Dynamical Systems and Chaos
Over the last four decades there has been extensive development in the theory of dynamical systems. This book starts from the phenomenological point of view reviewing examples. Hence the authors discuss oscillators, like the pendulum in many variation inc
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Henk Broer Floris Takens
Dynamical Systems and Chaos
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Henk Broer University of Groningen Johann Bernoulli Institute for Mathematics and Computer Science The Netherlands [email protected]
Floris Takens University of Groningen Johann Bernoulli Institute for Mathematics and Computer Science The Netherlands
An earlier version of this book was published by Epsilon Uitgaven (Parkstraat 11, 3581 PB te Utrecht, Netherlands) in 2009. ISSN 0066-5452 ISBN 978-1-4419-6869-2 e-ISBN 978-1-4419-6870-8 DOI 10.1007/978-1-4419-6870-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938446 Mathematics Subject Classification (2010): 34A26, 34A34, 34CXX, 34DXX, 37XX, 37DXX, 37EXX, 37GXX, 54C70, 58KXX 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)
Iterations of a stroboscopic map of the swing displaying complicated, probably chaotic dynamics. In the bottom figure the swing is damped while in the top figure it is not. For details see Appendix C.
Preface
Everything should be made as simple as possible, but not one bit simpler Albert Einstein (1879–1955)
The discipline of Dynamical Systems provides the mathematical language describing the time dependence of deterministic systems. For the past four decades there has been ongoing theoretical development. This book starts from the phenomenological point of view, reviewing examples, some of which have guided the development of the theory. So we discuss oscillators, such as the pendulum in many variations, including damping and periodic forcing, the Van der Pol system, the H´enon and logistic families, the Newton algorithm seen as a dynamical system, and the Lorenz and R¨ossler systems. The doubling map on the circle and the Thom map (also known as the Arnold cat map) on the 2-dimensional torus are useful toy models to illustrate theoretical ideas such as symbolic dynamics. In the appendix the 1963 Lorenz model is derived from appropriate partial differential equations. The phenomena concern equilibrium, periodic, multi- or quasi-periodic, and chaotic dynamics as these occur in all kinds of modelling and are met both in computer simulations and in e
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