Classification and stability of simple homoclinic cycles in R^5

TL;DR

This paper classifies simple homoclinic cycles in five-dimensional systems with symmetry, providing conditions for their stability based on eigenvalues, and systematically lists all possible classes and symmetry groups involved.

Contribution

It offers a complete classification of simple homoclinic cycles in R^5 and derives stability conditions based on symmetry group actions and eigenvalues.

Findings

All symmetry groups allowing simple homoclinic cycles in R^5 are identified.

Necessary and sufficient conditions for stability are established.

A comprehensive list of cycle classes and stability criteria is provided.

Abstract

The paper presents a complete study of simple homoclinic cycles in R^5. We find all symmetry groups Gamma such that a Gamma-equivariant dynamical system in R^5 can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in R^n based on the action of the system symmetry group. For systems in R^5, we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearisation near the steady state involved in the cycle. For any action of the groups Gamma which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining conditions for asymptotic stability of these cycles.