Global Bifurcation of Periodic Solutions with Symmetry
This largely self-contained research monograph addresses the following type of questions. Suppose one encounters a continuous time dynamical system with some built-in symmetry. Should one expect periodic motions which somehow reflect this symmetry? And ho
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1309
Bernold Fiedler
Global Bifurcation of Periodic Solutions with Symmetry
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Bernold Fiedler Institute of Applied Mathematics, University of Heidelberg 1m Neuenheimer Feld 294 0-6900 Heidelberg, Federal Republic of Germany
Mathematics Subject Classification (1980): 34XX, 35XX, 47H 15, 58C27, 58F 14, 58F22, 76E30, 80A30, 80A32 ISBN 3-540-19234-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19234-4 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Fiedler, Bernold, 1956- Global bifurcation of periodic solutions with symmetry 1 Bernold Fiedler. p. cm.-(Lecture notes in mathematics; 1309) Bibliography: p. Includes index. ISBN 0-387-19234-4 (U.S.) 1. Bifurcation theory. 2. Singularities (Mathematics) 3. Nonlinear operators. 4. Differential equations-Numerical solutions. 5. Differential equations, Partial-Numerical solutions. I. Title. II. Title: Periodic solutions with symmetry. III. Series: Lecture notes in mathematics (SpringerVerlag); 1309. 0A3.L28 no. 1309 [OA372j510 s-dc 19 [514'.74J 88-12342 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright law. © Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
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Preface The inherent harmony of periodic motions as well as of symmetry has exerted its own fascination, as it seems, ever since the dawn of thought. Today, such a "harmonia mundi" is at least hoped for on just about any possible scale: from elementary particle physics to astronomy. In search of some harmony let us ask naive questions. Suppose we are given a dynamical system with some builtin symmetry. Should we expect periodic motions which somehow reflect this symmetry? And how would periodicity harmonize with symmetry? These almost innocent questions are the entrance to a labyrinth of intricacies. Probing only along some fairly safe threads we are lead from dynamics to topology, algebra, singularity theory, numerical analysis, and to some applications. A global point of view will be one guiding theme along our way: we are mainly interested in periodic motions far from equilibrium. For a method we rely on bifurcation theory, on transversality theory, and on generic approximations. As a reward we encounter known local singularities. As a central new aspect we study the global interaction and interdependence of these local singularities, designing a homotopy invariant. As a result, we ob
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