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 $\mathbf{Q}_8\times\mathbf{S}^1.$ We also analyze the stability of the heteroclinic cycle.