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

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Abstract

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.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick Zeeman Building, Coventry, CV4 7AL, UK

    V. Gelfreich

  2. Faculty of Physics, St. Petersburg State University, Ulyanovskaya ul. 3, St. Petersburg, 198504, Russia

    N. Gelfreikh

Authors
  1. V. Gelfreich
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  2. N. Gelfreikh
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Correspondence to V. Gelfreich.

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This work was partially supported by a grant from the Royal Society.

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Gelfreich, V., Gelfreikh, N. Unique normal forms for area preserving maps near a fixed point with neutral multipliers. Regul. Chaot. Dyn. 15, 300–318 (2010). https://doi.org/10.1134/S1560354710020164

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  • DOI: https://doi.org/10.1134/S1560354710020164

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