Texts in Applied Mathematics 19 Editors J.E. Marsden L. Sirovich M. Golubitsky W. Jager F. John (deceased) Advisor G.1ooss Texts in Applied Mathematics 1. Sirovich: Introduction to Applied Mathematics. 2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. 3. Hale/Kofak: Dynamics and Bifurcations. 4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed. 5. HubbardlWest: Differential Equations: A Dynamical Systems Approach: Part I: Ordinary Differential Equations. 6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems. 7. Perko: Differential Equations and Dynamical Systems. 8. Seaborn: Hypergeometric Functions and Their Applications. 9. Pipkin: A Course on Integral Equations. 10. Hoppensteadt/Peskin: Mathematics in Medicine and the Life Sciences. 11. Braun: Differential Equations and Their Applications, 4th ed. 12. Stoer/Bulirsch: Introduction to Numerical Analysis, 2nd ed. 13. Renardy/Rogers: A First Graduate Course in Partial Differential Equations. 14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. 15. Brenner/Scott: The Mathematical Theory of Finite Element Methods. 16. Van de Velde: Concurrent Scientific Computing. 17. Marsden/Ratiu: Introduction to Mechanics and Symmetry. 18. HubbardIWest: Differential Equations: A Dynamical Systems Approach: Higher Dimensional Systems. 19. Kaplan/Glass: Understanding Nonlinear Dynamics. 20. Holmes: Introduction to Perturbation Methods. 21. Curtain/Zwart: An Introduction to Infinite Dimensional Linear Systems Theory. Daniel Kaplan Leon Glass Understanding Nonlinear Dynamics With 294 Illustrations Springer Science+Business Media, LLC Daniel Kaplan Leon Glass Department ofPhysiology McGill University 3655 Drummond Street Montreal, Quebec, Canada H3G lY6 Series Editors J.E. Marsden L. Sirovich Department of Mathematics Division of Applied Mathematics University of California Brown University Berkeley, CA 94720 Providence, RI 02912 USA USA M. Golubitsky w. Jăger Department of Mathematics Department of Applied Mathematics University of Houston Universităt Heidelberg Houston, TX 77004 Im Neuenheimer Feld 294 USA 69120 Heidelberg, FRG Mathematics Subject Classifications (1991): 39xx, 92Bxx, 35xx Library of Congress Cataloging-in-Publication Data Kaplan, Daniel, 1959- Understanding nonlinear dynamics I Daniel Kaplan and Leon Glass. p. cm. - (Texts in applied mathematics; 19) Includes bibliographica1 references and index. ISBN 978-0-387-94440-1 ISBN 978-1-4612-0823-5 (eBook) DOI 10.1007/978-1-4612-0823-5 1. Dynamics. 2. Nonlinear theories. 1. G1ass, Leon, 1943-. II. Title. III. Series. QA845.K36 1995b 94-23792 003~85-dc20 Printed on aeid-free paper. © 1995 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1995 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), except for brief excerpts in connection with reviews or scholarly ana1ysis. Use in connection with any form of infonnation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone. Production managed by Frank Ganz; manufacturing supervised by JeffTaub. Photocomposed copy prepared from the author's W'JlX files. 987654321 ISBN 978-0-387-94440-1 To Maya and Tamar. - DTK To Kathy, Hannah, and Paul and in memory of Daniel. - LG Series Preface Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics ( TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. About the Authors Daniel Kaplan specializes in the analysis of data using techniques motivated by nonlinear dynamics. His primary interest is in the interpretation of irregular physiological rhythms, but the methods he has developed have been used in geo­ physics, economics, marine ecology, and other fields. He joined McGill in 1991, after receiving his Ph.D from Harvard University and working at MIT. His un­ dergraduate studies were completed at Swarthmore College. He has worked with several instrumentation companies to develop novel types of medical monitors. Leon Glass is one of the pioneers ofwhat has come to be called chaos theory, specializing in applications to medicine and biology. He has worked in areas as diverse as physical chemistry, visual perception, and cardiology, and is one of the originators of the concept of "dynamical disease." He has been a professor at McGill University in Montreal since 1975, and has worked at the University of Rochester, the University of California in San Diego, and Harvard University. He earned his Ph.D. at the University of Chicago and did postdoctoral work at the University of Edinburgh and the University of Chicago. Preface This book is about dynamics-the mathematics of how things change in time. The universe around us presents a kaleidoscope of quantities that vary with time, ranging from the extragalactic pulsation of quasars to the fluctuations in sunspot activity on our sun; from the changing outdoor temperature associated with the four seasons to the daily temperature fluctuations in our bodies; from the incidence of infectious diseases such as measles to the tumultuous trend of stock prices. Since 1984, some of the vocabulary of dynamics-such as chaos, fractals, and nonlinear-has evolved from abstruse terminology to a part of common lan­ guage. In addition to a large technical scientific literature, the subjects these terms cover are the focus of many popular articles, books, and even novels. These pop­ ularizations have presented "chaos theory" as a scientific revolution. While this may be journalistic hyperbole, there is little question that many of the important concepts involved in modem dynamics-global multistability, local stability, sen­ sitive dependence on initial conditions, attractors-are highly relevant to many areas of study including biology, engineering, medicine, ecology, economics, and astronomy. X PREFACE This book presents the main concepts and applications of nonlinear dy­ namics at an elementary level. The text is based on a one-semester undergraduate course that has been offered since 1975 at McGill University and that has been constantly updated to keep up with current developments. Most of the students enrolled in the course are studying biological sciences and have completed a year of calculus with no intention to study further mathematics. Since the main concepts of nonlinear dynamics are largely accessible using only elementary ar­ guments, students are able to understand the mathematics and successfully carry out computations. The exciting nature and modernity of the concepts and the graphics are further stimuli that motivate students. Mathematical developments since the mid 1970's have shown that many interesting phenomena can arise in simple finite-difference equations. These are introduced in Chapter 1, where the student is initiated into three important mathematical themes of the course: local stability analysis, global multistability, and problem solving using both an algebraic and a geometric approach. The graphical iteration of one-dimensional, finite-difference equations, combined with the analysis ofthe local stability ofsteady states, provides two complementary views of the same problem. The concept of chaos is introduced as soon as possible, after the student is able graphically to iterate a one-dimensional, finite-difference equation, and understands the concept of stability. For most students, this is the first exposure to mathematics from the twentieth century! From the instructor's point of view, this topic offers the opportunity to refresh students' memory and skills in differential calculus. Since some students take this course several years after studying geometry and calculus, some skills have become rusty. Appendix A reviews important functions such as the Hill function, the Gaussian distribution, and the conic sections. Many exercises that can help in solidifying geometry and calculus skills are included in Appendix A. Chapters 2 and 3 continue the study of discrete-time systems. Networks and cellular automata (Chapter 2) are important both from a conceptual and technical perspective, and because of their relevance to computers.