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.

Highlights

  • In a Hamiltonian system small oscillations around a periodic orbit are often described using the normal form theory [12, 2]

  • A sequence of coordinate changes is used to transform the map to a normal form

  • In the absence of resonances the normal form is a rotation of the plane, and the angle of the rotation depends on the amplitude

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Summary

Introduction

In a Hamiltonian system small oscillations around a periodic orbit are often described using the normal form theory [12, 2]. Returning to the original coordinates we conclude that the original map has a non-trivial formal integral H0 This integral provides a powerful tool for analysis of the local dynamics including the stability of the fixed point In general the series of the normal form theory are expected to diverge They provide rather accurate information about the dynamics of the original map. In the case of a generic one-parameter unfolding of a non-degenerate resonant elliptic fixed point, the normal form provides a description for a chain of islands which is born from the origin when the unfolding parameter changes its sign [2, 10, 12, 14].

Unique normal forms
Bifurcations of n-periodic points
7: Level sets for the limit
Simplification of a degenerate Hamiltonian
Families with a small twist

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