. Second, the density function has its main maxima (density 2 3) at and only at initial phases that roughly coincide with the aforementioned effective initial phases (compare Figs. 4.13(c) and 4.14). Third, the density function has its highest maximum at the particular suitable initial phase =K as predicted (cf. the first remark to Eq. (2.126)). Such a maximum is the most isolated in the sense that the range of initial phases around K with null density is the largest. This is coherent with the fact that the reduction of chaotic escape expected at Osuitable = 7r is hardly observable, it being sharply localized when in fact it does occur (see, e.g., Figs. 4.12(b) and 4.12(c)). It is expected that the mechanism discussed for the reduction of chaotic escape by incommensurate escape-suppressing excitations will work for amplitudes sufficiently lower than the theoretically predicted threshold amplitudes. It should be emphasized that the present theoretical approach as well as the conclusion that chaos can be reduced (or even completely suppressed) by incommensurate chaos-suppressing excitations are both general enough to be applied to many other dissipative non-autonomous systems. 4.2

Taming chaos in a driven Josephson junction

Josephson junctions are superconducting devices that have great technological promise as detectors (they can detect electric potentials as small as a femtovolt), fast switching devices for digital circuits, and amplifiers, to cite some of their main applications. The physics of a Josephson junction only can be understood by means of quantum mechanics but their dynamics can be described in classical terms: the equation governing a single junction is the same a s that for a pendulum. Indeed, a typical Josephson junction can be described by McCumber’s model which is equivalent (after a suitable change of variables) to the dynamic equation of a linearly damped pendulum subjected to a periodic driving. A more exact description of the junction’s dynamics, when subjected to radio-frequency currents, is achieved by including a term which represents the interference effects between the quasiparticle-pair and quasiparticle currents. 4.2.1 Model equation McCumber’s model for a two-current-fed Josephson junction taking into account the interference effects has the following equivalent circuit equation,

hC d2x h dx (1 y cos x) 41red.r~ k e & dr

--

+-

+

+ Icsin x = I0 sin (u,.fr.)+ PI0 sin (uifr+ O) , (4.34)

145

Taming chaos in a driven Josephson junction

where x denotes the total phase difference across the junction, T the time, e the electron charge, h Planck’s constant, C the junction capacitor, I, the critical supercurrent, & the junction resistor corresponding to I, = 0, y the coherence coefficient which is related to the frequencies, w T f ,w i f , and amplitudes, Io,pI0, of the external excitations, and 0 an initial phase. Equation (4.34) can be put into the dimensionless form

2 + sinz = -a (1 + y cosz) Z + F sin ( w t ) + PF sin (Rt + 0 ),

(4.35)

with

w

=

WTf

(””)

47reIc

112

’

47reI,

(4.36)

Note that i(= d z / d t ) is proportional to the difference of potential between the two superconductors. In the following the dissipation and excitation t e r m are regarded as weak perturbations (0 < a << 1,layl << 1 , 0 < F << 1 , 0 < p < 1) and PF sin (Rt + 0 ) is the chaos-suppressing excitation. 4.22 Suppression of homoclinic bifurcations The application of the Melnikov’s method to Eq. (4.35) involves calculating the ‘-’-iko- functions,

M i( t o ) = -a

1: +

[1 y cos X O , i ( t ) ] Z&(t)dt

00

+ io,+(t) sin IQ (t + t o ) + 01d t ,

+F 1,50,i(t) sin [w (t t o ) ] d t

+PF

/

00

-m

(4.37)

where the negative (positive) sign refers to the bottom (top) homoclinic orbit of the underlying integrable pendulum (cf. Eq. (2.33)). After substituting Eq. (2.33) into Eq. (4.37) and computing the resulting integrals by using standard integral tables, Eq. (4.37) can be written as

&(to)

=D

k A sin(wto)& C sin(Rt0 + O),

(4.38)

146

Applications: Low-dimensional systems

with

(4.39) Let us assume that for ,B = 0 the Josephson junction (4.35) exhibits chaotic behaviour such that the respective Melnikov function M t (to) = D f A sin(wto)has simple zeros, i.e., A+D = d 3 0, where the equals sign corresponds to the case of tangency between the stable and unstable manifolds. If one now lets the chaos-suppressing excitation act on the junction (p # 0) in such a way that C > d , i.e., A - C D < 0, this relationship provides a necessary condition for M* (to) not to change sign, i.e., M* (to) < 0, which is

+

(4.40) with

RE

y)

cosh ( cosh ( 7 )’

(4.41)

<

Clearly, for general Q and 0 (0 0 6 an),this condition is not sufficient to assure the negativity of M* (to). To obtain such a sufficient condition, we shall first need four lemmas. For the sake of clarity, the cases M* ( t o ) will be considered separately. Motion near the upper homoclinic orbit In this case one needs two lemmas. Lemma X. Let !2/w be irrational. Then there is some tg such that

+ E ) . Then taking tc =

A sin (uti) C sin (Rti + 0 ) > A - C. Proof. Let 0 =

(3 -

since w and Sl are incommensurate, i.e., cos (2.:) Then taking tc = 5 one obtains Asin(wti) since we have 0 +

one obtains

< 1. Let now 0 #

(2 ”2”

+ Csin(QZtG + 0 ) = A + C c o s - + - + 0) > A

2#

in this case.

-

( 3 - f).

C,

147

Taming chaos in a driven Josephson junction

Lemma XI. Let f l / w = p / q for some positive integers p and q. Then there exists satisfying sin(wt7) = - sin(Rtc* 0) = 1 if and only if = 4nf3-28 4n+l ?r for some integers m and n. Proof. If sin(wtc*) = 1 then there exists some integer n such that w t r = (4n + 1) 7r/2. Also, if sin(fltr + 0) = -1 then there exists some integer m such that Rt; 0 = (4m 3)7r/2. Solving for tG* in both relationships, one obtains

tr

+

+

+

P , 4

4m

+3 -2 0 / ~ 4n + 1

(4.42)

Clearly, for Eq. (4.40) to also be a sufficient condition for M + ( t o ) to be negative for all t o , one must have

+

A sin(wt0) C sin (ato+ 0 ) < A - C.

(4.43)

One now looks for the values of w,s2 and 0 permitting Eq. (4.43) to be satisfied for all to. From Lemma X, a resonance condition is required: R = p w / q . In such a situation, Lemma XI gives a condition for Eq. (4.43) to be fulfilled for an infinity of to values. Thus, let us assume that p , q and 0 satisfy Eq. (4.42). One can then rewrite Eq. (4.43) in the form

A 1 - cos ( p ~ / q ) -C2 1 - COS7 ’

(4.44)

+

with T = wto - (4n 1) n/2. Finally, for a subharmonic resonance (q = I), Lemma I11 provides a condition for Eq. (4.44) to be fulfilled for all T :

R PG?, P

(4.45)

where R is given by Eq. (4.41). Motion near the lower homoclinic orbit For this case one requires two additional lemmas. Lemma XII. Let R/w be irrational. Then there is some ti such that

+

-Asin(wti) - Csin(Rt(; 0) > A - C. Proof. Let 0 =

4 (1 + $) . Then taking t; =

-Asin(wti) - Csin(Rtg

+ 0) = A - Ccos

one obtains

(s)

>A-C,

148

Applications: Low-dimensional systems

since $2 and w are incommensurate, i.e., cos (2.:) Then taking t; = -& one obtains -Asin (wt;)

-

Csin (at;+ 0 ) = A

< 1. Let now 0 #

+ Ccos (:-

-

::+0)

4 (1 + E).

>A-C,

4

since we have 0 # in this case. w Lemma XIII. Let R/w = p / q for some positive integers p and q. Then a t r exists such that - sin(wtr) = sin(Rtr 0 ) = 1 if and only if 29 = 4mf1-2e 4n+3 for some integers m and n. Proof. If sin(wtr) = -1 then there exists some integer n such that w t r = (4n 3 ) ~ / 2 .Also, if sin(Rtr 0 ) = 1 then there exists some integer m such that R t r 0 = (4n 1)7r/2. Solving for t;;*in both relationships, one obtains

+

+ +

+

+

p - 4m+1-20/7r -9 4n 3

(4.46)

+

w

In the present case, for Eq. (4.40) to be also a sufficient condition for M - ( t o ) to be negative for all to, one must have

+

-A sin(wt0) - C sin($2to 0 ) 6 A - C.

(4.47)

One then seeks the values of w, 52, and 0 permitting Eq. (4.47) to be satisfied for all to. From Lemma XII, the resonance condition is again required: R = p w / q . With that, Lemma XI11 gives a condition for Eq. (4.47) to be verified for an infinity of to values. Let us assume that p , q , and 0 satisfy Eq. (4.46). It is then possible to rewrite Eq. (4.47) as Eq. (4.44) with T = wto - (4n 3)7r/2. Lastly, with q = 1, Lemma I11 provides the condition (4.45) for Eq. (4.44) to be satisfied for all T .

+

Suppression theorem

In brief, one has the following theorem for subharmonic resonances. Theorem IX. Let R = pw, p an integer, such that, for M+(to) (M-(to)), the relationship p = (4m 3 - 20/7r) / (4n 1) ( p = (4m 1 - 20/7r) / (4n 3)) is satisfied for some integers m and n. Then M*(to) always has the same sign, specifically M* ( t o ) < 0, if and only if the following condition is fulfilled:

+

+

+

+

(4.48)

149

Taming chaos in a driven Josephson junction

Remarks. First, one can test Theorem IX theoretically by considering the limiting Hamiltonian case ( a = 0). Note that, in the absence of dissipation, Eq. (4.48) must be rewritten as p,, 3 p 3 pmin,,Brnin = R,P,,, = R / p 2 , since Pmincannot now be zero. Thus, by using lemmas XI and XI11 one obtains R = w, 0 = 0,, = n-,/3 = 1 (cf. Eq. (4.41)) as a necessary and sufficient condition for suppressing stochasticity. (This finding can be trivially obtained, to first perturbative order, from Eq. (4.37) with LY = 0, i.e., with M*(to) = 0, Vto.) Second, note that Theorem IX requires having 0 = 0,, = n-,n-/2,0,3n-/2 (n-, 3n-/2,0,n-/2) for p = 4n - 3,4n - 2,4n - 1,4n (n = 1,2,3,...), respectively, in order to tame chaos when one considers orbits initiated near the upper (lower) homoclinic orbit, as expected (cf. Table 2.1). Thus, one obtains distinct suppressory values O, (at the same resonance order) for the two homoclinic orbits. These different values are those compatible with the surviving natural symmetry under the chaos-suppressing excitation, according to the general discussion in Chap. 3. Indeed, the dissipative, harmonically driven Josephson junction ( p = 0) is invariant under the transformation

x

t

+ +

-x,

(4.49)

n-

t+(2n+l)-,

W

where n is an integer, i.e., if [x( t ),i ( t ) ]is a solution of Eq. (4.35) with p = 0, then so is [-z(t (2n l)n/w), - i ( t (271 l)n-/w]. Observe that these two solutions may be essentially the same, i.e., they may differ by an integer number of complete cycles, so that nx ( t ) = -x (2n 1) 2n-m, (4.50)

+

+

+

+

[t + +

-3

W

+

with m an integer, and they are termed symmetric. If that is not the case, the sign reversal and time-shifting procedure yields a second solution of the junction, and the two solutions are termed broken-symmetric. When p # 0 and 0 is arbitrary the aforementioned natural symmetry is generally broken. It is straightforward to see that the reason for that breaking is sin (Rt

+ 0 ) # - sin [Rt + 0 + (271 + 1)n-R/w] ,

(4.51)

for arbitrary w, R, and 0. Assuming a resonance condition R = pw, the survival of the above symmetry implies sin(pwt

+ 0 )= (-l)P+'sin

(pwt + 0 ).

(4.52)

Clearly, this is only the case for p an odd integer. For p an even integer, one has the new transformation

x t 0

---f

-x,

+

t+(2n+l)L,

---f

Ofn-.

W

(4.53)

150

Applications: Low-dimensional systems

In other words, if [ z ( t )i,( t ) ]is a solution for a value 0 , then so is

[-z(t + ( 2 n + l)7r/w, - i ( t

+ ( 2 n + 1)7r/w]

for 0 & 7r. Thus, this explains the origin of the differences between the corresponding (at the same resonance order) allowed @,t values (from Theorem IX) for the two homoclinic orbits, as pointed out in the second remark to Theorem IX. Similar results were discussed for a damped, driven pendulum in Chap. 2.

Suitable initial phase difference intervals According to Eq. (2.72), the maximum deviation from Oqt, A@,,,, such that there still exists a value of p (pmm> p > Pmin)for which Mi ( t o ) < O,Vto, is written for the Josephson junction as

A@,,,

= arcsin

(T)

= arcsin

(cf. Eq. (4.39)). Plots of A@,, depicted in Fig. 4.15.

[

4a

+ y/3)

7rF

I,-.(

cash -

(4.54)

versus w for two values of cr(1 + y / 3 ) / F are

(u

+

vs w for cr (1 y/3) /F = 0.1,O.Ol.

Figure 4.15: Function A@,,,

One sees that the function AOmax(w)displays a monotonous increasing behaviour from its minimum value arcsin [4a(1 y/3) / (7rF)]at w = 0. For each choice of the remaining parameters, the largest possible frequency w1 satisfies the condition

+

wl

2

= - arccosh 7r

7rF

(1 + y/3)I

which reflects the situation of tangency between manifolds (101= A ) .

(4.55)

151

Taming chaos in a driven Josephson junction

4.2.3 Comparison with Lyapunov exponent calculations Computer simulations of the biharmonically driven Josephson junction (4.35) showed good overall agreement between the theoretical predictions and the numerical results. Correctly such a comparison must take into account that the Melnikov's method predictions (and hence those from Theorem IX) are concerned with homoclinic (and heteroclinic) bifurcations and horseshoes. It is well-known that the existence of a horseshoe does not imply that the typical trajectories will be asymptotically chaotic. The asymptotically chaotic behaviour is characterized by a positive leading (maximal) Lyapunov exponent. This means that in most cases Melnikov's method provides an approximate threshold for transient chaos but not necessarily for steady chaos. The following fixed parameters were selected in all the numerical experiments: F = 0.8, Q = 0.4, y = (-0.8, -0.95}, and the chaos-inducing frequency in the range w E ]0,1.2]. These values were chosen since they are representative values for a Josephson junction, even though they do not reasonably fit the Melnikov's method requirements of smallness. In order to appreciate to some extent the global regularization of the dynamics as the parameter p (@) is varied, global bifurcation diagrams were determined corresponding to the variable x and the leading Lyapunov exponent X for several values of 0 (p). 2 n

I

0.0

0.2

....................... -2 !Ill , ........... I

I

%

0

\

-2

I

I

0.6

II

I

I

I

............,........l.........~.~

...................1............................

;,:, .

. I

0.4

I

0.8

I

0.2

--

(b)

-

(c)

L

.....................

-

I

I

I

L

I

I

I

-I : (4

....................................................................................

0.0

1.O

I

I

I

L

0.4

0.6

0.8

1.o

P Figure 4.16: ( a ) 0 = 1, ( b ) @ = 2.36455, (c) @

= n, ( d )

0 = 3.91866, ( e ) @ = 5.

152

Applications: Low-dimensional systems

As a first example for the main resonance, Fig. 4.16 shows diagrams constructed by means of a Poincare map at F = 0.8, a = 0.4, y = -0.95, w = f2 = 0.6725 and five values of 0 . The corresponding diagrams showing the leading Lyapunov exponent are given in Fig. 4.17, respectively. Starting at ,B = 0, and taking the transient time as 500 (5000) excitation periods T, (=w / (27r)) after every increment of Ap = 0.01, one samples 20 excitation periods by picking up the first x (A) value of every excitation cycle. The same initial conditions (x (0) = 0, x (0) = i0) are set for every new ,B after A@ is added. For the above parameters one obtains 0,, = T , Oqt - A@,, 1: 2.36455, Oqt + A@,,, II 3.91863 from the second remark to Theorem IX and Eqs. (4.39) and (4.54).

.:

. ...

t

....

?

I

...

.....

I

...........

.. ,

'LE-

I

I

0.0

0.2

0.4

0.6

0.8

1.o

0.0

0.2

0.4

0.6

0.8

1.0

FI.

I

I

I

I

0.2

0.4

0.6

-0.2

0.0

0.8

I .o

I

....

..........

-0.2

'

0.0

......

I

I

0.2

0.4

......

