Saddle-center and periodic orbit: dynamics near symmetric heteroclinic connection

TL;DR

This paper analyzes the complex dynamics near symmetric heteroclinic connections in reversible Hamiltonian systems, revealing chaotic behavior, homoclinic and heteroclinic orbits, and elliptic periodic orbits through rigorous theorems.

Contribution

It provides new theorems demonstrating chaotic dynamics and criteria for homoclinic orbits in symmetric heteroclinic structures of Hamiltonian systems.

Findings

Existence of countable transverse homoclinic orbits

Presence of heteroclinic connections involving saddle periodic orbits

Families of elliptic periodic orbits and homoclinic tangencies

Abstract

An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure and therefore it can be met generally in one-parameter families of reversible Hamiltonian systems. There exist two possible types of such connections in dependence on how the involution acts near the equilibrium. We prove a series of theorems which show a chaotic behavior of the system and those in its unfoldings, in particular, the existence of countable sets of transverse homoclinic orbits to the saddle periodic orbit in the critical level, transverse heteroclinic connections involving a pair of saddle periodic orbits, families of elliptic periodic orbits, homoclinic tangencies, families of homoclinic orbits to saddle-centers in the unfolding, etc. As a byproduct, we get a criterion of the existence of homoclinic orbits to a saddle-center.