The Hopf Bifurcation And Its Applications.

J. E. Marsden M. McCracken Applied Mathematical Sciences 19 The Hopf Bifurcation and Its Applications Springer-Verlag New York Heidelberg Berlin Applied Mathematical Sciences EDITORS Fritz John Lawrence Sirovich Courant Institute of Mathematical Sciences New York University New York, N.Y. 10012 Division of Applied Mathematics Brown University Providence, R.J. 02912 Joseph P. LaSalle Gerald B. Whitham Division of Applied Mathematics Brown University Providence, R.J. 02912 Applied Mathematics Firestone Laboratory California Institute of Technology Pasadena,CA.91125 EDITORIAL STATEMENT The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of mathematicalcomputer modelling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will proVide an outlet for material less formally presented and more anticipatory of needs than finished texts or monographs, yet of immediate interest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. The aim of the series is, through rapid publication in an attractive but inexpensive format, to make material of current interest widely accessible. This implies the absence of excessive generality and abstraction, and unrealistic idealization, but with quality of exposition as a goal. Many of the books will originate out of and will stimulate the development of new undergraduate and graduate courses in the applications of mathematics. Some of the books will present introductions to new areas of research, new applications and act as signposts for new directions in the mathematical sciences. This series will often serve as an intermediate stage of the publication of material which, through exposure here, will be further developed and refined. These will appear in conventional format and in hard cover. MANUSCRIPTS The Editors welcome all inquiries regarding the submission of manuscripts for the series. Final preparation of all manuscripts will take place in the editorial offices of the series in the Division of Applied Mathematics, Brown University, Providence, Rhode Island. SPRINGER-VERLAG NEW YORK INC., 175 Fifth Avenue, New York, N. Y. 10010 Printed in U.S.A. Applied Mathematical Sciences I Volume 19 J. E. Marsden M. McCracken The Hopf Bifurcation and Its Applications with contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt, and S. Smale I] Springer-Verlag New York 1976 J. E. Marsden M. McCracken Department of Mathematics Department of Mathematics University of California University of California at Santa Cruz at Berkeley AMS Classifications: 34C15, 58F1 0, 35G25, 76E30 Library of Congress Cataloging in Publication Data Marsden, Jerrold E. The Hopf bifurcation and its applications. (Applied mathematical sciences; v. 19) Bibliography Includes index. 1. Differential equations. 2. Differential equations, Partial: 3. Differentiable dynamical systems. 4. Stability. I. McCracken, Marjorie, 1949- joint author. II. Title. III. Series. QA1.A647 vol. 19 [QA372] 510'.8s [515'.35] 76-21727 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1976 by Spring