Pseudo-simple heteroclinic cycles in $R^4$

Abstract

We study pseudo-simple heteroclinic cycles for a $\Gamma$-equivariant system in $R^4$ with finite $\Gamma \subset O(4)$, and their nearby dynamics. In particular, in a first step towards a full classification - analogous to that which exists already for the class of simple cycles - we identify all finite subgroups of $O(4)$ admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups $\Gamma \subset O(4)$, $\Gamma \not\subset SO(4)$, admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for $\Gamma \subset SO(4)$ pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.

49 pages, 7 figures

References in corpus (3)

• Stability in simple heteroclinic networks in ${\mathbb{R}}^4$
• Asymptotic stability of pseudo-simple heteroclinic cycles in R^4
• Construction of heteroclinic networks in ${\mathbb{R}}^4$

Cited by in corpus (3)

• Asymptotic stability of robust heteroclinic networks
• Simple heteroclinic networks in ${\mathbb{R}}^4$
• Robust heteroclinic cycles in pluridimensions