On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

TL;DR

This paper investigates how area-preserving maps with quadratic homoclinic tangencies undergo bifurcations, leading to the emergence of infinitely many stable elliptic periodic orbits under certain conditions.

Contribution

It provides new conditions for the existence of infinitely many KAM-stable elliptic orbits in both orientable and non-orientable area-preserving maps with homoclinic tangencies.

Findings

Conditions for infinitely many elliptic periodic orbits

Results applicable to both symplectic and non-orientable maps

Analysis of one and two parameter unfoldings

Abstract

We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.