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The dynamics of physical, chemical, biological or fluid systems generally must be described by nonlinear models, whose detailed mathematical solutions are not obtainable. To understand some aspects of such dynamics, various complementary methods and viewpoints are of crucial importance. In this book and its companion volume, Perspectives of nonlinear dynamics, volume 1, the perspectives generated by analytical, topological and computational methods, and interplays between them, are developed in a variety of contexts. The presentation and style is intended to stimulate the reader's imagination to apply these methods to a host of problems and situations. The text is complemented by copious references, extensive historical and bibliographical notes, exercises and examples, and appendices giving more details of some mathematical ideas. Each chapter includes an extensive section commentary on the exercises and their solution. Graduate students and research workers in physics, applied mathematics, chemistry, biology and engineering will welcome these volumes as the first broad introduction to this important major field of research.
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Perspectives of nonlinear dynamics Volume 2 VOLUME 2 Perspectives of nonlinear dynamics E. ATLEE JACKSON Professor of Physics, University of Illinois at Urbana-Champaign AMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521354585 © Cambridge University Press 1990 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1990 First paperback edition (with corrections) 1991 Reprinted 1993, 1994 Re-issued in this digitally printed version 2008 A catalogue record for this publication is available from the British Library ISBN 978-0-521-35458-5 hardback ISBN 978-0-521-42633-6 paperback -7; 1 Contents of Volume 2 Preface Acknowledgements 6 Models based on second order difference equations 6.1 Some origins of maps in R2: Delayed and coupled logistic maps; Poincare surface of section in extended phase space (R2 x I); area-preserving maps; nonconserva- tive vs conservative maps; Levinson-Smith relaxation oscillator; Henon and Heiles Hamiltonian 6.2 Rotation and winding numbers: Maps and flows; knots, algebraic constants of the motion xv xviii 1 1 9 6.3 The Cartwright-Littlewood, Levinson and Levi analyses: The extraordinary family of solutions, KO, of the forced self-exciting oscillator; equivalence to Bernoulli sequences; Levi's extensions 16 6.4 Some abstract nonconservative maps in R2: Henon's map; strange attractor; contracting map; geometrically wild vs dynamically-wild sets; Lyapunov exponents; period-three does not imply chaos in R2; a fractal boundary between basins of attraction; Julia and Mandelbrot sets; Newton map 22 6.5 The standard map; twist maps: Ball on a vibrating plate; the microtron accelerator; harmonic lattice in a periodic potential; toroidal magnetic fields; twist maps; characteristic multipliers 6.6 'Near-integrable' systems: Poincare's last geometric theorem; hyperbolic and 33 elliptic fixed point pairs; stable and unstable manifolds; homoclinic and heteroclinic points; KAM surfaces; Poincare's chaotic tangle; Bernoulli sequence 41<