# Pseudo-simple heteroclinic cycles in $R^4$

## TL;DR

This paper classifies pseudo-simple heteroclinic cycles in four-dimensional space with symmetry, introduces a method to construct systems with such cycles, and explores their stability and nearby dynamics.

## Contribution

It provides a classification of symmetry groups admitting pseudo-simple cycles, introduces a constructive method for such systems, and analyzes their stability and periodic orbits.

## Key findings

- Identified all finite subgroups of O(4) with pseudo-simple cycles.
- Developed a constructive method to build equivariant systems with heteroclinic cycles.
- Found subgroups with fragmentarily asymptotically stable pseudo-simple heteroclinic cycles.

## 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.