Control of Homoclinic Chaos by Weak Periodic Perturbations

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CONTROl OF HOMOCN lC l CHAOS BY WEAK PERIODIC PERTURBATIONS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon 0. Chua University of California, Berkeley Series A.

MONOGRAPHS AND TREATISES

Volume 35:

Introduction to Control of Oscillations and Chaos A. L. Fradkovd A. Yu. Pogromsky Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska,K. Czolczynski, T. Kapitaniak & J. Wojewoda

Volume 36: Volume 37:

Invariant Sets for Windows - Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & 0.V. Malysheva

Volume 38:

Nonlinear Noninteger Order Circuits & Systems - An Introduction P. Arena, R. Caponetto, L. Fortuna & D. Port0

Volume 39:

The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda

Volume 40:

Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin

Volume 41:

Synchronization in Coupled Chaotic Circuits and Systems c. w. w u

Volume 42:

Chaotic Synchronization: Applications to Living Systems E. Mosekilde, Y. Maistrenko & D. Postnov

Volume 43:

Universality and Emergent Computation in Cellular Neural Networks R. Dogaru

Volume 44:

Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems Z. T. Zhusubaliyev & E. Mosekilde

Volume 45:

Bifurcation and Chaos in Nonsmooth Mechanical Systems J. Awrejcewicz & C.-H. Lamarque

Volume 46:

Synchronization of Mechanical Systems H.NJmeJer & A. Rodriguez-Angeles

Volume 47:

Chaos, Bifurcations and Fractals Around Us W. Szemplinska-Stupnicka

Volume 48:

Bio-Inspired Emergent Control of Locomotion Systems M. Frasca, P. Arena & L. Fortuna

Volume 49:

Nonlinear and Parametric Phenomena V. Damgov

Volume 50:

Cellular Neural Networks, Multi-Scroll Chaos and Synchronization M. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle

Volume 51:

Symmetry and Complexity K. Mainzer

Volume 52:

Applied Nonlinear Time Series Analysis M. Small

Volume 53:

Bifurcation Theory and Applications T, Ma & S. Wang

Volume 54:

Dynamics of Crowd-Minds

A. Adamatzky

Series Editor: Leon 0. Chua

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

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore. 596224

USA office: 27 Warren Street, Suite 401-402,Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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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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Printed in Singapore by B & JO Enterprise

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.


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


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


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.

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


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


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


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


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


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


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


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.


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

<
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


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


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 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


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 A - ID1 G Cmin. N

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


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


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 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


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

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


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.

&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)

<
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 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 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


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


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-


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.


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 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


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


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*

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 0. Another relativistic effect is the modification of the fixed points existing in the classical (Newtonian) regime: (4.103)

174


Two limiting cases may be distinguished: E 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-


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


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 w , where (5.20) (5.21) (5.22)

R w; = -w1, W

(5.23) (5.24)

and

(5.25)


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


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

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