0.6

I

L

0.8

1.o

P Figure 4.17: ( a ) 0 = 1, (b) 0 = 2.36455, (c) 0 = 7 r , ( d ) 0 = 3.91863, ( e ) 0 = 5. Thus, for 0 = T , {2.36455,3.91863} the interval of suitable values of p for suppression of chaos is approximately 10.3,l.l ,]0.42,0.71], respectively (cf. Eqs. (4.48) and (2.84) with D+ = D, respectively.) Figures 4.16(a), 4.17(a) and 4.16(e), 4.17(e) show the diagrams associated with the values 0 = 1 and 0 = 5, respectively, for which inhibition of homoclinic bifurcation in not expected. However, after removing chaotic transients, periodic response is obtained within an interval of values of ,B roughly included in the maximum-range permitted interval ]0.42,0.71]. From Figs. 4.16(b)-(d), 4.17(b)-(d) one sees that the numerically obtained ranges of suppression of chaos for 0 = 2.36455,O = 3.141593 z T and 0 = 3.911863, respectively, are larger than

153

Taming chaos in a driven Josephson junction

those predicted theoretically due to the chaotic transients being eliminated before constructing the bifurcation diagrams. As a second example for the main resonance, Figs. 4.18 and 4.19 show the corresponding diagrams (i vs 0 and X vs 0 , respectively) for the parameters F = 0 . 8 , ~=~ 0 . 4 , ~= - 0 . 8 , ~ = R = 0.675, and three values of ,B. Now one has OWt = 7r and the theoretical range [O,t - A@,,, 0, A@,,] N [2.28,4.] (cf. Eqs. (4.39) and (4.54)). The maximum-range permitted interval of p (i.e., for 0 = 7r = is now 10.24,l.l.

+

eTt)

I

I

I

I

0.0 2 , I

0.5

1 .O

1.5

2.0

0.0 I1

0.5

1.o

1.5

2.0

I

I

I

0.0

0.5

1.o

1.5

I

I

I

I 1

I 1

2.0

@In: Figure 4.18: ( a ) ,B = 0.1, ( b ) ,B = 0.25, ( c ) ,B = 0.5. Figures 4.18(a)-(c), 4.19(a)-(c) correspond to the cases ,B = 0.1,0.25, and 0.5, respectively. Note that, although for ,B = 0.1 inhibition of homoclinic bifurcation is not expected, diverse periodic states were reached by the Josephson junction in certain ranges of 0. Consider now some illustrative examples for the second resonance. In Fig. 4.20, a,y,F, and w are taken to be the same as in Fig. 4.16, R = 2w, and the initial condition (x (0) = 0, x (0) = 20) which is near the upper homoclinic orbit and the bifurcation diagram with respect to ,B is plotted for the values 0 = 0 (Fig. 4.20(a)), 0 = 1.570796 N 7r/2 = 0, (Fig. 4.20(b)), and 0 = 3.141593 2: 7r (Fig. 4.20(c)).

154

Applications: Low-dimensional systems

0.0

>

g

0.0

0.5

1.o

1.5

2.0

0.0

0.5

1.o

1.5

2.0

0.0

a

3 -0.2

3

0l.n Figure 4.19: ( a ) p = 0.1, ( b ) p = 0.25, ( c ) p = 0.5. Since for this case A@,,, = 0.855353, elimination of the horseshoe is not expected for 0 = O,T as indeed is numerically confirmed. Now ]0.641,0.654] is the predicted interval of suppressory values of p associated with 0 = OOpt= ~ / (cf. 2 Eq. (4.48)). The diagrams showing the leading (maximal) Lyapunov exponent X vs p corresponding to the respective cases of Fig. 4.20 are depicted in Fig. 4.21. A similar sequence of bifurcation diagrams (ivs p and X vs p ) for the same parameters as in Fig. 4.20, but for the respective initial condition which now is near the lower homoclinic orbit (x(0) = 0, x (0) = -Zo), are shown in Figs. 4.22 and 4.23, respectively. Again, as A@,, = 0.855353, one cannot expect complete regularization (including the elimination of chaotic transients) for 0 = 3.7,5.7. Finally, Figs. 4.24 and 4.25 show the bifurcation diagrams x vs 0 for three values of p and the same remaining parameters as in Figs. 4.20 and 4.22, respectively. In these cases, for 0 = OVt = 7r/2 (37r/2), associated with the upper (lower) homoclinic orbit, one calculates ]0.641,0.654] to be the suitable values of /3. Therefore, reliable regularization is expected for p = 0.647 (Figs. 4.24(b) and 4.25(b)) and for values of 0 close to the optimal values. The diagrams showing the leading Lyapunov exponent X vs 0 corresponding to the cases of Figs. 4.24, 4.25 are depicted in Figs. 4.26, 4.27, respectively. In comparing Fig. 4.16(b) with Figs. 4.24(b) and 4.25(b) one observes the shrinkage of the stabilization range when the resonance order changes from 1 to 2, as predicted in Chap. 2.

155

Taming chaos in a driven Josephson junction

2

,I

I

I

I

I

I

I

I

0.0

0.2

0.4

0.6

0.8

1.0

I

I

I

I

I

0.0

0.2

0.4

0.6

0.8

I

I

P Figure 4.20: ( u ) 0 = 0, ( b ) 0 = ~ / 2 (, c ) 0 = T .

I)

'1

1.0

156

Applications: Low-dimensional systems

2

11

,I

I

I

I

I

0.0

0.2

0.4

0.6

0.8

1.0

I

1

I

I

d

0.2

0.4

0.6

0.8

I1

I

I

I

I

~

0.0

1.a

P Figure 4.22: ( u ) p = 0.3, ( b ) p = 0.647, (c) p = 0.9.

02

*

......

0.0

. . . . .

E

*0 9)

-0.2

(4 0.0

0.2

0.4

0.6

0.8

P Figure 4.23: ( u ) 0 = 0, ( b ) 0 = ~ / 2 (c) , 0 = 7r.

1.o

157

Taming chaos in a driven Josephson junction

2 , )

LI 0.0 I I

-4 L 0.0

I

I

I

I

I

I

I

I

0.5

1 .o

1.5

2.0

I

I

I

I

0.5

I

I

1 .o

1.5

I I

I 1

2.0

o/n Figure 4.24: ( a ) /3

I

= 0.3,

( b ) /3 = 0.647, (c) /3 = 0.9.

I

I

I

0.0

0.5

1 .o

1.5

1

I

I

I

411 0.0

Figure 4.25:

I

0.5

I

I

1.o

1.5

o/x ( a ) /3 = 0.3, ( b ) /3 = 0.647, (c) p = 0.9.

2.0 I1

I

2.0

158

Applications: Low-dimensional systems

0.0

0.5

1.5

1.0

2.0

0ln: Figure 4.26: ( u ) p = 0.3, ( b ) /3

= 0.647,

(c) /3 = 0.9.

(C)

0.0

05

1.o

1.5

2.0

0l.n Figure 4.27: ( u ) p = 0.3, ( b ) = 0.647, (c) p = 0.9. The regularization ranges of the initial phase differences typically connect the two

Suppression of chaos of charged particles in an electrostatic wave packet

159

zones of chaos by period doubling on the one hand, or inverse period doubling on the other, in a roughly symmetric form (especially noticeable in versions (b) of Figs. 4.16, 4.18). Lastly, it is worth mentioning that the regularized responses were in most cases a period-1 attractor (main resonance) and a period-2 attractor (second resonance).

4.3

Suppression of chaos of charged particles in an electrostatic wave packet

The interaction of charged particles with an electrostatic wave packet is a basic and challenging problem appearing in many fundamental fields including astrophysics, plasma physics, and condensed matter physics. As is well known, the structure of the wave packet reflects the particular physical situation that is being considered in each case. In this respect, a widely studied particular case is that of a charged particle solely subjected to an infinite set of electrostatic waves having the same amplitudes, same wave numbers, zero initial phases, and integer frequencies. This Hamiltonian limiting case is suitably described by a simple mathematical model: the sc-called standard map. While the Hamiltonian approach to the aforementioned general problem is suitable in many physical contexts, the consideration of dissipative forces seems appropriate so as to take into account such diverse phenomena as electron-ion collisions, energy losses through synchrotron radiation, and stochastic heating of charged particles in plasma physics. In any case, stochastic (chaotic) dynamics already appears (can appear) when the wave packet consists solely of two electrostatic plane waves. Such non-regular behaviour of the charged particles may have undesirable effects in a number of technological devices such as the destruction of magnetic surfaces in tokamaks. Thus, apart from its general intrinsic interest, the problem of regularization of the dissipative dynamics of charged particles in an electrostatic wave packet by a small-amplitude uncorrelated wave (which is added to the initial wave packet) is especially relevant in plasma physics, and is considered as follows. 4.3.1 The three wave case Let us consider first the simplest equation to examine the problem: e

x + yx = -- [Eosin (koz - wot) + E, sin ( k c z- w c t ) + En,sin ( k s z- w,t - @)I, m

(4.56) where the amplitudes Eo,E,, E,, wave numbers ko, k,, k,, and frequencies wo, w,, w, correspond to the “main,” chaos-inducing, and chaos-suppressing waves, respectively, @ is an initial phase, and where weak dissipation (0 < y < 1) and non-uniform amplitudes (0 < E,,,/EO < 1) are assumed. Equation (4.56) describes the dissipative dynamics of a particle having charge -e and mass rn in the field of three general electrostatic plane waves. From a reference frame moving along with the main wave, Eq. (4.56) transforms into the equation d2e + sine dr2

-

=

-6

-

7-4 - &,sin (K,E - f12,-r)- E,, sin (ITs[ - fl,r), dr

(4.57)

160

Applications: Low-dimensional systems

where

(4.58) (4.59) (4.60) (4.61) (4.62) (4.63) (4.64) (4.65) are all dimensionless variables and parameters. Also,

(4.66) where woc, wo, are the relative phase velocities with respect to the main wave, Voc

=

wos

(4.67) (4.68)

wo/ko, w,/k, -wo/ko;

wc/a

-

while A, and A, are the wavelengths of the chaos-inducing and chaos-suppressing waves, respectively. Parameters ko, W O , Eo, and 00 are held constant throughout. Physically, Eq. (4.57) can be used to describe a number of important problems in plasma physics: the dissipative dynamics of a charged particle in the field of three electrostatic plane waves, the bounce dissipative motion of a charged particle trapped in a toroidal magnetic field which is perturbed by two (one chaos-inducing and the other chaos-suppressing)superposed electrostatic waves, and the suppressory effect of a (secondary) resonant magnetic perturbation on the destruction of magnetic surfaces by a (primary) resonant magnetic perturbation in a tokamak. Since Eq. (4.57) represents a perturbed pendulum (0 < 6,77, E,, E, < l ) , one can apply Melnikov’s method to obtain analytical estimates of the ranges of the parameters (E,, k,, w,, @) for suppression of the chaos existing in the absence of the chaos-suppressing wave. Thus, one obtains the Melnikov function

M*(To)= -D*

+ A* sin

(!&TO)

+ L?*

sin (0,-ro

+ @) ,

(4.69)

with

D* A* B’ L* ( K ,R)

= = =

87*2d, ~E,L*(K,,R,), 4&,L*(K,,R,),

G

f

l=

sechT cos [2Karctan (sinh T )

(4.70) (4.71) (4.72)

RT] d r ,

(4.73)

Suppression of chaos of charged particJes in an electrostatic wave packet

161

where the positive (negative) sign refers to the top (bottom) homoclinic orbit of the underlying conservative pendulum (cf. Eq.(2.33)). It is straightforward to deduce the following properties of the threshold functions L" ( K ,R):

L+ ( K ,f R )

=

-L- ( K ,TO), 2

L* (n,0)

=

0, n = 0,1, ...)

(4.74) (4.75) (4.76) (4.77)

One also finds that L f ( K ,R) vanishes completely in the region ( K ,52 < 0) but in a very narrow neighbourhood of the line R = 0, and that there exists a particular relationship, R = a K with a N (1 &) /2 (i.e., the golden ratio), which determines the direction of the main tonguelike region where L+ ( K ,R) presents its largest values (i.e., for small K values).

+

Figure 4.28: Function Lf ( K ,R), Eq. (4.73). Figure 4.28 shows a plot of L f ( K ,S2) . Remarks. First, without loss of generality, one can assume Ro = wo = 1, and hence D* > 0 (cf. Eqs. (4.61), (4.61), and (4.70)). Second, in the absence of any chaos-suppressing wave ( E , = 0), the limit WO, -+ 03 implies K , + 0, i.e., the chaos-inducing wave becomes a time-dependent harmonic forcing. In order to analyze the threshold of chaos, consider first the case with no chaos-suppressing wave (E, = 0). With fixed parameters 6, Q,E,, one sees that for

162

Applications: Low-dimensional systems

suficiently small values of k, the best chance for the occurrence of a homoclinic bifurcation, i.e ., (4.78) occurs when (wo,~is close t o J v o , , ~ ~where ~J, (4.79) is the most chaotic relative phase velocity.

0

1

3

2

4

5

K

Figure 4.29: Contour plot of ILf ( K ,R)1 (cf. Eq. (4.73)).

A contour plot of IL+(K,R)I with a gray scale from white (1.0 contour) to black (0.0 contour) is shown in Fig. 4.29. This theoretical prediction can be compared with the Lyapunov exponent calculations of Eq. (4.57). For the parameters S = 77 = 0.1,= ~ ~0.7, Fig. 4.30 shows a grid of 200 x 200 points in the K, - R, parameter plane where cyan, magenta, and black points indicate that the respective Lyapunov exponent was larger than 0.0, 0.07, and 0.14, respectively. One sees that the largest values of the maximal Lyapunov exponent (black points) lie inside a narrow tongue like region which is near the theoretical estimate R = (YK(yellow line in Fig. 4.30). Notice that the maximal (positive) Lyapunov exponent increases as K, is increased, which is coherent with the corresponding growth of the chaotic threshold function IL* (K,, R,)I. With fixed S,', E,, chaotic motion is possible when R, = 0 (i.e., when the relative phase velocity vanishes) and k, # nko, n positive integer, although this possibility decreases as A, -+ 0. Also, as expected, property (4.74) means that the chaotic threshold depends upon the absolute value of the relative phase velocity, but not on its sign. Let us suppose in the following that, in the absence of any chaossuppressing wave ( E , = O), the associated Melnikov function A 4: ( T O ) = -D*

+

163

Suppression of chaos of charged particles in an electrostatic wave packet

A* sin (R,To) changes sign at some T O ,so that the charged particle exhibits (at least transient) chaotic behaviour. I

,

*

I

’

I

’

I

’

I

’

I

I

K Figure 4.30: Lyapunov exponent distribution in the K, - R, parameter plane.

To study the taming effect of the chaos-suppressing wave in the most favourable situation, the case of a subharmonic (R, = PO,, p a positive integer) resonance is analyzed below. Notice that for this case one has (4.80) which permits one to identify two different physical situations (or mechanisms) for the added chaos-suppressing wave to tame chaotic charged particles: I. Chaos-inducing and chaos-suppressing waves having both commensurate wavelengths (X,/X, = m/n, m, n positive integers) and commensurate relative phase velocities ( V O , / V ~ ,= m’/n’, m’, n’ positive integers) such that p = m’n/(n’m). 11. Chaos-inducing and chaos-suppressing waves having both incommensurate wavelengths and incommensurate relative phase velocities such that the quotient ( V O ~ / U O/~ (X,/X,) ) satisfies Eq. (4.80). Now, one can directly apply the theory developed in Chap. 2 to the Melnikov function (4.69).

164

Applications: Low-dimensional systems

Theorem X. Let R, = pa,, p a positive integer, such that the relationships 9 = 9,t 9 = 9,t = 9,t

9

9 = 9,t

= 7r [4m + 3 - p (4n + l)]/2, = 7r [4m + 5 - p (4n + l)]/2,

+ 3 - p (4n - l)]/2, = 7r [4m + 5 - p (4n - l)]/2, ii 7r [4m

(4.81)

are independently satisfied for some positive integers m and n for the parameter regions (K, R) where

[L* (Kc,Rc) > 0, L* (Ks, Rs) > 01 , [L' (L524 > 0, Li (KS, 0 s ) < 03 , [L' (Kc,flc) < 0, Li (Ks, Rs) > 01 [L* (Kc,Rc) < 0, L* (Ks, 0,) < 01 I

1

(4.82)

respectively. Then the frustration of a homoclinic bifurcation occurs (i.e,, M* always has the same sign) if and only if the conditions & ni,:

E,:

R*

<

<

E s , , , fE

=

(1 - ID*/A*I) R*,

5

=

( T ~ )

R*/p2, E~ IL* (Kc,a,)/L* (Ks, 0,)I

(4.83)

are fulfilled for each region (K,R), respectively. Remarks. First, for the above ( K ,52) regions, this result requires having (Q = 7r, 7r/2,0,37r/2), (9= 0,37r/2, 7 r , 7r/2), (Q = 0,7r/2, T ,37r/2), (9 = 7 r , 3~/2,0,7r/2),respectively, for p = 4m - 3, 4m - 2, 4m - 1, 4m ( m = 1,2, ...), respectively in each case. Second, the effectiveness of the chaos-suppressing wave decreases as the resonance order p is increased. This is relevant in experimental realizations of the control where the main resonance case is then expected to be the most favourable to reliably tame the chaotic dynamics. Third, for the main resonance case (0, = Rc), one has a more accurate and complete estimate of the regularization boundary in the 9 - E , parameter plane: E:

=

{

cos Q f dcos2 Q - [l - (Di/Ai)2]} R*,

(4.84)

for Bk > 0 and A*, respectively, and E:

=

{

~

C

O

+ 4 ~ 09s-~[I - ( D * / A * ) 2 ] }R*, S

~

(4.85)

165

Suppression of chaos of charged particles in an electrostatic wave packet

for B* < 0 and A*, respectively. The two signs before the square root apply to each of the sign superscripts of E:, which, in its turn, is independent of the sign of D*. Also, the area enclosed by the boundary functions is given by 4 (lD*/A*l)R*. The relevance of the above theoretical findings to strange chaotic attractor elimination is confirmed by means of Lyapunov exponent calculations of Eq. (4.56). Figures 4.31 and 4.32 show the results corresponding to illustrative examples of the mechanisms of type I ( k s = k,, woS = UO,) and type I1 ( k , = (& - 1) k,/2, voS = (h 1) v0,/2), respectively, for the main resonance case. In the absence of the chaos-suppressing wave ( E , = 0), Eq. (4.56) presents a strange attractor with a maximal Lyapunov exponent A+ ( E , = 0) = 0.073 f 0.001 (bits/s).

