Introduction to Applied Nonlinear Dynamical Systems and Chaos

weixin_42381005-10508357-4744300845214388540.pdf-第1页.png

第1页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第2页.png

第2页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第3页.png

第3页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第4页.png

第4页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第5页.png

第5页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第6页.png

第6页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第7页.png

第7页 / 共860页

weixin_42381005-10508357-4744300845214388540.pdf-第8页.png

第8页 / 共860页

Texts in Applied Mathematics 2 Editors J.E. Marsden L. Sirovich S.S. Antman Advisors G. Iooss P. Holmes D. Barkley M. Dellnitz P. Newton

Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures

Stephen Wiggins School of Mathematics University of Bristol Clifton, Bristol BS8 1TW UK S.Wiggins@bristol.ac.uk Series Editors J.E. Marsden Control and Dynamical Systems, 107–81 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA chico@camelot.mssm.edu S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA ssa@math.umd.edu Mathematics Subject Classification (2000): 58Fxx, 34Cxx, 70Kxx Library of Congress Cataloging-in-Publication Data Wiggins, Stephen. Introduction to applied nonlinear dynamical systems and chaos / Stephen Wiggins. — 2nd ed. p. cm. — (Texts in applied mathematics ; 2) Includes bibliographical references and index. ISBN 0-387-00177-8 (alk. paper) 1. Differentiable dynamical systems. 2. Nonlinear theories. 3. Chaotic behavior in I. Title. systems. QA614.8.W544 2003 003′.85—dc21 II. Texts in applied mathematics ; 2. 2002042742 Printed on acid-free paper. ISBN 0-387-00177-8 © 2003, 1990 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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 in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 10901182 www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

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 re- search 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 numeri- cal 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 to 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 Mathe- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California Providence, Rhode Island College Park, Maryland J.E. Marsden L. Sirovich S.S. Antman

Preface to the Second Edition This edition contains a signiﬁcant amount of new material. The main rea- son for this is that the subject of applied dynamical systems theory has seen explosive growth and expansion throughout the 1990s. Consequently, a student needs a much larger toolbox today in order to begin research on signiﬁcant problems. I also try to emphasize a broader and more uniﬁed point of view. My goal is to treat dissipative and conservative dynamics, discrete and con- tinuous time systems, and local and global behavior, as much as possible, on the same footing. Many textbooks tend to treat most of these issues separately (e.g., dissipative, discrete time, local dynamics; global dynamics of continuous time conservative systems, etc.). However, in research one generally needs to have an understanding of each of these areas, and their inter-relations. For example, in studying a conservative continuous time system, one might study periodic orbits and their stability by passing to a Poincar´e map (discrete time). The question of how stability may be aﬀected by dissipative perturbations may naturally arise. Passage to the Poincar´e map renders the study of periodic orbits a local problem (i.e., they are ﬁxed points of the Poincar´e map), but their manifestation in the continuous time problem may have global implications. An ability to put together a “big picture” from many (seemingly) disparate pieces of information is crucial for the successful analysis of nonlinear dynamical systems. This edition has seen a major restructuring with respect to the ﬁrst edition in terms of the organization of the chapters into smaller units with a single, common theme, and the exercises relevant to each chapter now being given at the end of the respective chapter. The bulk of the material in this book can be covered in three ten week terms. This is an ambitious program, and requires relegating some of the material to background reading (described below). My goal was to have the necessary background material side-by-side with the material that I would lecture on. This tends to be more demanding on the student, but with the right guidance, it also tends to be more rewarding and lead to a deeper understanding and appreciation of the subject. The mathematical prerequisites for the course are really not great; ele- mentary analysis, multivariable calculus, and linear algebra are suﬃcient. In reality, this may not be enough on its own. A successful understanding of applied dynamical systems theory requires the students to have an inte-

