Unique normal forms for area preserving maps near a fixed point with neutral multipliers

We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers ±1 at ɛ = 0. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system....

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Bibliographic Details
Published in:Regular & chaotic dynamics 2010-06, Vol.15 (2-3), p.300-318
Main Authors: Gelfreich, V., Gelfreikh, N.
Format: Article
Language:English
Subjects:
Dynamical Systems and Ergodic Theory
L.P. Shilnikov-75
Mathematics
Mathematics and Statistics
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Summary:We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers ±1 at ɛ = 0. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain non-degeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.
ISSN:1560-3547
1560-3547
1468-4845
DOI:10.1134/S1560354710020164