Asymptotic stability of pseudo-simple heteroclinic cycles in R^4

TL;DR

This paper investigates the asymptotic stability of pseudo-simple heteroclinic cycles in four-dimensional equivariant dynamical systems, revealing complex stability properties due to their unique geometric structures.

Contribution

It introduces and analyzes pseudo-simple heteroclinic cycles in R^4, expanding understanding beyond simple cycles and exploring their stability characteristics.

Findings

Pseudo-simple heteroclinic cycles have at least one equilibrium with a 2D unstable manifold.

The paper provides conditions for asymptotic stability of these cycles.

Complex geometric structures influence stability properties in R^4 systems.

Abstract

Robust heteroclinic cycles in equivariant dynamical systems in R^4 have been a subject of intense scientific investigation because, unlike heteroclinic cycles in R^3, they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyse the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in R^4.