+

1.o

0.8 0.6 s'

0.4

0.2 nn I

0.0

.

I

0.4

.

4

t

I

0.8

1.2

,

I

1.6

,

, I

2.0

Yln:

Figure 4.31: Lyapunov exponent distribution (type I mechanism, see the text).

0.8 -

0.6-

0.0

0.4

0.8

1.2

1.6

2.0

Yln

Figure 4.32: Lyapunov exponent distribution (type I1 mechanism, see the text).

166

Applications: Low-dimensional systems

The diagrams in these figures were constructed by only plotting points on the grid when the respective Lyapunov exponent was larger than 0 (gray points) or than A+ ( E ~= 0) (black points) for the parameters ko = wo = 00= 1, 0, = fl,, E, = 0.7, k, = 1.22,w, = 2.26, and y = 0.1, and with the black dashed-line contour denoting the theoretical boundary function (cf. Eq. (4.84)) which is symmetric with respect to the optimal suppressory value 9, 7r. Regarding the mechanism of type I, one typically finds that complete regularization (A+ ( E , > 0) 0) mainly appears inside the island which symmetrically contains the theoretically predicted area, while regularization by a type I1 mechanism seems to be insensitive to the initial phase P. In this latter case, the lowest value of the theoretical boundary function roughly coincides with the regularization threshold value. The regularization regions in the P - E, parameter plane under the two types of mechanisms present different characteristics. For type I, one typically finds that inside the regularization area which contains the predicted area, the two non-null Lyapunov exponents are identical and constant (A+ = A- = -y/2). Figure 4.33 shows the maximal Lyapunov exponent A+ versus the initial phase 9/7rfor six values of the suppressory amplitude E, = 0.2,0.4,0.6,0.8,1.0,1.2 corresponding to the type I mechanism for the remaining parameter as in Fig. 4.31.

<

1.4

1.2 1.o Lo*

++

<

0.8 0.6 0.4

0.2

.o

0.5

1.0

1.5

2.0

Ylrc Figure 4.33: Lyapunov exponent vs initial phase (type I mechanism, see the text). This symmetry property of the contraction of the phase space volume for the predicted regularization areas in the P - E, parameter plane is an inherent feature of the type I mechanism. This property does not hold for the type I1 mechanism, since in this case the regularization region is not completely “clean” (there exist is@ lated points corresponding to chaotic behaviour) and furthermore the distribution

167

Suppression of chaos o f charged particles in an electrostatic wave packet

of the non-null (negative) Lyapunov exponents is not perfectly uniform, as can be appreciated in the example of Fig. 4.34. This figure shows the maximal Lyapunov exponent A+ versus the suppressory amplitude E, for six values of the initial phase 9 = 0 , ~ / 3 , 2 ~ / 3 , ~ , 4 n / 3 , 5 . lcorresponding r/3 to the type I1 mechanism for the remaining parameters as in Fig. 4.31. 60

I

I

I

I

,

.

I

.

,

.

I

50k! 4 . 0 --.

-i,

_-

1

,,&

- - - - _ _.

\

, .

n

-.1 0 K

m

9-

0.0 -10

:

, 1. "

l

a

"

"

"

I

'

Figure 4.34: Lyapunov exponent vs initial phase (type II mechanism, see the text). 4.3.2 Case of a general electrostatic wave packet During the past quarter of century the Hamiltonian and dissipative versions of the so-called standard map have been widely studied as basic models of the dynamics of charged particles in the field of an electrostatic wave packet: e

N

En sin (knz - writ) , x + yx = -me n=-N

(4.86)

where En, kn, and w, are the amplitudes, wave numbers, and frequencies, respectively, of the (2N 1) plane waves, and y,e, and me are the damping coefficient, the charge, and the mass of the particle, respectively. Specifically, the standard map describes the particular case of an infinite set of waves having the same amplitudes, same wave numbers, and integer frequencies. While the standard map is one of the basic models used in chaos theory and also has diverse physical applications, its assumption of an infinite and uniform amplitude distribution seems quite restrictive since it does not permit one to study the sensitivity of the dynamics to changes in the wave packet width. Physically, this sensitivity yields changes in the properties and structures of

+

168

Applications: Low-dimensional systems

the phase space, as can be appreciated when comparing, for example, the cases of two waves and infinite waves. It is therefore pertinent to consider a generalized model of the wave packet structure to take into account such a finite-size effect on the particle dynamics. Specifically, it is assumed that

En

=

En (m) = EOsech [nrK (1 - m) / K (m)] ,

kn = ko, w, = w o + n A w ,

(4.87)

where K(m) is the complete elliptic integral of the first kind, i.e., a sech distribution is assumed for the amplitudes such that the effective width is controlled by a single parameter: the elliptic parameter m. This specific form of En (m) is motivated by the following properties: i) En(m = 0) = Eob,~, with bnO being the Kronecker delta, i.e., one recovers the non-chaotic limiting case of a single plane wave. ii) En(m = 1) = Eo,Vn, i.e., one recovers the limiting case described by the standard map. iii) For any m E [0, l), one may define an efSectiwe number of harmonics forming the wave packet as follows. Let us choose quite freely a real number C E ( 0 , l ) such that Neff is the largest integer satisfying EN^^^ /Eo 3 then En/Eo < c , V n > Neff, i.e.,

c1

(4.88) where the brackets stand for the integer part. Also, the assumption kn = ko corresponds to the situation where the group velocity is much greater than the phase velocity (w << vg = A w / A k ) and then the wave packet is called time-like. Thus, for this choice of the wave packet structure, one has N

Neff

Ensin ( k n z - writ) = sin (koz - wot) n=-N

En (m) cos ( n A w t )

(4.89)

n=-N ef f

and, after extending the summation from -00 to 00, Eq. (4.86) transforms into the form eEo . (4.90) x + yx = -- sin (koz - wot) D ( t ;T ,m) , me where T = 21r/Aw is the characteristic period of the field and

[2Kg)t;

D ( t ;T ,m) = 2K(m) dn -m] ,

(4.91)

with dn being the Jacobian elliptic function of parameter m. Notice that the extension of the summation horn -00 to 00 does not imply a loss of generality unlike in the case

Suppression of chaos of charged particles in an electrostatic wave packet

169

of the standard map. In a reference frame moving with the main wave, Eq. (4.90) transforms into the equation

d2t

dt

+ sin< = -6 - 17-d r dr2

- [D(7; a , m) - 11sin[,

(4.92)

where

(4.93) are all dimensionless variables and parameters. The parameters ko, W O , Eo, and Ro are held constant throughout. Physically, Eq. (4.92) represents a damped pendulum subjected to a periodic string of finite pulses having an effective width and an amplitude controlled by m. Thus model (4.92) permits one to study the structural stability of the system under changes in the width of the wave packet by solely varying the parameter m (and hence N , f f ) between 0 and 1. In particular, the case of a single plane wave ( m = 0) is described by a purely damped pendulum, while the case of an infinity of plane waves (standard map: m = 1) is described by a delta-kicked rotator. Consider first the case of weak dissipation (0 < y << 1). Since

(4.94) Eq. (3) may be regarded as a perturbed pendulum (0 < S,q << 1) for m E [0, 1) and then one can apply Melnikov’s method to obtain an analytical estimate of the edge of chaos in the parameter space. The application of this method to Eq. (4.92) gives the Melnikov function

M*

(TO)

= -D*

2

+67r3

n2En (m)b, ( a )sin

(4.95)

with

D*

s 817f27r6,

b, ( a ) s a-’ csch

(S) 1

(4.96)

and where the positive (negative) sign refers to the top (bottom) homoclinic orbit of the underlying conservative pendulum. From M* ( T O ) one sees that a homoclinic

170

Applications: Low-dimensional systems

bifurcation (signifying the possibility of chaotic behaviour) is guaranteed for trajectories whose initial conditions are sufficientlyclose to the separatrix of the underlying conservative pendulum if 1211f 7r6/21

47r3

< U (m,a ) = -x n 2 E n( m )b, ( a ) , Eo

O0

(4.97)

n=l

where U(m, a ) is the chaotic threshold function (see Fig. 4.35).

Figure 4.35: Chaotic threshold function U (m,a ) (Eq. (4.79)). It is straightforward to obtain the following properties: i) U (m,a ) increases, as a function of m, as m is increased, i.e., the possibility of chaos increases as the spectral width is increased. ii) U(m, a -+ 0, co) = 0, i.e., for any spectral width, the possibility of chaos diminishes when the small-amplitude frequency of the non-perturbed equivalent pendulum ( d ’ J / d ~ ~sinJ = 0) is much higher or much lower than the characteristic spectral frequency. iii) U (m,a ) presents a maximum, as a function of the parameter a , at a,, = a,,, ( m ),V m E (0, l ) ,such that a,, (m)is a monotonously increasing function and exhibits the asymptotic behaviour amax ( m -+ 1) N l n ( l - m)-’”. iv) U ( m + 1,a) = 47r3 0 0 n2b, ( a ) > U (m,a ) ,Vm E [0, l), i.e., it is expected that the limiting case m = 1 be maximal with respect to the extension of dissipative chaos in parameter space. Observe that properties (iii) and (iw) mean that the dissipative standard map is non-universal in the framework of the wave particle interaction, because this map corresponds to an infinite set of waves having the same amplitudes, and hence it does not present the aforementioned maximum. Lyapunov exponent calculations of Eq. (4.92) are consistent with properties (2)-(iw). One typically finds that the maximal Lyapunov exponent, A’, presents a maximum as a function of a at aka, = aka, ( m ) ,and that both A+ (a;,,) and atax increase

+

171

Suppression of chaos of charged particles in an electrostatic wave packet

with the spectral width in accordance with the Melnikov-method-based predictions. Figure 4.36 shows an illustrative example for three increasing values of the spectral width and 6 = 7 = 0.1. Vertical arrows indicate the position of the respective maxima.

-5

0.4

R

@

0.3

bE

0.2

$

4 U .B3 2

0.1

2 -,

0.0 -0.1

'

'

I

I

I

I

I

3

6

9

12

15

a Figure 4.36: Maximal Lyapunov exponent vs parameter a for three m values. When dissipation is negligible (y = 0), system (4.92) is generated by the Hamiltonian (4.98)

/cox/fl0, and Melnikov's method provides an estimate of the width of the stochastic layer generated around the unperturbed separatrix:

pg

(4.99) Thus, the aforementioned properties of the chaotic threshold function hold for the width of the stochastic layer and hence one expects an intensification of the deterministic diffusion as the spectral width is increased. Figure 4.37(b, d, f ) provides an illustrative sequence for three increasing values of m at a N amax (m),respectively. Moreover, one typically finds the gradual disappearance of the invariant curve region inside the separatrix cell (cf. Figs. 4.37(c, b, d, f)). Numerical simulations also confirm that the stochastic layer exhibits a maximal width as a function of a at a value close to amax (m). This can be appreciated in the sequence of Figs. 4.37(a, c, e). Another difference with respect to the standard map is that the phase space of the Hamiltonian H (<, p c , 7 ) is bounded by Kolmogorov-Arnold-Moser (KAM) tori at an9

172

Applications: Low-dimensional systems

values of a and m E (0, l), i.e., global stochasticity is not possible for any arbitrary but finite spectral width (notice that, contrary to the standard map, the phase space of the Hamiltonian H ( < , p < T, ) is not periodic in pc). One thus concludes that, in the context of the wave-particle interaction, such a transition to global stochasticity is a peculiarity of the standard map.

10,

7

2

0

0

-1

1

0

1

Figure 4.37: Poincar6 section (pl vs < / T ) for the system (4.92) with 6 = 7 = 0. Similarly to the case of the standard relativistic map, the relativistic generalization of model (4.90) (y = 0) is necessary if the acceleration of particles is sufficiently large. The relativistic equations corresponding to model (4.90),

x =

PC2

Jm’

p = -eEo sin (Icoa: - wot) D (t;T ,m) ,

(4.100)

where p , meO, and c are the momentum and the rest mass of the particle, and the light velocity, respectively, may be conveniently transformed into the dimensionless

173

Suppression of chaos of charged particles in an eiectrostatic wave packet

form

* dr

= -sinED(qa,m),

(4.101)

where E = ( n O / / ~ o ) /~c 2 is the relativistic parameter, p~ = /Cop/ (m,oQo)is the dimensionless momentum, and r = wo/Ro.

3

0 -1 1

0

1 I

3

0 -1

Figure 4.38: Poincark section (d
(/T)

0

1

for the system (4.102) (see the text).

To also characterize the relativistic dynamics in the coordinate-velocity phase space, ([, dJ/d.r), Eq. (4.101) is rewritten as a second-order differential equation:

A relativistic effect is the reduction of the system symmetry: Eq. (4.101) presents two mirror symmetries with respect to ( = 0 and pt = r, respectively, for non-relativistic particles, while it solely presents the former symmetry for relativistic particles when I' > 0. Another relativistic effect is the modification of the fixed points existing in the classical (Newtonian) regime: (4.103)

174

Applications: Low-dimensional systems

Two limiting cases may be distinguished: E << and E 5 Physically, the resonance condition E = F2means that the phase velocity of the main wave (wo/lco) is equal to the velocity of light. In the former case, when the phase velocity differs significantly from the velocity of light, the stochastic motion is restricted to the neighbourhood of the fixed points, such that the corresponding chaotic layers are bounded by KAM tori at any values of CY and m E ( 0 , l ) . For relativistic particles just beyond the Newtonian regime ( E 2 0), it is found numerically that the extension of the stochasticity regions in the coordinate-momentum phase space for relativistic and non-relativistic particles, respectively, is approximately the same, while it diminishes relatively in the coordinate-velocity phase space for relativistic particles, as can be appreciated in the examples shown in Figs. 4.38 and 4.39. Figure 4.38 shows the trajectories in the stroboscopic Poincare section (&/& vs [/n)for the relativistic system (Eq. (4.102), CY = 5 . 3 , r = 0), and (a) m = 0 . 6 , ~ = 0, (b) m = 0 . 6 , ~ = 1/9, (c) m = 0.9, E = 0, and (d) m = 0.9, E = 1/9. I

I

4

Figure 4.39: Poincare section (pc vs 6-n) for the system (4.101) (see the text). Figure 4.39 shows the corresponding trajectories in the stroboscopic Poincare section (pc vs [/T) for the relativistic system (Eq. (4.101), a = 5 . 3 , r = 0) and the same parameters, respectively, as in Fig. 4.38. Clearly, the aforementioned shrinkage is a consequence of the relativistic constraint

(4.104) i.e., 121

< c.

Note that a proper comparison of the structures appearing in the two

Suppression of chaos of charged particles in an electrostatic wave packet

175

phase spaces would require taking the initial momentum as (4.105) where (dJ'/d.r)(0) is the corresponding initial velocity. A preliminary quantitative estimate of such a relativistic effect in the coordinate-velocity phase space can be obtained from the equation giving the first-order relativistic correction: (4.106)

Similarly to Eq. (4.92), Eq. (4.106) may be considered as a perturbed pendulum for m E [0, 1) and, after applying Melnikov's method to it, one straightforwardly obtains the following expression for the width of the stochastic layer: (4.107)

where (4.108) and UR (m,a, E) is the first-order relativistic threshold function. One obtains that, for fixed m,a, the relativistic width d R decreases as E is increased, and that this decrease is ever more noticeable as m is increased. Numerical results confirm these two predictions, as in the example shown in Fig. 4.38. Proceeding similarly t o obtain a first-order relativistic correction in the coordinate-rnomentum phase space by expanding the square root in Eq. (4.101), Melnikov's method yields a null correction to the Newtonian width because the corresponding integrals vanish. Numerical results confirm this prediction, as in the example shown in Fig. 4.39. Figure 4.40 shows the trajectories in the stroboscopic Poincare section ( p vs ~ ( I T ) for the relativistic system (Eq. (4.101), m = 0 . 5 , = ~ 5.3,r = l), and (a) E = 0.8, and (b) E = 1. As the phase velocity approaches the velocity of light ( E 5 rP2), the momentum of the fixed points increases continuously, and the KAM torus limiting the stochastic region from above expands into the higher-momentum region (see Fig. 4.40(a)). When the resonance condition E = is met exactly, the particles can be accelerated to very high energies, which is indicated by the vertical traces in Fig. 4.40(b). This mechanism of particle acceleration is also observed in the relativistic standard map for the case wo = 2 d / T , with 1 being an integer. Notice that, far from the resonance condition,

176

Applications: Low-dimensional systems

the analysis of the relativistic standard map indicated a relativity-induced decrease of the stochasticity in the coordinate-momentum phase space. However, the above findings clearly show that such a property does not hold for finite wave packets.

