Transient Chaos Complex Dynamics on Finite Time Scales
This book represents the first comprehensive treatment of Transient Chaos. It gives an overview of the subject based on three decades of intensive research. One special emphasis is on applications, and the fact that certain interesting dynamical phenomena
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Ying-Cheng Lai · Tam´as T´el
Transient Chaos Complex Dynamics on Finite-Time Scales
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Ying-Cheng Lai Department of Electrical Engineering Arizona State University Tempe Arizona USA [email protected]
Tam´as T´el Department of Theoretical Physics Institute of Physics E¨otv¨os University 1117 Budapest Hungary [email protected]
ISSN 0066-5452 ISBN 978-1-4419-6986-6 e-ISBN 978-1-4419-6987-3 DOI 10.1007/978-1-4419-6987-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011920691 Mathematics Subject Classification (2010): 37-XX c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
In a dynamical system, transients are temporal evolutions preceding the asymptotic dynamics. Transient dynamics can be more relevant than the asymptotic states of the system in terms of the observation, modeling, prediction, and control of the system. As a result, transients are important to dynamical systems arising from a wide range of disciplines such as physics, chemistry, biology, engineering, economics, and even social sciences. Research on nonlinear dynamical systems has revealed that sustained chaos, as characterized by a random-like yet structured dynamics with sensitive dependence on initial conditions, is ubiquitous in nature. A question is, then, can chaos be transient? A common perception, as conveyed in many existing books on nonlinear dynamics, is that chaos is an asymptotic property that manifests itself only after a long observation. Indeed, standard characteristics of chaos, such as the Lyapunov exponents that measure the exponential separation rates of nearby trajectories and hence quantify the degree of the sensitivity to initial conditions, are defined in the infinite time limit. These features seem to be incompatible with the possibility of chaotic transients. Research on nonlinear dynamics has shown, however, that the essential feature of chaos is the existence of so-called chaotic sets in the phase space, and quantitative characterization of chaos is meaningful with respect to the dynamics on such sets only. Since this does not imply that trajectories from random initial condi
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