It's a Nonlinear World
Drawing examples from mathematics, physics, chemistry, biology, engineering, economics, medicine, politics, and sports, this book illustrates how nonlinear dynamics plays a vital role in our world. Examples cover a wide range from the spread and possible
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Series Editors: J. M. Borwein, Callaghan, NSW, Australia H. Holden, Trondheim, Norway Editorial Board: L. Goldberg, Berkeley, CA, USA A. Iske, Hamburg, Germany P.E.T. Jorgensen, Iowa City, IA, USA S. M. Robinson, Madison, WI, USA
For other titles in this series go to: http://www.springer.com/series/7438
Richard H. Enns
It’s a Nonlinear World
1C
Richard H. Enns Department of Physics Simon Fraser University Burnaby, BC V5A 1S6 Canada [email protected]
ISSN 1867-5506 e-ISSN 1867-5514 ISBN 978-0-387-75338-6 e-ISBN 978-0-387-75340-9 DOI 10.1007/978-0-387-75340-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938298 Mathematics Subject Classification (2010): 34A34, 97Mxx © 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)
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This text is dedicated to my loving wife, Karen, who lights my path through this nonlinear world.
Contents xi
Preface
Part I: WORLD OF MATHEMATICS 1
World of Nonlinear Systems 1.1 Introduction to Nonlinear ODE Models 1.2 Introduction to Difference Equation Models 1.3 Solving Nonlinear ODEs on the Computer ..
1 3
4 9 12
2 World of Nonlinear ODEs 2.1 Breakdown of Linear Superposition. 2.2 Some Analytically Solvable Examples . 2.3 Fixed Points and Phase-Plane Analysis .. 2.4 Bifurcations . 2.5 Hysteresis and the Jump Phenomena .. 2.6 Limit Cycles . 2.7 Strange Attractors and Chaos . 2.8 Fractal Dimensions . . . 2.9 Poincare Sections . 2.10 Power Spectrum .. . .
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3
71
World of Nonlinear Maps 3.1 Fixed Points of One-Dimensional Maps . 3.2 Stability Criterion . . . . . . . . . . . .... 3.3 Cobweb Diagram. . . . . . 3.4 Period Doubling to Chaos . . . . . . ... . . 3.5 Creating Lorenz Maps. . . . . . 3.6 Lyapunov Exponent . . . . . . . . . 3.7 Two- Dimensional Maps .. Mandelbrot and Julia Sets . 3.8 . . . . . 3.9 Chaos versus Noise . . 3.10 Controlling Chaos vii
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44 47 49 53 56 58 59
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77 80 82 84
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CONTENTS
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4 World of Solitons 4.1 Korteweg-deVries Solitons .. 4.2 Sine-Gordon Solitons . . . . . . . . . . . . . . . . . . 4.3 Similarity Solutions. . . . . 4.4 Numerical Simulation . . . 4.4.1 Finite Difference Approximations .. 4.4.2 The Zabusky-Kruskal Algorithm . . . 4.4.3 Method of Characteristics 4.4.4 Numerica