viii Preface to the Second Edition grated knowledge of these prerequisites in the sense that they can ﬂuidly manipulate and use the ideas between the subjects. This means they must possess the quality often referred to as “mathematical maturity.” A study of dynamical systems theory can be a good way to obtain this. In addi- tion, an ordinary diﬀerential equations course from the geometric point of view (e.g., the material in the books of Arnold [1973] or Hirsch and Smale [1974]) would be ideal. Chapters 1-17 form the core of the ﬁrst term material. It provides stu- dents with the basic concepts and tools for the study of dynamical systems theory. I tend to cover chapters 7, 11 and 12 at a brisk pace. The main point there is the ideas and main results. The details can be grasped over time, and in other settings. Chapters 13-17 could be viewed as belonging to the common theme of “dynamical systems with special structure.” Chap- ter 14 is the most important of these chapters. The relation, and contrasts, between Hamiltonian and reversible systems is useful to understand, and is the reason for including chapter 16. I often just assign selected back- ground reading from chapter 13, but knowledge of the relation between Lagrangian and Hamiltonian dynamical systems is of growing importance in applications. Gradient dynamical systems arise in numerous applica- tions (e.g., in biologically related areas) and knowledge of the nature of their dynamics, and how it contrasts with, e.g., Hamiltonian dynamics, is important. Chapter 17 is short, but I have always felt that students should be aware of these results because there are numerous examples of systems arising in applications that experience a “transient temporal disturbance.” Throughout the early chapters I discuss a number of results and theoretical frameworks for general nonautonomous vector ﬁelds (i.e., time-dependent vector ﬁelds whose time dependence is not periodic). This area tradition- ally has not been a part of dynamical systems from a geometric point of view, but this situation is changing rapidly, and I believe it will play an increasingly important role in applications in the near future. Chapters 18-22 are covered in the second term. The subject is “local bifurcation theory.” The two key tools for the local analysis of dynami- cal systems are center manifold theory and normal form theory, covered in chapters 18 and 19. The chapter on normal form theory is greatly expanded from the ﬁrst edition. The main new material is the normal form work of Elphick, Tirapegui, Brachet, Coullet and Iooss, a discussion of Hamilto- nian normal form theory (following Churchill, Kummer, and Rod), and some material on symmetries (whose possible existence, and implications, should be considered in the course of study of any dynamical system). Pos- sibly sections 19.1-19.3 could have been omitted in this edition; however it has been my experience that students understand the later (and more diﬃcult) material more easily once they have been exposed to this more pedestrian introduction. In chapters 20 and 21 I tend not to cover in much detail the material related to the codimension of a bifurcation and versal deformations. This is a standard language used in discussing the subject

Preface to the Second Edition ix and it is important that the students have all the details available to them for background reading and see it in the context of the material I lecture on. New material on Hamiltonian bifurcations and circle maps is included. The inclusion of introductory material on Hamiltonian bifurcations is an example of the eﬀort to have a broader and more uniﬁed point of view as discussed earlier. For example, we ﬁrst describe the “generic saddle-node bifurcation at a single zero eigenvalue.” It is then natural to ask about the saddle-node bifurcation in a Hamiltonian system, which turns out to be rather diﬀerent. Chapter 22 mainly serves as a warning that the way in which bifurcation phenomena are discussed in applications may not agree with the mathematical reality, and appropriate pointers to the literature are given. Chapters 23-33 are covered in the third term. The subject is “global dynamics, bifurcations, and chaos.” There is a sprinkling of new material throughout these chapters (e.g., a proof of a simple version of the lambda lemma and a proof of the shadowing lemma), but the structure is basically the same as the ﬁrst edition. There is not a great deal of overlap between the material in the individ- ual terms, and with the appropriate prerequisites, each of these one term courses could be viewed as an independent course in itself. The textbook provides the necessary background for the students to make this a possi- bility. Some material has been left out of this edition; in particular, material on averaging, the subharmonic Melnikov function, and lobe dynamics. The reason is that over time I have begun to cover averaging and the subhar- monic Melnikov function as topics in a course solely devoted to perturbation methods. I cover lobe dynamics in a course devoted to transport phenom- ena in dynamical systems, which has developed in the last ten years to the point that it now justiﬁes an independent course of its own, with applica- tions taken from many diverse disciplines. It has been my experience over time that a signiﬁcant obstacle for stu- dents in their study of the subject is the sheer amount of (initially) unfa- miliar jargon. In order to make this a bit easier to deal with I have now included a glossary of frequently used terms. The bibliography has also been updated and greatly expanded. I would also like to take this opportunity to express my gratitude to the National Science Foundation and to Dr. Wen Masters and Dr. Reza Malek-Madani of the Oﬃce of Naval Research for their generous support of my research over the years. Research and teaching are two sides of the same coin, and it is only through an active and fruiful research program that the teaching becomes alive and relevant. Bristol, England 2003 Stephen Wiggins

资料库

Introduction to Applied Nonlinear Dynamical Systems and Chaos - ....pdf

相关推荐

课程资源

热门标签

最新资料