The Hamiltonian Hopf Bifurcation
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Jan-Cees van der Meer Centre for Mathematics and Computer Science (CWI) Konislaan 413. 1098 SJ Amsterdam The Netherlands
Mathematics Subject Classification (1980): 58F05. 58F 14, 58F22. 58F30. 70F07 ISBN 3-540-16037-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16037-X Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data. Meer. Jan-Cees van der, 1955-. The Hamiltonian Hopf bifurcation. (Lecture notes in mathematics; 1160) Bibliography: p.lncludes index. 1. Hamiltonian mechanics. 2. Bifurcation theory. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1160. 0A3.L28 no. 1160 [0A614.83] 510 s [514'.74] 85-27646 ISBN 0-387-16037-X (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to 'Verwertungsgesellschaft Wort", Munich.
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Preface The research laid down in these notes began several years ago with some questions about a particular bifurcation of periodic solutions in the restricted problem of three bodies at the equilibrium
L 4 • This
particular bifurcation takes place when, for the linearized system, the equilibrium
L4
changes from stable to unstable. This kind of bifur-
cation is called a Hamiltonian Hopf bifurcation. During the research it became apparent that new methods had to be developed and that existing methods had to be reformulated in order to deal with the specific nature of the problem. The development of these methods together with their application to the Hamiltonian Hopf bifurcation is the main topic of these notes. As a result a complete description is obtained of the bifurcation of periodic solutions for the generic case of the Hamiltonian Hopf bifurcation. This research was carried out at the Mathematical Institute of the State University of Utrecht. I am very grateful to Prof. Hans Duistermaat and Dr. Richard Cushman for their guidance and advice during the years I worked on this subject. I also thank Richard Cushman for his careful reading of the earlier drafts of the manuscript. Thanks are also due to Prof. D. Siersma of the University of Utrecht for the discussions we had on chapters
3
and
4
, and to
Prof. F. Takens of the University of Groningen for his remarks concerning the final manuscript. Finally, I would like to thank Drs. H. van der Meer for his assistence in plotting fig. 4.1 4. 14, and Jacqueline Vermeij and Jeannette Guilliamse for their excellent typing of the manuscript. JanCees van der Meer June 1985
CONTENTS INTRODUCTION
1
CHAPTER I, Preliminaries. O. Introduction
4
1. Hamiltonian systems
4
2. Symm
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