Applied Mathematical Sciences

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Applied Mathematical Sciences

Volume 64

Editors

F. John J. E. Marsden L. Sirovich Advisors

M. Ghil J. K. Hale J. Keller K. Kirchgassner B. Matkowsky J. T. Stuart A. Weinstein

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c.s. ^Hsu

Cell-to-Cell Mapping

A Method of Global Analysis for Nonlinear Systems

With 125 Illustrations

Springer Science+Business Media, LLC

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C. S. Hsu

Department of Mechanical Engineering University of California

Berkeley, CA 94720 U.S.A.

Editors F.John

Courant Institute of Mathematical Sciences New York University New York, NY 10012 U.S.A.

AMS Classification: 58FXX

J. E. Marsden Department of

Mathematics University of

California Berkeley, CA 94720 U.S.A.

Library of Congress Cataloging-in-Publication Data Hsu, e.S. (Chieh Su)

Cell-to-cell mapping.

(Applied mathematical sciences; v. 64) Bibliography: p.

Includes index.

1. Global analysis (Mathematics) 2. Mappings (Mathematics) 3. Nonlinear theories. I. Title.

II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.); v. 64.

QA1.A647 vol. 64 [QA614] 510 s [514'.74] 87-12776

L. Sirovich Division of Applied

Mathematics Brown University Providence, RI 02912 U.S.A.

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 of general descriptive names, trade names, trademarks, etc. in this publication, even if the 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 by anyone.

Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 5 4 3 2 1

ISBN 978-1-4419-3083-5 ISBN 978-1-4757-3892-6 (eBook) DOI 10.1007/978-1-4757-3892-6

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Preface

For many years, I have been interested in global analysis of nonlinear systems.

The original interest stemmed from the study of snap-through stability and jump phenomena in structures. For systems of this kind, where there exist multiple stable equilibrium states or periodic motions, it is important to examine the domains of attraction of these responses in the state space. It was through work in this direction that the cell-to-cell mapping methods were introduced. These methods have received considerable development in the last few years, and have also been applied to some concrete problems. The results look very encouraging and promising.

However, up to now, the effort of developing these methods has been by a very small number of people. There was, therefore, a suggestion that the published material, scattered now in various journal articles, could perhaps be pulled together into book form, thus making it more readily available to the general audience in the field of nonlinear oscillations and nonlinear dynamical systems. Conceivably, this might facilitate getting more people interested in working on this topic. On the other hand, there is always a question as to whether a topic (a) holds enough promise for the future, and (b) has gained enough maturity to be put into book form. With regard to (a), only the future will tell. With regard to (b), I believe that, from the point of view of both foundation and methodology, the methods are far from mature.

However, sufficient development has occurred so that there are now practical and viable methods of global analysis for certain problems, and the maturing process will certainly be hastened if more people do become interested.

In organizing and preparing the manuscript, I encountered several difficult problems. First, as cell mapping methods are quite different from the classical modes of analysis, there is not a set pattern of exposition to follow. Also, the two cell mapping methods, simple and generalized, make use of techniques

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vi Preface

from different areas of mathematics; it is, therefore, difficult to set up a smooth flow pattern for the analytical development. To help the reader, I have elected to begin with an Introduction and Overview chapter, which explains how the various parts of the book fit together. The coverage within this chapter is such that Section l.N gives an overview of the material in Chapter N.

The other difficult problems were connected with the mathematical background needed by the reader for the development of the methods. For simple cell mapping, the concept of simplexes from topology is important. For generalized cell mapping, the theory of Markov chains is essential. Both topics are very well covered in many excellent mathematical books, but, unfortu- nately, they are usually not parts of our regular engineering and applied science curricula. This situation leads to a somewhat arbitrary choice on my part with regard to how much elementary exposition of these topics to include in the manuscript and how much to refer to the outside source books. It is hoped that sufficient introductory material has been provided so that a reader with the mathematical background of a typical U.S. engineering graduate student can follow the development given in this book. As far as the background knowledge on nonlinear systems is concerned, some acquaintance with the results on multiple solutions, stability, and bifurcation from classical analysis is assumed.

This book is intended to introduce cell-to-cell mapping to the reader as a method of global analysis of nonlinear systems. There are essentially three parts in the book. The first part, consisting of Chapters 2 and 3, discusses point mapping to provide a background for cell mapping. The second part, from Chapters 4 to 9, treats simple cell mapping and its applications. General- ized cell mapping is then studied in the third part, from Chapters 10 to 13.

The discussions on methodologies of simple and generalized cell mapping culminate in an iterative method of global analysis presented in Chapter 12.

As far as the reading sequence is concerned, my suggestions are as follows.

A reader who is very familiar with point mapping can skip Chapters 2 and 3.

He can return to these chapters later when he wants to compare the cell mapping results with the point mapping results for certain problems. A reader who is not interested in the index theory of simple cell mapping can skip Chapter 6. A reader who is familiar with Markov chains can skip Section 10.4.

Since cell mapping methods are fairly recent developments, they have not received the rigorous test of time. Therefore, there are possibly inadequacies and undetected errors in the analysis presented in the book. I shall be most grateful if the readers who notice such errors would kindly bring them to my attention.

During the last few years when working on cell mapping, I have benefited greatly through discussions with many people. They include Dr. H. M. Chiu (University of California, Berkeley), Professors H. Flashner and R. S. Guttalu (University of Southern California), Mr. R. Y. Jin (Peking), Professor E. 1.

Kreuzer (University of Stuttgart), Dr. W. H. Leung (Robert S. Cloud Associ- ates), Dr. A. Polchai (Chiengmai University, Thailand), Dr. T. Ushio (Kobe

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Preface vii

University), Mr. W. H. Zhu (Zhejiang University, China), and Mr. J. X. Xu (Sian Jiaotong University, China). I am particularly grateful to M. C. Kim, W. K. Lee, and J. Q. Sun (University of Cal ifomi a, Berkeley) who, in addition, read the manuscript and made numerous suggestions for improvements. I also wish to acknowledge the continual support by the National Science Founda- tion for the development of much of the material on cell mapping presented in this book. The aid received during preparation of the manuscript from IBM Corporation through its DACE Grant to the University of California, Berkeley is also gratefully acknowledged.

c.

S. Hsu

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Applied Mathematical Sciences