Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic ${\mathbf{Q}}_8$ Group

In this paper we analyze the heteroclinic cycle and the Hopf bifurcation of a generic dynamical system with the symmetry of the group ${\mathbf{Q}}_8,$ constructed via a Cayley graph. While the Hopf bifurcation is similar to that of a ${\mathbf{D}}_8$–equivariant system, our main result comes from analyzing the system under weak coupling. We identify the conditions for heteroclinic cycle between three equilibria in the three–dimensional fixed point subspace of a certain isotropy subgroup of ${\mathbf{Q}}_8\times {\mathbf{S}}^1\mathrm{.}$ We also analyze the stability of the heteroclinic cycle.