0

-1

1

I

I

1000

/I

j/

0

-1

-1

0

Figure 4.40: Poincar6 section (pc vs < / T ) for the system (4.101) (see the text). Consider now the possible suppressory effect of adding an initial phase to a single wave of the wave packet (4.87). Specifically, the dynamics of the charged particles is now governed by the equation e

z + y i = --

N

C E, sin (knx - wnt + hon(pn),

(4.109)

me n=-N

where h,, is the Kronecker delta, and En, kn, w, are given by Eq. (4.87), i.e., only the initial phase of the main wave is assumed to be non-null. Thus, for this choice of the wave packet structure, one has N

C E, sin (knx - writ) n=-N

Neff

=

sin (k0a:- wot)

1 En(m)cos (nawt)

n=-N e f f

+2Eosin ((p0/2) cos (kox- wot

+ (p0/2),

(4.110)

and, after extending the summation from -co to 00, Eq. (4.109) transforms into the

177

Suppression of chaos of charged particles in an electrostatic wave packet

form (4.111)

ZeEo . sin ((p0/2) cos (koa: - wot

_-

me

+ (po/2)

In a reference frame moving with the main wave, Eq. (4.111) transforms into the equation

c

dT2 d2 + sine = -6

(c

- p0)[ I - D (7;a , m)], (4.112) d.r where = ( yo, the remaining dimensionless variables and parameters being given by Eq. (4.93). Consider again the case of weak dissipation (0 < y << 1) in order to apply Melnikov’s method to Eq. (4.112) for m E [0,1). One thus obtains the Melnikov function

c

- 7-dC +sin

+

16n3 M * ( ~ o )= - D * + - ~ n 2 E n ( m ) b , ( a ) c o s ( r p o ) s i n EO n=l

[zn7]

(4.113)

with (4.114)

D*,b,a given by Eq. (4.96), and where the positive (negative) sign refers to the top (bottom) homoclinic orbit of the underlying conservative pendulum. From M* ( T O ) one sees that a homoclinic bifurcation (signifying the possibility of chaotic behavior) is guaranteed for trajectories whose initial conditions are sufficiently close to the separatrix of the underlying conservative pendulum if (4.115) d, (a,y o )

=

dl

- sech’ ( n n z / a sin2 ) (yo),

where U’ (m,a , y o ) is the chaotic threshold function. It is straightforward to obtain the following properties: i) U’ (m,a, yo = 0 ,n)= U(m,a ) ,i.e., the onset of chaos is the same for propagations of the main wave in phase and in opposition with respect to the remaining waves. ii) The threshold function U’(m,a , yo) presents minima at yo = n/ZI37r/2 and maxima at yo = 0, n. With fixed m, a , one thus expects that the existing chaos at rpo = 0 can, in some case, be diminished (in the sense of lowering of the maximal Lyapunov exponent) or completely suppressed as rpo approximates f n / 2 .

178

Applications: Low-dimensional systems

4.4 Notes and references The present discussion of the control of chaotic escape from a potential well follows [l-41. Some additional works appear in [5-81. The example of the Helmholtz oscillator [9] is used in [5] to demonstrate that parametric modulations of the linear or quadratic potential terms inhibit chaotic escape when certain resonance conditions are met, while the case of additional external modulations is discussed in [6]. Reference [7] studies the robustness of the inhibition of bidirectional chaotic escape of a harmonically driven oscillator from a quartic potential well by the application of weak parametric excitations. The structural stability of the Helmholtz oscillator under changes of the shape of periodic nonlinear perturbations is investigated in 181. For an introduction to Josephson junctions see, e.g., [lo]. An application of Melnikov’s method to McCumber’s model for the Josephson junction px [l y cos 212 sin z = Q kw sin (wt)is given in [ll].Previous application of the Melnikov’s method to the prediction of horseshoe chaos in a Josephson junction with no interference effects and subjected to a single harmonic excitation is discussed in [12,13]. Many of the results concerning suppression of chaos in Josephson junctions in this chapter appear in [14]. The interaction of charged particles with a wave packet is a basic and challenging problem appearing in many fundamental fields such as plasma physics, astrophysics, and condensed matter physics [ 15,161. Many interesting notes concerning the history of this problem can be found in [17].

+

+

+ +

[l] Chacbn, R., Balibrea, F., and Lbpez, M. A,, (1997) “Role of parametric resonance

in the inhibition of chaotic escape from a potential well,” Phys. Lett. A 235, pp. 153-158.

[Z] Chacbn, R., Balibrea, F., and Lbpez, M. A., (2001) “Role of nonlinear dissipation in the suppression of chaotic escape from a potential well,” Phys. Lett. A 279,pp. 38-46. [3] Chacbn, R., (2001) “Maintenance and suppression of chaos by weak harmonic perturbations: A unified view,” Phys. Rev. Lett. 86,pp. 1737-1740. [4] Chac6n, R. and Martinez, J. A., (2002) “Inhibition of chaotic escape from a potential well by incommensurate escape-suppressing excitations,” Phys. Rev. E 65,pp. 03621311-7. 15) Chacbn, R., Balibrea, F., and Lbpez, M. A., (1996) “Inhibition of chaotic escape from a potential well using small parametric modulations,” J. Math. Phys. 37,pp. 5518-5523.

[6] Balibrea, F., C h a c h , R., and Lbpez, M. A., (1998) “Inhibition of chaotic escape by an additional driven term,” Int. J. Bifurcation and Chaos 8,pp. 1719-1723. [7] Chacbn, R., SBnchez-Bajo, F., and Martinez, J. A., (2002) “Robustness in the suppression of bidirectional chaotic escape from a potential well by weak parametric excitations,” Phys. Lett. A 303,pp. 190-196. 18) Balibrea, F., Chacbn, R., and Lbpez, M. A., (2005) “Reshaping-induced order-chaos

Notes and references

179

routes in a damped driven Helmholtz oscillator,” Chaos, Solitons and Fractals 24, pp. 459-470. [9] Helmholtz, H. L. F., (1954) O n the Sensation of Tone as a Physiological Basis f o r the Theory of Music, Dover, New York. [lo] Barone, A. and Paterno, G., (1982) Physics and Applications of the Josephson Effect, Wiley, New York. [ll]Sanders, J. A., (1982) “Melnikov’s method and averaging,’’ Celestial Mechanics 28, pp. 171-181. [12] Bartucelli, M., Christiansen, P. L., Pedersen, N. F., and Soerensen, M. P., (1986) “Prediction of chaos in a Josephson junction by the Melnikov-function technique,” Phys. Rev. B 33,pp. 4686-4691. [13] Schecter, S., (1987) “Melnikov’s method at a saddlenode and the dynamics of the forced Josephson junction,” SIAM J. Math. Anal. 18, pp. 1699-1715. [14] C h a c h , R., Palmero, F., and Balibrea, F., (2001) “Taming chaos in a driven Josephson junction,” Int. J. Bifurcation and Chaos 11, pp. 1897-1909. [15] Lichtenberg, A. J. and Lieberman, M. A., (1983) Regular and Stochastic Motion, Springer-Verlag, New York, Heidelberg, Berlin. [16] Sagdeev, R. Z., Usikov, D. A., and Zaslavsky, G. M., (1988) Nonlinear Physics: From the Pendulum to Turbulence and Chaos, Harwood Academic Publishers, New York. [17] Zaslavsky, G. M., Sagdeev, R. Z., Usikov, D. A., and Chernikov, A. A., (1991) Weak Chaos and Quasi-Regular Patterns, Cambridge University Press, Bristol.

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Chapter 5

APPLICATIONS: HIGH-DIMENSIONAL SYSTEMS This chapter discusses some important applications of the results of Chap. 2 to particular, but broad and important, classes of high-dimensional systems, in particular, to arrays of coupled, nonlinear, dissipative oscillators as well as to partial differential equations. It is shown that the application of such results is useful and reliable in taming both extended spatio-temporal chaos and spatially localized chaos.

5.1

Controlling chaos in chaotic coupled oscillators

Control of chaos in spatially extended systems is today one of the most interesting and challenging problems in the field of nonlinear dynamics. Examples of the many possible applications include the stabilization of superconducting Josephson-junction arrays, semiconductor laser arrays, and periodic patterns in optical turbulence. Since the general problem of controlling spatio-temporal chaos leading up to that of controlling turbulence is extremely complicated (and certainly extremely important), to gain insight, a popular first approach is to investigate simplified models of realworld complex systems, such as coupled map lattices and arrays of coupled nonlinear oscillators.

5.1.1 Localized control of spatio-temporal chaos A fundamental question in this field is whether effective control of the whole system can be achieved by solely influencing some of its degrees of freedom. In order to demonstrate the effectiveness of the findings of Chap. 2 in taming spatio-temporal chaos of chains of identical chaotic coupled oscillators, consider a homogeneous chain of damped driven two-well Duffing oscillators governed by the equation

where n = 1,..., N, and where only the free ends are additionally subjected to the two chaos-taming excitations -Pq1z3 cos (wt pl) yq2 cos (wt pz). It is assumed that, in the absence of any chaos-taming excitation, the entire chain exhibits spatiotemporal chaos for a given set of parameters. Choosing chaos-taming excitations to act on the chain’s ends according to the theoretical predictions in Chap. 2, intensive numerical simulations indicated that, for each N ( N 6 128), there exists at least one

+

+

+

181

182

Applications: High-dimensional systems

critical coupling, kc, for which the entire chain regularizes to a period-1 response, even in the case of a single chaos-taming excitation such as that depicted in Fig. 5.l(c).

- 2n --

2 -

0

5990

5992

5994

5996

5998

6000

t/T

Figure 5.1: Time series of the velocities 2, from Eq. (5.1) (see the text).

183

Controlling chaos in chaotic coupled oscillators

Figure 5.1 shows time series of the velocities xn, n = 1,4i (i = 1,..., 16) from bottom to top, versus drive cycles (7' = 27r/w) for the chain (5.1) with the parameters: (a) 711 = 0.169525,91 = 3.1415 2 T 9 1 0 p t , r/2 = 0; (b) 71 = 0.169525, 9 1 = 2.5, ~2 = 0; and (c) q1 = 0.169525,'pl = 2.5, q2 = 0.1, 'pz = 3.1415 N 7r = 920pt; the remaining parameters are k = 5.68, p = 4,6= 0.154, y = 0.095,w = 1.1, and N = 128. However, the synchronization of the global regularized response is not complete: correlation in the amplitudes and correlation in the phases only appear for groups of oscillators. Also, the critical coupling k, is not a threshold value, in that one typically finds windows of regularization interspersed with windows of chaos as k increases, the remaining parameters being held constant. An illustrative example is shown in Fig. 5.2, where bifurcation diagrams for the correlation function (empty circles)

C ( t )= N(N - 1)

c

cos (z$) - xj ( t ) )

(4

and the variable xl - x64versus k are plotted for the chain (5.1) with q1 = 0.169525, 'pl = 3.1415 N 7r = plopt,q2 = 0, and N = 128. In Eq. (5.2), (ij) indicates that the summation is over all pairs of (respective) excitations. The application of a second chaos-taming excitation typically reinforces the inhibitory effect of the first chaostaming excitation in two ways: (i) it regularizes the response that remained chaotic (except at the two ends of the chain) with only a single chaos-taming excitation (as in the case shown in Fig. 5.1(b)), and (ii) it improves the synchronization of the regularized global response (as in the case shown in Fig. 5.3; compare the correlation functions corresponding to Figs. 5.l(a) and (c)). 1.2,

-0.8

,

.

'

.

2

,

.

,

'

"

'

4

6

.

'

,

'

R

.

,

10

k Figure 5.2: Bifurcation diagrams for C ( t )(Eq. (5.2), top) and the variable 2 1 The underlying mechanism is simple: A second chaos-taming excitation can (in some

184

Applications: High-dimensional systems

case) make the resonant interactions between (some of the) uncorrelated (in the aforementioned sense) groups of oscillators effective, such that they become nearly perfectly synchronized, thus improving the regularization-induced synchronization of the entire chain.

0.80 5990

5992

5994

5996

5998

6000

t/T

Figure 5.3: Correlation function vs drive cycles (see the text). The fact that complete synchronization occurs by groups of oscillators suggests that the complete synchronization of the entire chain could be achieved by controlling other oscillators besides the two ending the chain. Indeed, one typically finds that the application of (a suitable number of) multiple chaos-taming excitations (even a single one) to several coupled oscillators (besides the two ending the chain) yields a coherent oscillation of the entire chain, as can be appreciated in Fig. 5.3 where the correlation function is plotted versus the drive cycles for the parameters as in Fig. 5.l(a) (thin grey line), Fig. 5.l(b) (thin black line), and Fig. 5.l(c) but with chaos-taming excitations acting every eighth oscillator (thick black line).

5.1.2 Application to chaotic solitons in Fkenkel-Kontorova chains Although localized control of spatietemporal chaos has been investigated in diverse extended systems including coupled oscillators, coupled map lattices, and plasmas, much less effort has been devoted to controlling chaotic localized solutions (such as solitons and breathers) arising from both nonlinear partial differential equations and lattices of nonlinear oscillators. A general theoretical approach t o controlling chaotic solitons in damped, noisy, and driven Frenkel-Kontorova chains by weak periodic excitations is discussed as follows. As is well known, the Frenkel-Kontorova model provides a fairly accurate description of diverse biological and physical systems and phenomena, including ladder networks of discrete Josephson junctions, charge density wave conductors, and DNA dynamics. For the sake of simplicity, the application of the results of Chap. 2 is discussed in the simple case of additive noise, linear damping,

185

Controlling chaos in chaotic coupled oscillators

and two harmonic excitations uj

K +sin (2xuj) = 27l

-ahj

+u

~ -+2uj ~ + uj-l

(wt)

+ PF cos (Rt + cp) + t( t ),

(5.3)

where uj is the phase of the j t h oscillator, K measures the strength of the substrate potential, cx is the damping coefficient, F cos ( w t ) is the cham-inducing excitation, and (PI0,'p) are the as yet undetermined suitable parameters of the chaos-suppressing excitation PF cos (Rt + cp). The term

t( t )= X sin [R't + aB(t)+ I?]

(5.4)

represents bounded noise with zero mean, where X and R' are the amplitude and averaged frequency, respectively, B ( t ) is a unit Wiener process, u represents the intensity of a random frequency, and r is a random initial phase uniformly distributed in [ 0 , 2 ~ [ . Also, a finite chain of N particles with the boundary condition u j + N = uj N + 1 is assumed to keep the analysis close to experimental realization (for example, a circular array of Josephson junctions). As is well known, a collective coordinate formalism permits one to describe the motion of the soliton centre of mass, X ( t ) , by means of an effective ordinary differential equation, which is a perturbed pendulum for the Frenkel-Kontorova model. Thus, the application of collective coordinate formalism to Eq. (5.3) by assuming a sine-Gordon profile for the discrete soliton,

+

(5.5)

7l

yields the perturbed pendulum equation

= ~- T Xr,= Q p N t , 6 = aR&,

-

-

y = T ~ ~ ~ F RW; = $ ,wR&, R = RR&, X = 7r310XR&, R' = R'R,;, and (T = uR2,Z2, where f l p and ~ 10 are the Peierls-Nabarro frequency and the soliton width, respectively [R22PN = (7r4 + 2 ~ ~ 1/(31: : ) sinh ( 7 r 2 1 0 ) ) ] . To apply the results of Chap. 2 , let us assume in the following that the noise, excitation, and damping terms in Eq. (5.6) are small amplitude perturbations of the underlying integrable pendulum, i.e., they satisfy Melnikov's method requirements, and that, in the absence of both cham-suppressing excitations (P = 0) and noise (A = 0), the perturbed pendulum exhibits homoclinic chaos which corresponds to a chaotic soliton existing in the F'renkel-Kontorova model (5.3). In such a situation, one lets the chaos-taming excitation act on each pendulum of the chain (and hence on the associated perturbed pendulum (5.6)) (P > 0) in the presence of noise (A > 0) where z

186

Applications: High-dimensional systems

and deduces analytical estimates of the regularization regions in the parameter space (/3,6,‘p). Next, one conjectures, notwithstanding the perturbative and approximate character of both the collective coordinate formalism and Melnikov’s method, that such analytical predictions could remain reliable to some extent (in the corresponding regions of the control parameter space (p,0,p)) to effectively tame chaotic solitons arising from the uncontrolled Frenkel-Kontorova model. Numerical results confirm that this is the case. For the sake of clarity, the cases with and without noise will be considered separately. In the absence of noise X = = 0 , the application of Melnikov’s method to Eq. (5.6) yields the Melnikov function

(

M;=o

(TO)

= -C

F ACOS( G T ~ ) B cos

(- + 0

~

1

0‘p

(5.7)

0

where C = 86, A = 27rysech(7rG/2), and B = 27rpysech 1&/2 , where the plus (minus) sign corresponds to the upper (lower) homoclinic orbit of the unperturbed pendulum. Now the results of Chap. 2 directly apply to the Melnikov function (5.7). In particular, for the main resonance case O/w = fill2 = 1, one obtains the boundary function in the ’p - p parameter plane,

p = - cos ’p 3~ Jcos2

p - (1 - C 2 / A 2 ) )

(5.8)

where homoclinic chaos is suppressed. Recall that Eq. (5.8) represents a sufficient condition for ( T O ) to be negative or null for all T O , and that the area enclosed by such a boundary is 4 ICJ/ A . Next, one can compare the analytical predictions and Lyapunov exponent calculations of both the Frenkel-Kontorova model and the associated perturbed pendulum. Notice that one cannot expect too good a quantitative agreement between the two kinds of results because the Lyapunov exponent provides information concerning solely steady chaos, while Melnikov’s method is a perturbative method generally related to transient chaos. The maximal Lyapunov exponent, A+, was calculated for each point of a dense grid, with normalized initial phase ‘p and amplitude p along the horizontal and vertical axes, respectively, for both systems. An illustrative example is shown in Fig. 5.4, where the maximal Lyapunov exponent distribution in the ‘p - p parameter plane is plotted for the soliton in the Frenkel-Kontorova model (top panel) and for the associated perturbed pendulum (bottom panel) in the absence of noise (A = 0). Cyan, green, and magenta regions indicate that the respective maximal Lyapunov exponent, A+ ( p > 0) is non-positive, belongs to the interval [0,A+ ( p = O)], and is larger than A+ ( p = 0) N 0.03 bits/s, respectively, for the parameters a = 0.1, K = l, F = 0.0017, w = O.O47r, and l~ = 1.0118179.

