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Unique normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps

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Published 24 June 2014 • © 2014 IOP Publishing Ltd & London Mathematical Society
, , Citation Vassili Gelfreich and Natalia Gelfreikh 2014 Nonlinearity 27 1645 DOI 10.1088/0951-7715/27/7/1645

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Author e-mails

v.gelfreich@warwick.ac.uk

gelfreikh@mail.ru

Author affiliations

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

2 Department of Higher Mathematics and Mathematical Physics, Faculty of Physics, and Faculty of Liberal Arts and Sciences, St Petersburg State University, Russia

Dates

  1. Received 6 December 2013
  2. Accepted 22 May 2014
  3. Published

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0951-7715/27/7/1645

Abstract

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric family. The normal forms are used to analyse bifurcations of n-periodic orbits. In particular, for n ⩾ 6 we find regions of parameters where the normal form has 'meandering' invariant curves.

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10.1088/0951-7715/27/7/1645