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

TL;DR

This paper investigates heteroclinic cycles and Hopf bifurcations in dynamical systems with quaternionic symmetry, revealing conditions for heteroclinic connections and their stability, especially under weak coupling.

Contribution

It provides a detailed analysis of heteroclinic cycles in systems with quaternionic symmetry, including stability conditions and effects of weak coupling, extending previous symmetry-based bifurcation studies.

Findings

Identifies conditions for heteroclinic cycles between three equilibria.

Analyzes the stability of heteroclinic cycles.

Shows the Hopf bifurcation behavior similar to D8-symmetric systems.

Abstract

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.