187

Controlling chaos in chaotic coupled oscillators

"1

1.2

1.4

1.6

1.8

cpln Figure 5.4: Lyapunov exponent distribution in the absence of noise (see the text). The solid black contour indicates the predicted boundary function (cf. Eq. (5.8)). Only Lyapunov exponents corresponding to values p E [ T ,27r] are depicted because of symmetry with respect to the optimal suppressory value popt = 7r. Typically, one finds for both systems that complete regularization (A+ ( p > 0) 0) mainly appears inside maximal islands which symmetrically contain the theoretically predicted areas where even the chaotic transients are inhibited. The great similarity of both the two Lyapunov exponent distributions in their respective p - p parameter planes should be stressed. In the presence of noise A, > 0 , one has a random Melnikov process:

<

0

where

R ( T O )= 2'

1:

sech(r)( ( r

+TO) d r

(5.10)

is the component of the random Melnikov process due to noise. As is well known, the simple zeros of a (deterministic) Melnikov function imply transversal intersections of stable and unstable manifolds, giving rise to Smale horseshoes and hence hyperbolic invariant sets. However, the random Melnikov process in Eq. (5.9) can be considered

188

Applications: High-dimensional systems

solely in some statistical sense. Thus, one has to deduce an ejfectiwe Melnikov function (and hence to discuss the existence of its simple zeros) to obtain approximate predictions concerning the random Melnikov process. 3,

1.5

p 1 0.5 n

Figure 5.5: Lyapunov exponent distribution in the presence of noise (see the text). To this end, let us assume that in the absence of periodic excitations (y = 0), the threshold amplitude of bounded noise excitation for the onset of chaos occurs when the random Melnikov process has a simple zero in the mean-square sense, i.e.,

(5.11) where

S(G)=

ij2

x 2 -U 2

+

47r

+ E4/4 2 54/4) + E4G2

El2

~

( i j 2 - El2 -

(5.12)

is the spectral density of [ ( T ) . One finds that this estimate is reasonably confirmed by numerical simulations. Now, in the presence of periodic excitations ( F , p > 0), one defines an effective Melnikov function

(5.14) so that

M;>o

(To)

< M&

(70)

1

YTO.

(5.15)

Notice that Eq. (5.15) connects the effective Melnikov function with the random Melnikov process. Thus, one can again apply the results of Chap. 2 to the above

189

Controlling chaos in chaotic coupled oscillators

effective Melnikov function. In particular, for the main resonance case discussed above for a purely deterministic situation, one obtains a new boundary function

p

= - cos cp f Jcosz cp - (1 - Czff/A2),

(5.16)

which represents a sufficient condition for A4$, (TO)< 0, Wro, and hence for A4k (TO)< X>O 0, Wro (cf. Eq. (5.15)). Therefore, a first prediction is that the theoretical boundaries of the regularization regions associated with the random and deterministic cases have identical form and are symmetric with respect to the same (single) optimal suppressory value popt = 7r, while the respective enclosed areas are smaller for the former case than for the latter case (cf. Eqs. (5.8) and (5.16)). A second prediction is that there exists a critical amount of noise, ( T R , = ~ C , beyond which regularization is no longer possible, and that this critical value depends on the dissipation intensity, as expected. Figure 5.5 shows an illustrative example of the comparison between the theoretical predictions and Lyapunov exponent calculations, where the maximal Lyapunov exponent distribution in the cp - /3 parameter plane is plotted for the soliton in the Frenkel-Kontorova model (left panel) and for the associated perturbed pendulum (right panel) in the presence of noise X = 1.5, fl' = 0.4n, iT = 0.2 . Cyan and magenta regions indicate that the respective maximal Lyapunov exponent, A+ (p > 0), is non-positive and positive, respectively. The solid black contour represents the predicted boundary (cf. Eq. (5.16)), while the remaining parameters are as in Fig. 5.4. As for the deterministic case, one typically obtains extraordinary agreement between the two kinds of findings.

(-

)

Another interesting point is the robustness of the above theoretical predictions against a discrete (not a global) application of the chaos-taming excitations. Since Frenkel-Kontorova solitons present a very sharp spatial localization (typically, lo 1 in the present numerical simulations), one could expect that reliable soliton control may be achieved by solely applying the soliton-taming excitation to a few pendula of the Frenkel-Kontorova chain. Numerical simulations confirmed this conjecture. Figure 5.6 shows an illustrative example of a chain of 200 pendula with solitontaming excitations acting on every fiftieth pendulum in the absence of noise. One finds that the regularization region in the cp - p parameter plane has very nearly the same size as in the case of a global control. Figure 5.6 represents the maximal Lyapunov exponent vs cp/. for the Frenkel-Kontorovamodel (top panel) and temporal series of the soliton centre of mass (bottom panel), X ( t ) , for X = 0 , p = 0.6, and the remaining parameters as in Fig. 5.4. To understand the mechanism underlying the regularization of the chaotic soliton, temporal series of the soliton centre of mass were calculated for a constant /3 while the control initial phase changed according to cp ( t )= N ( t ) / N T ,where NT and N ( t ) = w t / (27r) are the total number of driving cycles and the number of cycles after time t , respectively. N

190

Applications: High-dimensional systems

A+-0,04 O ' " 0,08 0

T " i r " l

0.5

1

1.5

2

Cplx 200 X

150

100

50

'0 10000

30000

50000

The

Figure 5.6: Lyapunov exponent and time series of the soliton centre of mass. The bottom panel in Fig. 5.6 shows a representative example for NT = 200. Starting from 'p = 0, one sees that the soliton moves chaotically along the chain at 'p values that are out of the predicted regularization region, as expected. For cp values belonging to the predicted regularization region, one typically observes that the soliton moves t o be pinned t o the nearest (with respect to its position when crossing the chaotic threshold) pendulum subjected to the soliton-taming excitation where it remains regularized. Also, for certain 'p values which are above the predicted regularization range, one finds that the soliton moves with a definite (mode-locked) velocity along the chain while its behaviour remains chaotic. Finally, because of the generality of the present theoretical approach one expects it can also be applied to other types of lattices as well as to the cases of multiplicative noise and parametric chaos-taming excitation. 5.2

Controlling chaos in partial differential equations

During recent decades a great deal of effort has been devoted to the study of perturbed versions of exactly integrable partial differential equations that play an outstanding role in physical problems, such as the sine-Gordon, Korteweg-de Vries, and nonlinear Schrodinger equations, to cite only the most popular. A widely studied aspect in such a context is the persistence of localized solutions (moving and oscillating solitons) under perturbations which can be both dissipative and Hamiltonian. Generally speaking, this question is relevant in discerning how sensitive certain solutions are t o small changes in the integrable nonlinear equations, a notion that is commonly termed structural stability. In this regard, an ubiquitous and challenging phenom-

Controlling chaos in partial differential equations

191

enon, which appears when suitable dissipative and time-periodic perturbations act upon spatially extended nonlinear systems, is spatio-temporal chaos. At present, the control of spatio-temporal chaos implies a number of fundamental questions: What are the essential aspects of the system which need to be considered to develop a reliable control method? Can the complete system be controlled by solely acting on some of its degrees of freedom? Are there optimal regions in the parameter space of the chaos-controlling excitations and any optimal spatial distributions for such excitations? Is the theory discussed in previous chapters applicable (to any extent) to some of the aforementioned (among other) nonlinear evolution equations? Regarding the last question, a preliminary answer could be based on: i) The application of the method of phase space analysis for stationary waves, be they periodic, soliton, or shock like. Indeed, such stationary waves arising from the nonlinear partial differential equation can be studied in a co-moving reference frame as solutions to a set of ordinary differential equations, which (in various cases) are capable of being studied with the aid of the Melnikov's method. Thus, diverse solutions appearing in the phase space of the ordinary differential equations correspond to solutions of partial differential equations. For example, equilibria, periodic orbits, homoclinic and heteroclinic orbits, correspond to homogeneous solutions, traveling waves, pulses (solitons) to a homogenous state and fronts (shock waves) connecting two different homogeneous solutions, respectively. Additionally, homoclinic and heteroclinic bifurcations arising from the interaction of stable and unstable manifolds of equilibria in the phase space of the ordinary differential equation could be closely related to the onset of spatic-temporally chaotic dynamics in the original partial differential equation. ii) The consideration of separable solutions of partial differential equations which can exhibit spatially chaotic behaviour. Indeed, after separation of variables, one obtains an ordinary differential equation for the spatial part, which could be capable of being investigated with the help of Melnikov's method. iii) The use of a collective coordinate approach for the motion of a breather in partial differential equations capable of exhibiting such a soliton solution, which leads to a system of ordinary differential equations capable of being studied by means (of some version) of Melnikov's method. To illustrate such theoretical approaches, different physically relevant instances will be discussed as follows.

5.2.1 Damped sine-Gordon equation additively driven by two spatio-temporal periodic fields The sine-Gordon equation Utt--U,,+sinU=d'(~),

(5.17)

where d'(U) represents a general perturbation, covers a strikingly broad area of physical applications ranging from the dynamics of quasi-onedimensional ferromagnets

192

Applications: High-dimensional systems

with easy plane anisotropy, the theory of long Josephson junctions, dislocations in solids (as put forward by F'renkel and Kontorova), and liquid crystals, to the dynamics of a charged-density-wave system when the Peierls wave number is commensurate with the inverse spacing of an underlying ionic lattice. In this section, the following damped sineGordon equation, additively driven by two spatio-temporal fields in the form of monochromatic waves, will be considered: Utt

-

U,,+ sin U = -aUt

+ r sin (wt - k,x) + $sin

(Rt - khz - @) ,

(5.18)

where the amplitudes I?, $, wave numbers k, = 2 m / L , kh = 27rn'/L, and frequencies w, R correspond to the chaos-inducing and chaos-suppressing fields, respectively, @ is an initial phase, L is the total length of the system, and where weak dissipation (0 < a < 1) and small amplitudes (0 < I? < 1 , 0 < 7 < 1) are assumed. Physically, Eq. (5.18) describes, for example, the dynamics of the orientation angle U ( z , t ) of the magnetization vector lying in the easy plane of a quasi-onedimensional easy-plane ferromagnet in the presence of a strong constant magnetic field H (lying in the easy plane) and two additional weak variable magnetic fields in the form of monochromatic waves rsin(wt - knx),$sin(Rt - kkx - @), both being perpendicular (n/2 rad.) to H. In the absence of any chaos-suppressing field (7 = 0) and for the conditions exp[iU(x,t)] = exp[iU(x+ L,t)], U ( x , t = 0) = U,(x,t = 0) = 0, two different regimes have been identified (see below in Notes and references) characterized by the conditions k, < w and k, > w , respectively. Since periodic wave trains locked to the wave field I? sin (wt - knx) are numerically observed in both regimes (for certain ranges of the parameters), it is natural to consider an ansatz of the form U ( w t - knx) to the complete sineGordon equation. Thus, Eq. (5.18) is reduced, after requiring that both waves have the same phase velocity, to the perturbed pendulum equations ucc for k,

+ sinu = aiuc + r sin (Wl<) + $-sin (Wit + O) ,

(5.19)

> w , where (5.20) (5.21) (5.22)

R w; = -w1, W

(5.23) (5.24)

and

(5.25)

Controlling chaos in partial differential equations

193

for w > kn, where

w;

E

[ E

w2 - k i

= kg(W;

wc - knx

-

-

(5.26)

l),

IT

(5.27)

’

WZ

(5.28) (5.29) and where (5.30) (5.31) are the common initial phase and phase velocity, respectively. Note that Eq. (5.25) describes a damped driven pendulum, while Eq. (5.19) represents a driven pendulum with an antidamping term. Since both equations represent perturbed pendula (0 < I?, qr < 1), one can apply Melnikov’smethod to obtain analytical estimates of the ranges of the parameters (q,R, k;, 0 ) for inhibition of the chaos existing in the absence of the chaos-suppressing field. Thus, one readily achieves the Melnikov functions A4*(Co) = D If A1 sin (W1C0)f BI sin (W;Co 0 ), (5.32)

+

Mi([,)

=

-D2 7 A2 sin (W2t0)F B2 sin (W;l0+ 0 ),

(5.33)

for k, > w and kn < w , respectively, and where (5.34) (5.35) (5.36) (5.37)

A2

=

(“T)

21rrsech -

(5.38) (5.39)

The positive (negative) sign of the Melnikov functions refers to the top (bottom) homoclinic orbit of the underlying integrable pendulum (cf. Eq. (2.33)). Now one

194

Applications: High-dimensional systems

can directly apply the theory developed in Chap. 2 to the Melnikov functions (5.32), (5.33). In particular, for the case of subharmonic resonances between the (frequencies of the) two wave fields involved, one has the following suppression theorems. Theorem XI. Let R = pw,p an integer, such that

37r 7r @ = O opt f = q +opt - 7 r 7 TI 0, -, 2

(5.40)

for p = 472 - 3,4n - 2,4n - 1,4n ( n = 1,2, ...), respectively. Then M+ (Co) always has the same sign if and only if the following condition is fulfilled: ”lmin

<

6 Vmaxr

(5.41) (5.42) (5.43) (5.44)

Theorem XII. Let 52

= pw,p

an integer, such that

(5.45) for p = 4n - 3,4n - 2,4n - 1,4n ( n = 1 , 2 , ...), respectively. Then M - ( c 0 ) always has the same sign if and only if Eqs. (5.41)-(5.44) are satisfied. Theorem XIII. Let R = pw,p an integer, such that

(5.46) for p = 4n - 3,472 - 2,471 - 1,4n(n = 1 , 2 , ...), respectively. Then M+(Eo)always has the same sign if and only if the following condition is fulfilled:

(5.47) (5.48) (5.49) R9

C O S ~l(

i~

~W p2 /2) cosh(~W2/2)

(5.50)

‘

Theorem XIV. Let R

=pw,p

an integer, such that

(5.51)

195

Controlling chaos in partial differential equations

for p = 4 n - 3 , 4 n - 2 , 4 n - 1 , 4 n ( n = 1,2, ...), respectively. Then M - ( t 0 ) always has the same sign if and only if Eqs. (5.47)-(5.50) are satisfied. Remarks. First, in the absence of any chaos-suppressing field ( q = 0), the perturbed pendula (5.19), (5.25) can exhibit homoclinic chaos if

r 2 U;h(kn,Wf)=

-

IT

a!

r 2 Uih (kn,W f ) =

-

a

4wf

(1 - w;)

I,n

cosh

4wf 112 cosh 1)

7l (w; -

($ Jq) ,

(5.52)

($ d-)

(5.53)

,

respectively, where U$(kn, w f ) are chaotic threshold functions (cf. Eqs. (5.32) and (5.33), respectively). Second, one straightforwardly obtains the following limit:

(5.54)

lim u$(kn, w f ) = 00,

Vf+l

i.e., in such a limit chaotic behaviour is not possible. Third, for the limiting case of a purely temporal forcing, kn = 0, one has

4 lim Uih(knr w f ) = - cosh Vf+M

7r

( y )=

U~'(W),

(5.55)

(cf. Eq. (5.53)). Observe that the chaotic threshold function is a monotonously increasing function of the driving frequency. Also, Theorems XI1 and XI11 hold with the substitutions W, -+ w , W; + 0,a; a. Fourth, for each homoclinic orbit of the integrable pendulum, the optimal suppressory values O,t are the same for both regimes (i.e., w f < 1 and u f > 1, cf. Eq. (5.30)), for each resonance order p , respectively (cf. Eqs. (5.40), (5.45), (5.46), and (5.51)). 5.2.2 Damped sine-Gordon equation additively and parametrically driven by two spatio-temporal periodic fields Consider now the following damped sineGordon equation, addatawely and parametrically driven by spatio-temporal fields in the form of monochromatic waves: ---f

U,, - U,,

+ [l+ qrcos (0t - kLx - Q)] sinU = -aUt + rcos ( w t - k n x ) ,

(5.56)

where the amplitudes I', qr, wave numbers kn = 2 m / L , k; z 21rn'/L, and frequencies w , R correspond to the chaos-inducing and chaos-suppressingfields, respectively, Q is an initial phase, L is the total length of the system, and where weak dissipation (0 < (Y < 1) and small amplitudes (0 < r < 1 , 0 < q < 1) are assumed. Physically, Eq. (5.56) describes, among other cases, the dynamics of the orientation angle U ( x ,t )

196

Applications: High-dimensional systems

of the magnetization vector lying in the easy plane of a quasi-one-dimensional easyplane ferromagnet in the presence of a strong constant magnetic field H (lying in the easy plane) and two additional weak variable magnetic fields in the form of moncchromatic waves, with 7r/2,0 (rad) being the angles between H and rcos(wt - knz), qrcos(C2t - kkx - Q), respectively, in the easy plane. Consider again, as in the preceding section, an ansatz of the form U(wt - knx) for Eq. (5.56), i.e., after requiring that both waves have the same phase velocity, Eq. (5.56) is reduced t o the perturbed pendulum equations

ucc+[ 1 + r l r ~ O ~ ( W I C + @ ) ] s i n u = a ’ , u ~ + r c o s ( W ~ C ) , (5.57) for kn > w , where u, W,,(, W:,ai are given by Eqs. (5.20) to (5.24), respectively, and where n @.T--Q, (5.58) W

and

Utt

+ [I + qrcos (W;< + @)I sin U = -‘“hut - r cos (W2<),

(5.59)

for w > kn, where W2, t,W;, a;, and @ are given by Eqs. (5.26)-(5.29), and (5.58), respectively. Proceeding similarly to the case of two additional wave fields (cf. preceding section), one applies Melnikov’s method to the perturbed pendula (5.57) and (5.59) to obtain estimates of the ranges of the parameters (q,R, kk, @) for suppression of the chaos existing with no chaos-suppressing field. It is straightforward to calculate the associated Melnikov functions

for kn > w and kn < w , respectively, and where (5.62)

(T) ,

0 3

2 d - sech TWl

A3

B3

=

2.rrqrn2~,2 W2 csch

0 4

E

84,

A4

=

2 d ? sech

B4

E

2Tqrn2 w; W2

(z) ,

(5.64) (5.65)

(T) ,

csch

(5.63)

(5.66)

(T) .

