Normal forms for three-parametric families of area-preserving maps near an elliptic fixed point

TL;DR

This paper develops simplified normal forms for three-parameter families of area-preserving maps near elliptic fixed points, enabling detailed analysis of bifurcations and dynamics in these systems.

Contribution

It introduces normal forms for co-dimension 3 fixed points and applies them to study bifurcations in generic three-parameter families of area-preserving maps.

Findings

Normal forms for co-dimension 3 fixed points are derived.

Normal form theory is used to describe bifurcations of periodic orbits.

The approach simplifies the analysis of complex dynamical behaviors.

Abstract

We study dynamics of area-preserving maps in a neighbourhood of an elliptic fixed point. We describe simplified normal forms for a fixed point of co-dimension 3. We also construct normal forms for a generic three-parameter family which contains such degeneracy and use the normal form theory to describe generic bifurcations of periodic orbits in these families.