Methods of Bifurcation Theory
An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To
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Editors
M. Artin s. E. Heinz F. W. Magnus W. Schmidt
S. Chern J. L. Doob A. Grothendieck Hirzebruch L. Hormander S. Mac Lane C. C. Moore J. K. Moser M. Nagata D. S. Scott J. Tits B. L. van der Waerden
Managing Editors
B. Eckmann S. R. S. Varadhan
Shui-Nee Chow
Jack K. Hale
Methods of Bifurcation Theory With 97 Illustrations
Springer-Verlag New York
Berlin
Heidelberg
Shui-Nee Chow Department of Mathematics Michigan State University East Lansing, Michigan 48824 U.S.A. Jack K. Hale Division of Applied Mathematics Brown University Providence, Rhode Island 02912 U.S.A.
AMS Subject Classifications 34Bxx, 34Cxx, 34Kxx, 35Bxx, 47A55, 47H15, 47H17, 58Cxx, 58Exx Library of Congress Cataloging in Publication Data Chow, Shui-Nee. Methods of bifurcation theory. (Grundlehren der mathematischen Wissenschaften; 251) Bibliography: p. Includes index. I. Functional differential equations. 2. Bifurcation theory. 3. Manifolds (Mathematics) I. Hale, Jack K. II. Title. III. Series. QA372.C544 515.3'5 81-23337 AACR2
© 1982 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1982 All rights reversed. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. 9 8 765 432 I
ISBN -13:978-1-4613-8161-7 e- ISBN -13 :978-1-4613-8159-4 DOl: 10.1007/978-1-4613-8159-4
To Marie and Hazel
Preface
An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To accomplish this objective and to make the book accessible to a wider audience, we have presented in detail much of the relevant background material from nonlinear functional analysis and the qualitative theory of differential equations. Since there is no good reference for some of the material, its inclusion seemed necessary. Two distinct aspects of bifurcation theory are discussed-static and dynamic. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. If the function is not a gradient or if more detailed information is desired, the general theory is usually local. At the same time, the theory is constructive and valid when several independent parameters appear in the function. In differential equations, the equilibrium solutions are the zeros of the vector field. Therefore, methods in static bifurcation theory are directly applicable. Dynamic bifurcation theory is concerned with the changes that occur in the structure of the limit sets of solutions of differential equations as parameters in the vector field are varied. For example, in addition to discussing the way that the set of zeros of the vector field (the