(5.67)

The positive (negative) sign of the Melnikov functions refers to the top (bottom) homoclinic orbit of the underlying integrable pendulum (cf. Eq. (2.33)). Now, by

197

Controlling chaos in partial differential equations

applying the theory discussed in Chap. 2 to the case (for example) of subharmonic resonances between (the frequencies of) the two wave fields involved, one obtains the following suppression theorems. Theorem XV. Let 0 = p w , p an integer, such that (5.68) for p = 2 n - 1 , p = 2 n ( n = 1 , 2 , ...), respectively, and then _= 3 n / 2 , Vp (cf. Eq. (5.58)). Then M + (cf. Eq. (5.60)) always has the same sign if and only if the following condition is fulfilled

(co)

“lmin

< -

“lmin

= (1 -

“lmax

=

R3

G

-

Theorem XVI. Let R

2)

R3i

R3 P2 ’

r sinh (7rpW1/2) p2W: cosh(7rW1/2) ‘

=pw,p

(5.69) (5.70) (5.71) (5.72)

an integer, such that

for all p , and then @lopt 5 7r/2,37~/2for p odd, even, respectively (cf. Eq. (5.58)). Then M (cf. Eq. (5.60)) always has the same sign if and only if Eqs. (5.69)(5.72) are satisfied. Theorem XVII. Let R = p w , p an integer, such that

(co)

(5.74) for p = 272 - 1 , p = 2n(n = 1 , 2 , ...), respectively, and then = 7r/2,Vp (cf. Eq. (5.58)). Then M+([,) (cf. Eq. (5.61)) always has the same sign if and only if the following condition is fulfilled: (5.75) (5.76) (5.77) R4

r sinh ( n p W 2 / 2 ) p2W: sin h (T W 2 /2 ) .

-

(5.78)

198

Applications: High-dimensional systems

Theorem XVIII. Let 52 = p w , p an integer, such that

(5.79) for all p , and then U & F 3 ~ / 2 , 7 r / 2for p odd, even, respectively (cf. Eq. (5.58)). Then M - (to) (cf. Eq. (5.61)) always has the same sign if and only if Eqs. (5.75)(5.78) are satisfied. Remarks. First, in the absence of any chaos-inhibiting wave (7= 0), the perturbed pendula (5.57), (5.59) can exhibit homoclinic chaos if

(5.80) respectively, where the threshold functions U$ are given by Eqs. (5.52), (5.53), respectively. Therefore, the second and third remarks in the previous section hold, but the fourth remark does not due to the parametric character of the chaos-suppressing field. 5.2.3 Damped sane-Gordon equation additively driven by two temporal periodic excitations In this section an alternative approach is discussed for the suppression of chaos arising from the equation

Utt - U,,

+ sin U = -6Ut + r sin ( w t ) + $'sin

(Rt + U) ,

(5.81)

which was briefly considered (as a limiting case of spatiotemporal fields), in the third remark to Theorems XI1 and XI11 . Here, a singlebreather excitation is assumed to be present for 6 = r = 0. In the absence of any chaos-suppressingac driving (77 = 0), it is well known that the simultaneous action of dissipative (S # 0) and spatially uniform driving forces (r # 0) can break the breather into a kink-antikink pair, which can then recombine into a breather soliton. This process may occur repeatedly yielding an intermittent chaotic sequence. The role to be played by the chaos-suppressing ac driving Vrsin ( O t U) is then to control such competition between the breather and the kink-antikink pair. To t h s end, one makes a severe mode truncation to the breather collective coordinates, which yields a set of ordinary differential equations capable of being studied with the aid of Melnikov's method. It is worth noting that such an effective equation exhibits a separatrix dynamics which is equivalent to the separatrix dynamics of the full sineGordon equation. In particular, homoclinic bifurcations correspond to (chaotic) intermittent binding and unbinding of the kinkantikink pair. In order to determine the equations of motion for the breather collective modes, one first needs the breather and kink-antikink solutions to the unperturbed sine-Gordon equation (6 = I? = 0 in Eq. (5.81)).

+

199

Controlling chaos in partial differential equations

Breather: UB(x,t)

=

tan 8 sin [w( t ) ] cash [2k (X - ~ o ) ]

4tan-'

w ( t ) = wo + tcose, Ic

E

1 -sine, 2 16sin8.

= =

(5.82)

Kink-antikink pair ( K K ) :

U,K(X,t)

=

4tan-l

sinh [u(t - t o ) /(1 - u2)ll2] ~COSh[(Z-Xo)/(l- u ~ ) ' / ~ ] (5.83)

Here, u is the t --+ 00 velocity of the kink while --u is the t --+ 03 velocity of the antikink, and W O ,Z O ,(0 ~ < 8 < 27r) are constants. Also, wE = cos 8 is the breather's internal frequency and E , EKE are the breather and kink-antikink-pair energies, r e spectively. Now, one assumes the ansatz that the solution of the perturbed sineGordon equation has the same form as the unperturbed breather (5.82), but now allowing 8 and wo to be temporal functions:

~(= t )wo(t) +

f cos8 (t')dt'.

(5.84)

This approximation means that one is assuming a perturbation sufficiently weak so that its principal effect on the system is to continuously alter the frequency and the phase of the breather. Therefore, such an approximation is valid when W B << 1 since the breather energy is EB = 16(1 - w ; ) ' / ~ . Next, one can deduce the equations of motion for the breather collective modes (u, w), where

4dA ( t ) / d t

A(t)

=

tanosinw ( t ).

After the coordinate transformation z can be shown to be

(5.85)

= tan (u/4) , Z = w, such equations of motion

(5.86)

200

Applications: High-dimensional systems

with

(5.87) and the perturbation

(5.88) where

+

7r

+

p ( t ) = - [rsin (wt) qrsin (Rt @)I, 4 [(w”z~)z2-s{z4+w2[z2+(1-w2)(1+z2)]}]

sp(z,w) =

(1

+

+ sw2) (w2+ 9 )(1+ s) (1 + z y 2 + + w2s3 (1 + 222) (1+ 2))”

2w2z2s2(1

2’)

(l+sw2)(w2+z~)(l+s)(l+z2)1/2 ’ w [(2 2 )( w 2 2 )z2 - s {z4 - w 2[(2 2 )z4 - (1 - w 2 )(1 g6(z?w) = (1 sw2)(w2 z2) (1 2 2 )

+

+

+ w [w22(1 + 222) (1+ z”] + ( 1 + sw2)(w2+ 9 )(1 +

+

+

+

+ 2)]}]

22)’

S

E

sinh-l ( z ) z (1 + z y 2

(5.89) ’

Noting that the separatrix in the phaseplane u - u is given by

uO,*(t) = f 4 tan-l(t), vo,*(t) = f 4 / ( 1 + t2), one finds that on the separatrix in the phaseplane z

-

(5.90) w

(5.91)

where the positive (negative) sign refers to the upper (lower) homoclinic orbit in the phaseplane u - u , which is obtained for the energy E = 16 of the unperturbed energy functional E = 2 tan2(u/4) ( i ~ / 4sec2(u/4). )~ Thus, the corresponding Melnikov function is M*(to) = F D A sin (wto) B sin (Rto 9), (5.92)

+

+

+

+

20 1

Controlling chaos in partial differential equations

with

D - 6 pv

(1 - tanh’ z) [ (2 - tanh’ z) tanh2 z l+r 2.037276,

1,

A

+ r + r2 (1

-

tanh4 z)]

dx (5.93)

= 24F % ( w ) ,

(5.94)

(5.95) where

r O0

= 22 csch (22) ,

cos (w sinhz) sechz (1 - r ) (1 - tanh’ x)

+

1, O0

(5.96)

(1 r)’ cos (w sinh x) sech z [2r2tanh2 z

(I

+

+

T

dx

( +

+ 41 dz.

(5.97)

Now, by applying the theoretical results of Chap. 2 to the case of subhaxmonic resonances between (the frequencies of) the two drivings, one obtains the following suppression theorems. Theorem XM. Let R = p w , p an integer, such that 7T 37l 9 = 91t = 7r, -, 0, y, 2

(5.98)

for p = 4n - 3,4n - 2,4n - 1,4n (n = 1,2, ...), respectively. Then M+(to) (cf. Eq. (5.92)) always has the same sign if and only if the following condition is fulfilled:

(5.99) (5.100) -

rlmax

=

R

-

P2’

(5.101) (5.102)

Theorem XX. Let R = p w , p an integer, such that 9 = 9 & t = n , 2371 ’ 0 , - , 7r 2

(5.103)

for p = 4n - 3,4n - 2,4n - 1,4n ( n = 1,2, ...), respectively. Then M-(to) (cf. Eq. (5.92)) always has the same sign if and only if Eqs. (5.99)-(5.102) are satisfied.

202

Applications: High-dimensional systems

Remarks. First, in the absence of any chaos-suppressing ac driving ( q = 0), the perturbed system (5.86) can exhibit homoclinic chaos if 4

6

>, Uth(w) = -(2.03727)/%((~), 7r

(5.104)

where the chaotic threshold function Uth(w) is a monotonously increasing function of the driving frequency. Note the great similarity between the chaotic threshold functions Uth(w)and U;’((W) (cf. Eq. (5.55)). 5.2.4 Nonlinear Schrodinger equation subjected to dissipative and spatially periodic perturbations The nonlinear Schrodinger equation

iu,+ u,,+ 2 IUI2 u = EP(U)

(5.105)

is a universal partial differential equation describing the evolution of wave envelopes in a dispersive weakly nonlinear medium. The main applications of Eq. (5.105) are found in nonlinear optics, where the function U ( X t, ) represents a complex envelope of the electromagnetic field, in dynamics of quasi-onedimensional ferromagnets with easy-axis anisotropy, and in plasma physics, where it describes electron (Langmuir) waves. Consider perturbations of the form

E P ( U )= -SU,

+ y cos ( k x )eiwt+ qyU cos ( K x + E),

(5.106)

where the amplitudes y, qy and the wave numbers k , K correspond to the chmsinducing and chaos-suppressing fields, respectively, E is an initial phase, and where small parameters (0 < 6 , y , q < 1) are assumed. In order to characterize the s u p pression of spatially chaotic behaviour in certain parameter regions ( q ,K , Z), the separable solution V ( Xt,) = S ( X exp ) (iwt) is substituted into (5.105),(5.106)to yield

d2S dX2

ws+ 2s3 = -6- ddxS +

+ qyscos ( K +~Z) ,

cos (kX)

(5.107)

which represents a damped, biharmonically driven two-well D f i g oscillator and where w is a frequency. The application of Melnikov’s method provides the Melnikov function M* (XO) = -D A sin(kxo) B sin(Kzo Z), (5.108)

+

+

+

(5.109) (5.110)

203

Controlling chaos in partial differential equations

B G -"TJyK2csch ( K ) 2 2W'P '

(5.111)

and where the positive (negative) sign refers to the right (left) homoclinic orbit of the underlying integrable Duffing equation (6 = y = 0). As an illustrative application of the theory developed in Chap. 2, consider, for instance, the case of a subharmonic resonance between (the wave numbers of) the two wave fields involved. Thus, one readily obtains the following suppression theorems. Theorem XXI. Let K = pk,p an integer, such that

(5.112) for p = 4n - 3,4n - 2,4n - 1,471( n = 1,2, ...), respectively. Then M+(zo) always has the same sign if and only if the following condition is fulfilled: TJmin

(5.113)

< TJ G ~ ~ m a x r

(5.114)

R

TJmax

=P2'

(5.115)

sinh (.rrpk~-~/'/2) cosh (7Tkw-'/2/2) .

(5.116)

Theorem XXII. Let K = pk,p an integer, such that

- -- = 0, -,

" = Copt

7r

2

7r,

37r

-,

(5.117)

2

for p = 4n - 3,4n - 2,4n - 1,4n ( n = 1,2, ...), respectively. Then IM-(zo)always has the same sign if and only if Eqs. (5.113)-(5.116)are satisfied. Remarks. First, in the absence of any chaos-suppressing field (TJ= 0), the perturbed twc-well D f i n g oscillator (5.107) can present spatial homoclinic chaos if

2w3/2

2 6 2 U(W,k) G 37rk cash

(&)

'

(5.118)

where U(w, k ) is the chaotic threshold function, which indicates a coupling between the spatial and temporal driving periods.

204

Applications: High-dimensional systems

4

3 2 2

1

0

2

4

8

6

w Figure 5.7: Contour plot of the chaotic threshold function U ( w , k ) (cf. Eq. (5.118)).

