Pseudo-simple heteroclinic cycles in R

We study pseudo-simple heteroclinic cycles for a Γ-equivariant system in R4 withfinite Γ ⊂ O(4), and their nearby dynamics. In particular, in a first step towardsa full classification – analogous to that which exists already for the class of simplecycles – we identify all finite subgroups of O(4) admitting pseudo-simple cycles. Tothis end we introduce a constructive method to build equivariant dynamical systemspossessing a robust heteroclinic cycle. Extending a previous study we also investigatethe existence of periodic orbits close to a pseudo-simple cycle, which depends on thesymmetry groups of equilibria in the cycle. Moreover, we identify subgroups Γ ⊂ O(4),Γ 6⊂ SO(4), admitting fragmentarily asymptotically stable pseudo-simple heterocliniccycles. (Recall, that for Γ ⊂ SO(4) pseudo-simple cycles generically are completelyunstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.