Singularity Theory and Equivariant Symplectic Maps
The monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighborhood of a fixed point.The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the
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Thomas J. Bridges Jacques E. Furter
Singularity Theory and Equivariant Symplectic Maps
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Thomas J. Bridges Jacques E. Furter Mathematics Institute University of Warwick Coventry CV4 7AL, Great Britain
Mathematics Subject Classification (1991): 58C27, 58F14, 58F05, 58F22, 58F36, 39AlO, 70Hxx ISBN 3-540-57296-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57296-1 Springer-Verlag New York Berlin Heidelberg 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 2146/3140-543210 - Printed on acid-free paper
Table of Contents 1. Introduction
1
2. Generic bifurcation of periodic points 2.1 Lagrangian variational formulation 2.2 Linearization and unfolding 2.3 Symmetries 2.4 Normal form for bifurcating period-q points 2.5 Reduced stability of bifurcating periodic points 2.6 4D-symplectic maps and the collision singularity
10 13 17 19 23 27
3. Singularity theory for equivariant gradient bifurcation problems 3.1 Contact equivalence and gradient maps 3.2 Fundamental results 3.3 Potentials and paths 3.4 Equivalence for paths 3.5 Proofs
33 35 38 .40 .43 .48
4. Classification of Zq-equivariant gradient bifurcation problems 4.1 AZq-classification of potentials 4.2 Classification of Zq-equivariant bifurcation problems 4.3 Bifurcation diagrams for the unfolding (4.9)
63 64 69 74
5. Period-3 points of the generalized standard map 5.1 Computations of the bifurcation equation 5.2 Analysis of the bifurcation equations
85 86 87
6. Classification of Oq-equivariant gradient maps on R2 6.1 Dq-normal forms when q f= 4 6.2 D4-invariant potentials with a distinguished parameter path
89 89 90
7. Reversibility and degenerate bifurcation of period-q points of multiparameter maps 7.1 Period-3 points with reversibility in multiparameter maps 7.2 Period-4 points with reversibility in multiparameter maps 7.3 Generic period-5 points in the generalized standard map 8. Periodic points of equivariant symplectic maps 8.1 Subharmonic bifurcation in equivariant symplectic maps
9
101 103 110 116 119
121
VI
8.2 8.3 8.4 8.5 8.6
Subharmonic bifurcation when acts absolutely irreducibly on Rn O(2)-equivariant symplectic maps Parametrically forced spherical pendulum Reduction to the orbit space Remarks on linear stability for equivariant maps
130 132 140 144 146
9. Collision of multipliers at rational points for symplectic maps 9.1 Gene
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