A contour plot of U ( w ,k ) is shown in Fig. 5.7. Second, for fixed k ( w ) one has that U ( w ,k )

m when w + 00 ( k -+ m), i.e., in such limits chaotic behaviour is not possible. 5.2.5 44 model additively driven by two spatio-temporal periodic fields The 4*model has been extensively studied partly because of its diverse applications in condensed-matter physics. These applications include the description of topological excitations in quasi-onedimensional systems like biological macromolecules as well as the description of structural phase transitions in ferromagnetic and ferroelectric materials, etc. Consider the following q54 model equation, additively and parametrically driven by spatietemporal fields in the form of plane waves: $tt

+ d3 = -aqht + ~ C O (Sw t - knx) + ~

- $ J-~4 ~

-+

~ C (Rt O S-

kkx - Q ) ,

(5.119)

where the amplitudes, y,qy, wave numbers kn E 27rn/L, k; = 27rn'/L, and frequencies w , R correspond to the chaos-inducing and chaos-suppressingfields, respectively, Q is an initial phase, L is the total length of the system, and where weak dissipation (0 < a < 1) and small amplitudes (0 < y , q < 1) are assumed. Consider again, as in the case of the sineGordon equation, an ansatz of the form 4 ( w t - k,x) for Eq. (5.119). After requiring that both waves have the same velocity v ~ fEq. , (5.119) reduces to the perturbed oscillator equations

4CC + 4 - q53

= a:&

+ ycos (WIO - V Y 4 C O S (W:c+ 'PI,

(5.120)

for kn > w , where Wl, <,Wi,a; are given by Eqs. (5.21) to (5.24), respectively, v ~ is f

205

Controlling chaos in partial differential equations

given by Eq. (5.31), and where

R

(5.121)

@.-T-Q, W

and (#lcc-(#l+(#13

= -.~(#lc-yros(WzE)+qyCOS(W;E+ip),

(5.122)

for w > k,, where W2, t ,W;, a;, and ip are given by Eqs. (5.26)-(5.29), and (5.121), respectively. Note that Eq. (5.120) represents a perturbed one-well Duffing oscillator, while Eq. (5.122) represents a perturbed two-well D&ng oscillator. Now one applies Melnikov’s method to the perturbed Duffing oscillators (5.120) and (5.122) to obtain estimates of the ranges of the parameters (q,R, kk, Q) for suppression of the chaos existing in the absence of the chaos-suppressing field. It is straightforward to obtain the respective Melnikov functions

M* (Co)

= D1 f A1 sin (WlC,)

F B1sin (WiCo+ i p ) ,

Mi(to) = -D2 7 A2 sin (Wzt0)f B2 sin (WJt0+ a) ,

(5.123) (5.124)

for k, > w and k, < w , respectively, and where (5.125) (5.126) (5.127) 0 2

4 3

-.I,,

(5.128) (5.129) (5.130)

The positive (negative) sign of the Melnikov functions (5.123) and (5.124) refers to the top (bottom) and right (left) homoclinic orbits, respectively, of the respective integrable Duffing oscillators. Now, one can apply the theoretical findings obtained in Chap. 2. In the following, the case of subharmonic resonances between (the frequencies of) the two wave fields involved is considered. Theorem XXIII. Let R = pw, p an integer, such that (5.131)

206

Applications: High-dimensional systems

forp=4n-3,4n-2,4n-l,4n(n= 1 , 2 , . . . ) ,respectively. Then M+(C0)always has the same sign if and only if the following condition is fulfilled:

(5.132) (5.133) (5.134) (5.135)

Theorem XXIV. Let s2 = p w , p an integer, such that be (5.136) and then @opt EE ~ , ~ / 2 , 0 , 3 ~respectively, /2, for p = 4n - 3, 4n - 2, 4n - 1, 4n (n = 1,2, ...), respectively. Then M-(Co) always has the same sign if and only if Eqs. (5.132)-(5.135)are satisfied. Theorem XXV. Let s2 = p w , p an integer, such that

(5.137) and then = T , 3 ~ / 2 , 0 , ~ /respectively, 2, for p = 4n - 3,4n - 2,4n - 1,4n(n = 1,2, ...), respectively. Then M+([,) always has the same sign if and only if the following condition is fulfilled:

(5.138) (5.139) (5.140) R2

E

1 cosh(~pW2/2) p cosh(~W2/2)’

(5.141)

Theorem XXVI. Let s2 = p w , p an integer, such that (5.142) and then @Opt = T , 7r/2,0,37l/2, respectively, for p = 4n - 3,4n - 2,4n - 1,4n(n = 1,2, ...), respectively. Then M - ( t 0 ) always has the same sign if and only if Eqs. (5.138)-(5.141) are satisfied.

207

Controlling chaos in partial differential equations

Remarks. First, in the absence of any chaos-suppressing field ( q = 0), the perturbed D f f i g oscillators (5.120), (5.122) can exhibit homoclinic chaos if -

a

2

U3th (knGnluf)

2 2 Uih (kn,W f ) = a

2wf

37T (1 - w;) kn

sinh

4Vf cosh 3 & r( W j - 1) kn

(3d q )

,

(5.143)

(%,/v) ,

(5.144)

respectively, where U;f-(kn,wf) are chaotic threshold functions (cf. Eqs. (5.123) and (5.124), respectively. Second, one straightforwardly obtains the following limit:

(5.145)

, = m, lim ~ & ( l t - ~ wp)

Uf+l

i.e., in such a limit chaotic behaviour is not possible. Third, for the limiting case of a purely temporal forcing, kn = 0, one has lim Vf+Cc

uih(kn,wf)=

(5.146)

(cf. Eq. (5.144)). Note that Uih (w) is a monotonously increasing function of the driving frequency. Also, Theorems XXV and XXVI hold with the substitutions Wz -+ w,

w;+ R,a; + a.

44 model additiwely and parametrically driven by two spatio-temporal periodic fields Consider now the following 44 model additiuely and parametrically driven by spatiotemporal fields in the form of monochromatic waves: 5.2.6

g5tt

-

dZZ- 4 + qb3 =

+

~ C O (wt S

- knx)

+

~ ~ $ C O( R St

- k;x - a),

(5.147)

where the amplitudes y,qy,wave numbers kn = 27~nlL,k; = 2rn'/L, and frequencies w , R correspond to the chaos-inducing and chaos-suppressing fields, respectively, Q is an initial phase, L is the total length of the system, and where weak dissipation (0 < a < 1) and small amplitudes (0 < y < 1 , 0 < q < 1) are assumed. Consider again, as in the preceding section, an ansatz of the form #(wt - knx) for Eq. (5.147), i.e., after requiring that both waves have the same phase velocity wf,Eq. (5.147) is reduced to the perturbed oscillator equations

4cc + 4 - 43 = a:@< + YCOS (WIC) - V Y 4 C O S (

W C + @)

1

(5.148)

for kn > w , where Wl, <,Wi,a; are given by Eqs. (5.21) to (5.24), respectively, wf is given by Eq. (5.31), and where

R

E 7T-

W

-

Q,

(5.149)

208

Applications: High-dimensional systems

and

$Jet - 45 + $J3 = -+&

-

ycos (Wzt)

+ qy$Jcos(Wit + @) ,

(5.150)

for w > k,, where W2,J1W;.,crk,and @ are given by Eqs. (5.26)-(5.29), and (5.149), respectively. Proceeding similarly to the case of two additional wave fields, one applies Melnikov’s method to the Duffing oscillators (5.148) and (5.150) to obtain estimates of the ranges of the parameters (7, R, kh, a) for suppression of the chaos existing with no chaos-suppressing field. It is straightforward to calculate the associated Melnikov functions

M* ( ( 0 )

= D3

Mi(to)= -0

f A3 cos (WI C0) 4

+ B3 sin (WiC0 + @) ,

(5.151)

+ B4 sin (W;to + @) ,

(5.152)

T A4 sin ( W2t0)

for k, > w and k, < w , respectively, and where

(5.153) (5.154)

, 4 1 3

0 4

A4

(5.156)

- ~ 2 ,

=

&7ryW2sech

(5.155)

-

(5.157) (5.158)

The positive (negative) sign of the Melnikov functions (5.151) and (5.152) refers to the top (bottom) and right (left) homoclinic orbits, respectively, of the respective integrable Duffmg oscillators. Now one can apply the theoretical findings obtained in Chap. 2. In the following, the case of subharmonic resonances between (the frequencies of) the two wave fields involved is considered. Theorem XXVII. Let R = pw, p an integer, such that

(5.159) forp=2n-l,2n

(n = 1)2,...), respectively,mdthen@&

= 37r/2,Vpp. ThenM+(Co)

209

Controlling chaos in partial differential equations

always has the same sign if and only if the following condition is fulfilled:

(5.160) (5.161) (5.162) R3

Jz sinh ( V W d J z ) = p2W1 s i n h ( n W 1 / a )

(5.163) ’

Theorem XXVIII. Let R = p w , p an integer, such that

(5.164) and then Q’.pt = 7r/2,3~/2for p = 2n - 1, 2n (n = 1 , 2 , ...), respectively. Then A!-(&,) always has the same sign if and only if Eqs. (5.160)-(5.163)are satisfied. Theorem XXIX. Let R = p w , p an integer, such that

(5.165)

Qkt

and then = T , 3 n / 2 , 0 , ~ / 2respectively, , for p = 4n - 3,4n - 2,4n - 1,4n(n = 1,2, ...), respectively. Then A4+(&,)always has the same sign if and only if the following condition is fulfilled

(5.166) (5.167) (5.168) (5.169) Theorem XXX. Let R = pw,p an integer, such that

(5.170) and then = 0 , 3 ~ / 2T, , ~ / 2 respectively, , for p = 4n - 3,4n - 2,4n - 1,4n(n = 1,2, ...), respectively. Then M-(&,) always has the same sign if and only if Eqs. (5.166)-(5.169) are satisfied.

210

Applications: High-dimensional systems

Remarks. First, in the absence of any chaos-inhibiting wave (77 = 0), the perturbed Duffing oscillators (5.148), (5.150) can exhibit homoclinic chaos if

(5.171) respectively, where the threshold functions U$, are given by Eqs. (5.143), (5.144), respectively. Therefore, the second and third remarks in the previous section hold.

5.3

Notes and references

Localized control of spatio-temporal chaos has only recently been investigated in distributed systems. Examples exist in isotropic systems [l],plasma physics [2], continuous extended systems [3,4], and coupled oscillators [5,6,7]. A collective-coordinate Hamiltonian is derived for a sineGordon or 44 soliton in [8]. A complete Hamiltonian formalism for a kink on a onedimensional discrete lattice in which the position of the centre of the kink appears as one of the canonical variables is given in [9]. This derivation is valid for a particle chain in a periodic potential when there exists a solitary-wave solution in the continuum limit. Soliton motion on a damped, driven, one-dimensional lattice is proposed in [lo] by using a semi-discrete kink solution for which its centre of mass is considered as a collective variable. The authors show that the kink location is then governed by the equation of a damped sinusoidally driven pendulum in leading order. The collective coordinate formalism given in [9,10] is applied in [7,11] to damped driven F’renkel-Kontorova chains. An extensive review on collective coordinates in spatially inhomogeneous soliton-bearing equations is given in [12]. Suppression of temporal phase locked chaos in a damped sineGordon system subjected to two sinusoidal excitations having null initial phases is demonstrated numerically in [13]. Length-scale competition in a damped sineGordon system driven by the spatio-temporal force r sin (wt - k n z ) is studied in [14]. The authors show that there are two distinct regimes for the case of a flat initial condition: w > k, and w < k,. For w > k,, the system presents a periodic travelling wave locked to the driving for r below a certain critical value rc(n), while for I? > r,(n)it presents a translating two-breather excitation. Further increase of r gives rise to the appearance of spatio-temporal chaos. The discussion of many of the results of this chapter originated in work by the present author and coworkers [6,7]. [l] Hu, G., Qu, Y . , and He, K., (1995) “Feedback control of chaos in spatiotemporal systems,” Int. J. Bzjur. Chaos 81, pp. 901-iii.

[2] Aranson, I., Levine, H., and Tsimring, L., (1994) “Controllingspatiotemporal chaos,” Phys. Rev. Lett. 72,pp. 2561-2564.

Notes and references

211

[3] Boccaletti, S., Bragard, J., and Arecchi, F. T., (1999) “Controlling and synchronizing space time chaos,” Phys. Rev. E 59,pp. 6574-6578. [4] Grigoriev, R. 0. and Handel, A,, (2002) “Nonnormality and the localized control of extended systems,” Phys. Rev. E 66,pp. 065301(R)/1-4.

[5] Kocarev, L., JanjiC, P., Parlitz, U., and Stojanovski, T., (1998) “Controlling spatiotemporal chaos in coupled oscillators by sporadic driving,” Chaos, SoZit. FI-act. 9, pp. 283-293. [6] C h a c h , R., Preciado, V., and Tereshko, V., (2003) “Suppressing temporal and spatio-temporal chaos by multiple weak resonant excitations: Application to arrays of chaotic coupled oscillators,” Europhys. Lett. 63,pp. 667-673. [7] Martinez, P. J. and C h a c h , R., (2004) “Taming chaotic solitons in Frenkel-Kontorova chains by weak periodic excitations,” Phys. Rev. Lett. 93,pp. 237006/1-4.

[8] Rice, M. J., (1983) “Physical dynamics of solitons,” Phys. Rev. B 28, pp. 3587-3589. [9] Willis, C., El-Batanouny, M., and Stancioff, P., (1986) “SineGordon kinks on a discrete lattice. I. Hamiltonian formalism,” Phys. Rev. B 33,pp. 1904-1911. [lo] Sayadi, M. K. and Pouget, J., (1992) “Chaos transition of soliton motion on a microstructural lattice,” Physicu D 55,pp. 259-268. [ll] Martinez, P. J., Falo, F., Mazo, J. J., Floria, L. M., and Shnchez, A., (1997) “Mode locking in discrete soliton dynamics under ac forces,” Phys. Rev. B 56,pp. 87-90. [12] Shnchez, A. and Bishop, A. R., (1998) “Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations,” SIAM Rev. 40, pp. 579-615. [13] Filatrella, G., Rotoli, G., and Salerno, M., (1993) “Suppression of chaos in the perturbed sineGordon system by weak periodic signals,” Phys. Lett. A 178,pp. 81-84. [14] Cai, D., Bishop, A. R., and Sanchez, A., (1993) “Length-scale competition in the damped sine-Gordon chain with spatiotemporal periodic driving,” Phys. Rev. E 48, pp. 1447-1452.

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Chapter 6

FURTHER REMARKS AND OPEN PROBLEMS 6.1

Open problems

6.1.1 Beyond the main resonance Several extensions of the current theory of control by weak harmonic excitations remain to be developed. Among them: i) To obtain the boundaries of the regularization regions in the control parameter plane for the case of a general resonance (not just the main) between the excitations involved. ii) To develop the multiharmonic control theory beyond the main resonance case. iii) To extend the theoretical approach for the case of random excitations to other kinds of noise. iv) To extend the theory to (some family of) multidimensional systems capable of being studied by (generalized versions of) Melnikov’s method. v ) To obtain analytical approximations of the regularized responses for the deterministic case of a general resonance between the chaos-inducing and chaossuppressing excitations. These points clearly indicate that the present approach to control chaos by weak periodic excitations can be extensively developed. 6.1.2 Reshaping-induced control As emphasized in Chap. 1, the dynamics of generic, nonlinear and non-autonomous, systems is highly sensitive to changes of the waveform of the periodic excitations which are driving the system. In particular, the existence of generic reshaping-induced orderttchaos routes suggest that such a waveform effect should be taken into account in the control scenario described for harmonic functions (both chaos-inducing and chaos-controlling) in Chap. 2. In this regard, the use of Jacobian elliptic instead of harmonic functions could provide a useful first step towards this goal, since their waveform may be independently varied by solely changing the elliptic parameter, besides the fact that the former are a natural generalization of the latter. To illustrate this approach with a particular example, consider the generic system

213

214

Further Remarks and Open Problems

where U ( x ) is a nonlinear potential, - d ( x , k ) is a generic dissipative force, ysin ( w t ) is a chaos-inducing excitation, and 777 sn [R ( m )t O (m);m] is an as yet undetermined suitable chaos-controlling excitation, with sn (.; m) being the Jacobian elliptic function of parameter m and period T , R = 4 K ( m ) / T ,O = 2 K ( m ) 0 / ~(0 E [0,27r]), and where K ( m )is the complete elliptic integral of the f i s t kind. Also, the excitation and dissipation terms are weak perturbations of the underlying conservative system x d U ( x ) / d x = 0 which has a separatrix. For m = 0, one recovers the case of a harmonic chaos-controlling excitation. For this case, we have seen in Chap. 2 that chaos can be reliably tamed by choosing chaos controlling amplitudes 77 E (qmin,qmax] provided the frequencies w and 2n/T satisfy certain resonance conditions and the initial phase is suitably chosen. Now the challenge is to find the dependence of the regularization intervals on the shape parameter m : (qmin(m),qmax(m)].In this regard, one expects qmin( m ) to be a decreasing function of m ( m E [0,l ] ) ,since A(m)= J:”sn [Rt;m]dt increases over the interval [0, 11, but for the present this is just a conjecture.

+

+

6.1.3 Amplitude modulation control Regarding the way the chaos-controlling excitations are applied to each particular system, an interesting alternative to the cases studied in Chap. 2 consists in applying a parametric excitation to the amplitude of the chaos-inducing excitation. This procedure represents a new aspect of the flexibility of the general method and facilitates its experimental realization. Similarly to the discussion of the control of chaos by the previously studied procedures, we can assume any particular form of the Melnikov function (2.37) to discuss the effectiveness of the present one. Case of symmetric pulses Therefore, consider for instance the following nonlinearly damped, biharmonically driven, two-well Duffing oscillator:

3: - x

+ Px3 = -62

+ F [ 1 + ~ C O (Rt S + O)]cos ( w t ),

(6.2)

where 77, R, and O are the normalized amplitude factor, frequency, and initial phase, respectively, of the chaos-controlling parametric excitation (0 < 77 << l),and p, 6, n, F, and w are the normalized parameters of the potential coefficient, damping coefficient, damping exponent, chaos-inducing amplitude, and chaos-inducing frequency, respectively (0 < 6, F << 1, ,B > 0 , n = 0 , 1 , 2 , ...). The application of Melnikov’s method to Eq. (6.2) yields the Melnikov functions (cf. Eq. (2.88)),

M*(to)

=

77 D f A ( w )sin (wto)f -A (R - W ) sin [(R - W ) to + O] 2 77 *-A (R w ) sin [(R + w ) to + 01, 2

+

(6.3)

Open problems

215

with

(6.4) where B ( m ,n ) is the Euler beta function. Note that A ( w ) is a positive function with a single maximum, while D is a negative function, over the complete range of their parameters. Now note that for the main resonance case, R = w , the Melnikov functions (6.3) reduce to

M*(to)

=D

k A ( w ) sin ( d o ) & -77A (2w)sin (2wto + 0 ) , 2

(6.5)

to which the results concerning the subharmonic resonances can be directly applied (cf. Chap. 2 ) . Similarly, for the second resonance case, R = 2w, Eq. (6.3) reduces to

M*(to)

77 2

77

= DfA(w)sin(wto)f-A(w)sin(wto+O)f-A(3w)sin(3wto+O),

2

(6.6)

and the suppression of simple zeros of this Melnikov function (with A ( w ) 2 101) can be easily discussed with the aid of the previous results concerning subharmonic resonances (cf. Chap. 2). Case of asymmetric pulses Consider now the suppressory effect of a parametric excitation in the form of harmonic asymmetric pulses, X - 3:

+ Px3 = -SX

JZJn-l + F [l - ~ C O S ’ (Rt + O)] cos ( w t ) ,

(6.7)

with associated Melnikov functions

M*(to)

=

D f A ( w ) sin (wto) -77A

( w ) sin (wto) 2 F -77A ( 2 0 - w ) sin [ ( 2 R- w ) t o 201 4 $ A4 ( 2 0 w ) sin [ ( 2 ~w ) to 201 .

