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

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.

13 pages, 2 figures