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

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