+

+

One sees that, for the ultrasubharmonic resonance R (6.8) reduce to ~ * ( ( t ~ )=

+ +

=w/2,

(6.8)

the Melnikov functions

D f (1 - 2) A ( w ) sin (wto) 2 77 ~ -4A ( 2 w sin ) (2wto + 2 0 ) ,

(6.9)

216

Further Remarks and Open Problems

while for the main resonance (R = w ) one has

+ A ( w ) sin (wto + 20)

+

T ~ (3w) A sin ( b t o 20). (6.10) 4 One can again deduce the corresponding conditions for suppressing and enhancing chaos with the aid of the ideas and findings of Chap. 2. 6.2

Further applications

6.2.1 Ratchet systems The so-called ratchet systems are nonlinear systems that can extract usable work from unbiased non-equilibrium fluctuations. The control of directed energy transport by zerc-average forces in both point particles and spatially extended ratchet systems is a challenging problem which has attracted a great deal of attention, partly because of its possible applications in diverse fields such as biology and micro- and nanc-technology. Although the majority of the literature on ratchets considers the influence of noise, there also exists motivation to understand the transport properties of deterministic inertial ratchets, as we have seen in Chap. 1. These ratchets typically exhibit chaotic dynamics which determine their transport properties, and it has been shown that stabilization of the dynamics is typically accompanied by a suppression of the deterministic chaos, which may be related to a bifurcation from chaos to periodic motion or to phase locking phenomena. Therefore, it seems pertinent to consider the problem of controlling the transport of harmonically driven inertial ratchets by application of a secondary resonant excitation. Specifically, let us consider for example the one-dimensional problem of a particle driven by two periodic time-dependent external forces under the influence of an asymmetric periodic potential of the ratchet type. The equation of motion is (6.11) + y’cos (w’t) + qy’ cos (R’t + 9 ), dx where m is the mass of the particle, 6‘ is the friction coefficient, y‘ and w’ are the amplitude and the frequency, respectively, of the primary external force, while qy’, R‘, and 9 are the amplitude, frequency, and initial phase, respectively, of the ratchetcontrolling external force. The ratchet potential is

d V ( x ) = -6’x mx + -

[ (2y)

V(z) = Vo sin - +:sin(?)],

(6.12)

where L and Voare the period and amplitude of the potential, respectively. Equation (6.11), in dimensionless variables, is written as

d2.z dr2

1 2

- + sinz = --cos(2z)

dz dr

- 6-

+ y c o s ( w r ) + vycos(Rr + Q ) ,

(6.13)

217

Further applications

m,

+

where z = 27rx/L 7r/2, r = Rot, Ro = (27r/L) 6 5 6'/ (rnQo), y = y'/R$, = w'/Ro, and R = R'/Qo. There exists numerical evidence of the importance of the initial phase @ in the stabilization of the ratchet motion. Thus, one can conjecture that the parameter space regions (7,R, @) of the ratchet-controlling force where stabilization of the ratchet motion is found correspond to some extent to those where regularization of the chaos existing in its absence (7 = 0) is achieved. In order to apply Melnikov's method to Eq. (6.13), let us assume in the following that the damping, forcing, and breaking-symmetry potential terms may be considered as small amplitude perturbations of the underlying integrable pendulum, and that, in the absence of the ratchet-controlling external force (7 = 0), the perturbed pendulum exhibits homoclinic chaos which corresponds to a certain chaotic attractor existing in (6.13). Now we let the ratchet-controlling external force act on the system and deduce analytical estimates of the regularization regions in its parameter space. Then we conjecture that such theoretical predictions could correspond to some extent to the regions where the stabilization of the ratchet dynamics occurs. The application of Melnikov's method to Eq. (6.13) gives us the Melnikov function M i ( T O ) = Di f A cos ( W T O ) f B cos (QTO @) , (6.14) w

+

where (6.15) (6.16)

B

=

27r73ysech

(?)

(6.17)

7

where the plus (minus) sign corresponds to the upper (lower) homoclinic orbit of the unperturbed pendulum. Now the results from Chap. 2 directly apply t o the Melnikov function (6.14). Thus, one straightforwardly obtains the following results. Theorem XXXI. Let R = p , p an integer, 6 < 6, = 113 - ~ 1 1 6such , that @ = 9,t

= 7r, 0,

(6.18)

for p = 2n - 1,2n (n = 1,2, ...), respectively. Then M*(To)always has the same sign if and only if the following condition is fulfilled: Vmin

V m i. n -=

< v G Vmawr

(6.19)

(

(6.20)

I - - lDil

Vmax

A R

=P2'

) R,

(6.21)

218

Further Remarks and Open Problems

R E cosh ( 7 r p / 2 )

cosh (7rw/2) '

(6.22)

respectively. Theorem XXXII. Let R = p w , p an integer, 6 > 6,, such that

9 = 9,t for all p . Then satisfied.

A 4 + ( ~ 0 )always has

E 7r,

(6.23)

the same sign if and only if Eqs. (6.19)-(6.22) are

Remarks. First, for the main resonance case (R = w ) , the Melnikov function M*(To) always has the same sign (i.e., homoclinic chaos is suppressed) inside the region of the Q - q parameter plane limited by the boundary function q = - cos 9 f dcos2 9 - [1 - (D*/A)'].

(6.24)

Second, in the absence of any chaos-suppressing external force ( q = 0), the perturbed pendulum (6.13) can exhibit homoclinic chaos if

Y 3 IYfh*l7 where the chaotic threshold amplitudes

(6.25)

~ g are * given by (6.26) (6.27)

7;-

=

4 -

7r

(6,

+ 6) cosh

(6.28)

One sees that the breaking-symmetry potential term yields a difference between the chaotic threshold amplitudes associated with the upper (positive velocity) and lower (negative velocity) homoclinic orbits, and that such a difference increases as 6 -+ 6,, where the critical value seems to be closely related to the fine structure constant: 6, 137 x 10-3.

6.2.2 Coupled Bose-Einstein condensates Another possible application of the current theory is the control of chaotic population oscillations between two coupled Bose-Einstein condensates with time-dependent asymmetric potential and/or time-dependent tunneling amplitude for different damping forces. Within the mean-field approximation of the two-mode Gross-Pitaevskii equation, ignoring the finite-temperature and damping effects, two Josephson-coupled states of a two-state single-particle Bose-Einstein condensate are described by the nonlinear equations (6.29) (6.30)

219

Notes and references

where are the zero-point energies of the condensates, Ul,z are proportional to the mean-field energies, and K describes the tunneling dynamics between two condensates. This system of equations is valid in the approximation of a weak link between condensates. The wave functions are given by $ j = m e x p [ZOj ( t ) ]where , N j ( t )= and 0j (t) are the numbers of condensed atoms and phases of states, respectively ( j = 1,2). The total number of atoms NT = N l ( t ) N2(t) is conserved. Introducing the fractional population inbalance ~ ( t=) [ N l ( t )- Nz(t)]/ N T and the relative phase Q, ( t )= Ol(t)- & ( t ) ,one gets the following differential equations

l$jlz

+

i = - 2 ~ ( t d-sina ) Q,

=

uRz+-

2K(t)z

-

66,

(6.31)

cosa + A E ( t ) ,

(6.32)

where A = (Ul + Uz) NT and v = f l for the positive and negative atomic scattering length, respectively. The damping term dQ, appears when a non-coherent dissipative current of normal-state atoms, proportional to the chemical potential difference, is taken into account. Consider the case of periodic modulations of the tunnel coupling parameter K ( t )= KO K1 sin ( w t ) qK1 sin (OL 0 ), (6.33)

+

+

+

+

where K1 sin ( w t ) and qK1 sin (Ot 0) (0 < 17 < 1) are the chaos-inducing and chaossuppressing terms, respectively, while it is also assumed that the trap asymmetry A E remains constant. Assuming that the dissipative and time-periodic terms are perturbative terms, the system (6.31), (6.32) is capable of being studied by Melnikov's method when the underlying integrable system is a two-well Duffing oscillator (KO< R/8). Thus, for the right homoclinic orbit, the Melnikov function is M(t0)=

-D - A cos (wto) - B cos (Qto

+ 0 ),

(6.34)

where D = D (6,A,Ko),A = A (A,&, Kl,w), and B = B ( h , K o , q ,K1,R) are known positive functions. Once more, one can obtain the corresponding conditions for suppressing and enhancing chaos with the aid of the results of Chap. 2. Similarly, one can consider the case where the chaotic population oscillations between the two coupled BoseEinstein condensates are induced by a time-dependent asymmetric potential A E ( t ) = AEo AEl sin ( w t ) qAE1 sin (Rt 0 ) . (6.35)

+

6.3

+

+

Notes and references

Experimental and numerical evidence of amplitude modulation control of a single mode COz laser with an intra-cavity acousto-optic modulator is given in [1,2]. The ratchet effect induced solely by an unbiased biharmonic signal of the form €1 cos (wt) €2 cos (2wt + cp) was reported for the first time in the experimental work by Seeger and Maurer [3], and was independently re-discovered in [4]. The same

+

220

Further Remarks and Open Problems

type of biharmonic signal has been applied experimentally in a process called zerointegrated field gel electrophoresis t o separate chromosomal DNA [5-61. For additional references, see t h e extensive review by Reimann [7]. T h e discussion of the chaotic atomic population oscillations between two coupled BoseEinstein condensates follows the work by Abdullaev and Kraenkel [8]. T h e case of a time-dependent asymmetric potential is considered in [9].

[1] [2] [3] [4] [5] [6] [7] [8] [9]

Schwartz, I. B., Triandaf, I., Meucci, R., and Carr, T. W., (2002) “Open-loop sustained chaos and control: A manifold approach,” Phys. Rev. E 66,pp. 02621311-7. Meucci, R. et al., (2004) “Global manifold control in a driven laser: sustaining chaos and regular dynamics,” Physica D 189, pp. 70-80. Seeger, K. and Maurer, W., (1978) “Nonlinear electronic transport in TTF-TCNQ observed by microwave harmonic mixing,” Solid State Commun. 27,pp. 603-606. Ajdari, A., Mukamel, D., Peliti, L., and Prost, J., (1994) “Rectified motion induced by ac forces in periodic structures,” J. Phys. I France 4, pp. 1551-1561. Desruisseaux, C., Slater, G. W., and Kist, T. B., (1998) “Trapping electrophoresis and ratchets: a theoretical study for DNA-protein complexes,” Biophys. J. 75,pp. 1228-1236. Vidybida, A. K. and Serikov, A. A., (1985) “Electrophoresis by alternating fields in a non-Newtonian fluid,” Phys. Lett. A 108, pp. 170-172. Reimann, P., (2002) “Brownian motors: noisy transport far from the equilibrium,” Phys. Rep. 361,pp. 57-265. Abdullaev, F. Kh. and Kraenkel, R. A., (2000) “Coherent atomic oscillations and resonances between coupled Bose-Einstein condensates with time-dependent trapping potential,” Phys. Rev. A 62,pp. 02361311-9. Lee. C. et al., (2001) “Chaotic and frequency-locked atomic population oscillations between two coupled Bose-Einstein condensates,” Phys. Rev. A 64, pp. 05360411-8.

INDEX adiabatic invariant, 101 almost , 113 attractor strange chaotic, 38 autoresonance, 92, 94 bifurcation heteroclinic, 38 homoclinic, 38 Bose-Einstein condensate, 218 bounded random excitation additive, 81 multiplicative, 85 breather, 191 chaos heteroclinic , 1 homoclinic, 1 spatial, 203 spatic-temporal, 191 steady, 38, 56, 151 transient, 38, 151 before escape from a potential well, 125 chaos-controlling excitation multiple, 74 off-resonance, 4 parametric, 57, 214 periodic resonant character, 2 chaos-inducing excitation chaotic, 3, 136 multiple, 74 periodic, 3 quasiperiodic, 3 chaos-suppressing excitation

incommensurate, 63 localized, 181 multiple, 74 parametric incommensurate, 67 relative suppressory effectiveness, 69 area criterion, 71 chaotic escape model fractal character of the basin boundary, 125 Helmholtz oscillator, 64 with one-way escape, 125 chaotic threshold function for a driven nonlinear Schrodinger equation, 203 for a driven sine-Gordon equation, 202 collective coordinate approach for the motion of a breather, 191 for the motion of a soliton, 185 complete elliptic integral of the first kind, 5, 32, 168 of the second kind, 33 control of chaos definition, 1 feedback methods, 1 localized, 184, 189 non-feedback methods, 2 Curie’s principle, 22 delta-kicked rotator, 169 dissipative forces Coulombian, 39 Liknardian, 134 Newtonian, 39, 136 Stokesian, 39 22 1

222

electrostatic wave packet, 159 case of three waves, 159 chaotic threshold function, 161 escape-suppressing excitation “off-resonance”, 141 vs 1 : 1 and 1 : 2 parametric resonances, 131 vs 1 : 1 parametric resonance, 128 Euler beta function, 39, 58, 65, 127, 128, 215 excitation spatic- t emporal, 34 time-dependent, 2 Fibonacci numbers, Fi, 64 fine structure constant, 218 forcing waveform, 5 F’renkel-Kontorova chain. 184 golden ratio, 24, 99, 161 golden section, a, 64, 67 convergents, 68, 142 Helmholtz oscillator biharmonically excited, 64 Jacobian elliptic function cn, 5, 15 dn, 14, 168 sn, 14, 32 Josephson junction, 39 arrays, 1 Frenkel-Kontorova model, 184 McCumber model. 144 KAM tori, 171, 174 Korteweg-de Vries equation, 190 Lyapunov exponent, 32, 72 leading, 5, 72 for a biharmonically excited Josephson junction, 151 for a biharmonically excited twcwell Duffing oscillator, 72

INDEX

for a triharmonically excited twowell Duffing oscillator, 80 for the Frenkel-Kontorova model, 186 maximum symmetry principle, 48, 63 Melnikov function definition, 37 effective, 82, 83, 85, 188 equivalent, 76 for a biharmonically excited Josephson junction, 145 for a perturbed Helmholtz oscillator, 65 for a perturbed pendulum, 39, 193, 196, 208 for a two-well Duffing oscillator, 57, 71, 214 for a universal escape equation, 127 Melnikov’s method, 34 multiple-control scenario for the main resonance case, 76 area criterion, 77 optimal suppressory initial phase difference, 78 natural symmetry, 47 noise additive, 81 band-limited, 81 bounded, 185 fractal, 81 Gaussian, 81 multiplicative, 84 non-harmonic excitation chaos controlling, 214 vs harmonic excitation, 4 waveform effect, 4 nonlinear Klein-Gordon equation q54 model, 204 nonlinear Schrodinger equation perturbed, 202

INDEX

orbits heteroclinic, 38 homoclinic, 35, 38 oscillators, 1 chaotic coupled, 34 parametric resonance, 3 partial differential equations, 34 localized solutions, 190 separable solutions, 191 stationary solutions, 191 pendulum biharmonically excited, 39, 192 excited by symmetric pulses, 4 periodic &function, 5 perturbation dissipative, 190 Hamiltonian, 190 time-periodic, 191 phase space analysis for stationary waves, 191 random Melnikov process, 82-84 ratchet, 216 ratchet effect, 20 broken symmetries, 21 reshaping-induced basin boundary fractality, 15 crisis phenomena, 14 directed transport, 21 escape from a potential well, 16 resonance Galilean, 91 geometrical, 93 main, 69 nonlinear, 92 stochastic, 102 subharmonic, 44, 45, 215 ultrasubharmonic, 60, 215 sineGordon equation, 190 breather vs kink-antikink pair, 198

223

for quasi-onedimensional ferromagnets, 192, 195 Smale horseshoe, 38, 82 Smale-Birkhoff theorem, 38 soliton, 190 chaotic, 184 solutions broken-symmetric, 47, 149 symmetric, 47, 149 standard map, 159 relativistic, 172 stochastic heating of charged particles, 159 structural stability, 5, 169, 190 suitable initial phase difference maximum-range interval, 54 optimal suppressory values, 53 geometrical characterization, 51 threshold values, 51 suppression of chaos maximum resonance order, 55 synchronization regularization-induced, 184 system dissipative, 1 dissipative vs Hamiltonian, 31 Hamiltonian, 35 limiting, 45, 51, 53, 63, 149 periodically excited, 64 quasiperiodically excited, 64 universal escape model, 64 escaping basin, 132 non-escaping basin, 133 safe basin, 132 wave packet time-like, 168 Wiener process, 82, 185