Synchronization And Control Of Chaos: An Introduction For Scientists And Engineers

224 Pages·2004·3.715 MB·English

by  J. M. Gonzalez-Miranda

FFIRS.qxd 6/16/04 8:37 AM Page iv Quark03 Quark03:Desktop Folder:Chapter-FM:

Published by
Imperial College Press
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
SYNCHRONIZATION AND CONTROL OF CHAOS
Copyright © 2004 by Imperial College Press
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.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 1-86094-488-4
Printed in Singapore.
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A mi padre, Alfonso, y
a mi madre, Mar´ıa Angeles.
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Preface
Overthe lastfifteen yearsanimportantresearchactivity has been devoted
to the dynamics of coupled and driven chaotic oscillators. This is still an
active field of research; however, a considerable body of knowledge has
alreadybeenestablished. Thisismainly,butnotexclusively,intheformof
research papers in physics and mathematics journals. These results are of
interestforscientificanalysisandexplanationinallscientificdisciplines,as
well as for possible application in medicine and engineering. The purpose
of this book is to provide a systematic and broad account of that research
for a wide audience. This will be achieved by a selection of the most basic
theoreticalresults, as well as experiments andapplications to be presented
at a mathematicallevel accessible for people workingin non-hardsciences.
This, however,does not exclude physicists and mathematicians looking for
an introduction to the field.
Resonanceandnormalmodes ofvibrationarewellknownclassicalphe-
nomena observed in driven and coupled linear harmonic oscillators, which
have a great relevance in all the natural sciences and in engineering. Since
thelastquarterofthetwentiethcenturythestudyofchaoticoscillatorshas
emergedasanobjectofgreatinterestinphysicsandmathematics. Inview
of the importance that the classical results for the dynamics of driven, or
coupled harmonic oscillators, has achieved in science and technology, the
question of which phenomena emerge when chaotic oscillators are coupled
or somehow driven or perturbed has been, and is of great interest. The
most relevant phenomena studied until now are the synchronization and
the suppression of chaos. This book is aimed to provide a brief account of
theseinvestigations. Theapproachusedconsidersdifferentschemesofdriv-
ing or coupling. For each of these set-ups, the phenomena that occur are
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viii Synchronization and Control of Chaos
studied, and experiments that show that these phenomena are observable
in the real world are discussed.
The first two chapters, which deal with the theory of oscillators, har-
monic and chaotic, are aimed to provide the reader with basic concepts
on dynamical systems and chaos theory needed to follow the rest of the
book. In the next five chapters, the different scenarios which emerge when
chaotic oscillators are externally driven or mutually coupled are studied.
These include a variety of forms of chaos synchronization, which are sys-
tematically defined and characterized; as well as several conditions, and
procedures which lead to taming chaos; i.e., to turn the chaotic motion to
periodic. Moreover, throughout these seven chapters, by means of exam-
plesborrowedfromdifferentdisciplines,the multidisciplinarynatureofthe
subjectwillbeillustrated. Inthelastchapter,ageneralvisionofthefieldof
synchronization and control of chaos is given by means of a summary, and
examplesofapplicationofthisbodyofknowledgeinscienceandtechnology
are presented to sustain a discussion of perspectives for the future.
The didactical approach followed throughout the book combines tuto-
rial and review techniques. For each scientific issue studied, the basic con-
cepts and techniques are explained in detail and immediately exemplified
by means of numerical simulations made with simple, but significant mod-
els. This is followedby the presentationof a brief review and discussion of
experimentalrealizationsandnumericalsimulations. The former willshow
that the theory introduced is meaningful in the real world, and the latter
willexpandthescopeofthetheoryandexamplespreviouslyposed. Allthe
aboveisprovidedtogetherwithabibliographyaimedtoallowthereaderto
probe deeper into the particular issues that especially interest him or her.
I have been working on coupled and driven chaotic oscillators for the
lasttenyears. Theviewpresentedinthisbook,hasbeenshapedduringmy
experience with the development of my own research and the interaction
with many colleagues, whom I want to acknowledge and thank. I also
gratefully acknowledge the financial support from DGI which allowed the
scientific activity that has made this book possible.
Jesu´s Manuel Gonza´lez Miranda
Barcelona, Spain.
December 2003
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Contents
Preface vii
1. Dynamics of Coupled and Driven Harmonic Oscillators 1
1.1 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The free harmonic oscillator . . . . . . . . . . . . . . 2
1.1.2 Damped harmonic oscillator . . . . . . . . . . . . . . 5
1.2 Driven and coupled harmonic oscillators . . . . . . . . . . . 6
1.2.1 Periodically driven harmonic oscillator: resonance . . 7
1.2.2 Coupled harmonic oscillators: normal modes . . . . . 10
1.3 Nonlinear oscillators . . . . . . . . . . . . . . . . . . . . . . 12
2. Chaotic Oscillators 17
2.1 Nonlinear flows and maps . . . . . . . . . . . . . . . . . . . 17
2.2 Poincar´e maps and bifurcation diagrams . . . . . . . . . . . 24
2.3 Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Power spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Unstable periodic orbits . . . . . . . . . . . . . . . . . . . . 44
2.6 Time series analysis . . . . . . . . . . . . . . . . . . . . . . 49
3. Periodically Driven Chaotic Oscillators 53
3.1 Phase synchronization . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 Phase of a chaotic oscillator . . . . . . . . . . . . . . 54
3.1.2 Characterizationof phase synchronization . . . . . . 61
3.1.3 Numerical simulations and experimental evidence . . 67
3.2 Chaos suppression . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Weak resonant forces . . . . . . . . . . . . . . . . . . 72
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