Reversible Systems
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1211 M. B. Sevryuk
Reversible Systems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Mikhail B. Sevryuk Consulting Editor
Vladimir I. Arnol'd Leningrad Branch of V. A. Steklov Mathematical Institute Fontanka 27, 191011 Leningrad, 0-11, USSR
ISBN 3·540·16819·2 Springer-Verlag Berlin Heidelberg New York ISBN 0·387·16819-2 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
'l'ABLE OF CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . Chapter 1.
TORI OF PERTUEBATIONS OF INTEGRABLE REVERSIBLE DIFFEml0RPHISIIS AND VECTORFIELDS
22
Part 1. The discrete time case: Kolmogorov tori of perturbations of reversible diffeomorphisms
22
§ 1.1 . Preliminaries
22
§ 1.2. Pr inc ipal theorem
24
§ 1.3. I1ain lemma . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . .
28
§ 1.4. Termination of the proof of the principal theorem..
46
§ 1.5. Reversible diffeomorphisms of a plane
56
§ 1.6. Appendix . . . . . . . . . . . . . . • . . . . . . • . . . . . . . . . . . . . . . . . . . . .
58
Part 2. The continuous time case: Kolmogorov tori of perturbations of reversible vectorfields
66
§ 1.7. Preliminaries . . . . . . . . . . . . . . . . . . • • . . . . . . . . . . . . . . . . . .
66
§ 1.8. Principal theorem . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . ....
69
§ 1.9. Hain lemma.........................................
70
§ 1.10. Final remarks •.•..••..•........•.•.................
84
§ 1. 11. Appendix . . . • • . . . . . . . . . . . . . . • . . . . . . . . . . . . . . • . . . . . . . .
87
Chapter 2. NORHAL FORMS FOR REVERSIBLE DIFFEOHORPHISHS AND VECTORFIELDS NEAR AN EQUILIBRIUM AND THEIR KOLHOGOROV TORI . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . .
92
§ 2.1. Linear reversible and infinitesimally
reversible operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . §
92
Normal forms for slightly elliptic reversible diffeomorphisms near a fixed point
102
§ 2.3. Weakly reversible elliptic hyperbolic diffeo-
morphisms near a fixed point
110
IV
§ 2.4. Weakly reversible elliptic diffeomorphisms
near a fixed point .........•.•................••.•. 126 § 2.5. Normal forms for slightly elliptic reversible
vectorfields near an equilibrium
132
§ 2.6. Weakly reversible elliptic hyperbolic
vectorfields near
equilibrium
136
§ 2.7. Weakly reversible elliptic vectorfields
near an
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