Group Theoretic Methods in Bifurcation Theory
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Springer
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
762 D. H. Sattinger
Group Theoretic Methods in Bifurcation T'heory With an
Springer-veriag Berlin Heidelberg New York 1979
Author D. H. Sattinger School of Mathematics University of Minnesota Minneapolis, Minnesota 55455 USA
AMS Subject Classifications (1970): 35JXX, 35KXX, 47H15, 76-XX ISBN 3-540·09715-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387·09715-5 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data Sattinger, David H Group theoretic methods in bifurcation theory. (Lecture notes in mathematics; 762) Bibliography: p. Includes index. 1. Differential equations, Partial--Numerical solutions. 2. Bifurcation theory. 3. Representations of groups. I. Title. II. Series: Lecture notes in mathematics (Berlin); 762. OA3.L28 no. 762 [OA377] 510'.8s [515'.353] 79-23605 ISBN 0-387-09715-5
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PREFACE
This set of lectures was given in the winter and spring of 1978 while I was on sabbatical at the University of Chicago.
I have tried
to present the fundamental ideas involved in the combination of group representation theory and bifurcation theory.
In addition, there is
a chapter by Peter Olver on the derivation of the symmetry group of a differential equation by algebraic methods. I would like to thank the Mathematics Department of the University of Chicago for their kind hospitality, Peter Olver for his useful remarks during the lectures and his contribution to these notes, and F. Flowers for an excellent job of typing. My sabbatical was supported by the University of Minnesota, the National Science Foundation (MCS 73-08535) and the U.S. Army Research Office (DA AG 29-77-G-0122), whose support is much appreciated.
David H. Sattinger June 1978
TABLE OF CONTENTS
I
PHYSICAL EXAMPLES OF BIFURCATION .•.............•....
II
MATHEMATICAL PRELIMINARIES..........................
18
III STABILITY AND BIFURCATION . . . . . . . • . . • . . . . . . . . . . • . . . . .
37
IV
BIFURCATION AT MULTIPLE EIGENVALUES . . . . . . . . . . . . . . . . .
70
V
ELEMENTS OF GROUP REPRESENTATION THEORY.............
96
VI
APPLICATIONS • . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . .
1 51
VII APPENDIX: HOW TO FIND THE SYMMETRY GROUP OF A DIFFERENTIAL EQUATION
200
(by Peter Olver) Subject Index
.... .................................... . ~
240
PHYSICAL EXAMPLES OF
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