Hopf bifurcation and heteroclinic cycles in a class of $\mathbb{D}_2-$equivariant systems

TL;DR

This paper investigates Hopf bifurcation and heteroclinic cycles in a class of $ ext{D}_2$-equivariant dynamical systems, providing conditions for periodic solutions and heteroclinic cycle stability.

Contribution

It introduces conditions for Hopf bifurcation and heteroclinic cycles in $ ext{D}_2$-equivariant systems, especially under weak coupling, and analyzes their stability.

Findings

Conditions for Hopf bifurcation and periodic solutions identified.

Criteria for heteroclinic cycle formation between equilibria established.

Stability analysis of heteroclinic cycles conducted.

Abstract

In this paper we analyze a generic dynamical system with $\mathbb{D}_2$ constructed via a Cayley graph. We study the Hopf bifurcation and find conditions for obtaining a unique branch of periodic solutions. Our main result comes from analyzing the system under weak coupling, where we identify the conditions for heteroclinic cycle between four equilibria in the two-dimensional fixed point subspace of some of the isotropy subgroups of $\mathbb{D}_2\times\mathbb{S}^1.$ We also analyze the stability of the heteroclinic cycle.