Stability of a heteroclinic network and its cycles: a case study from Boussinesq convection

TL;DR

This paper investigates the stability of a complex heteroclinic network in six-dimensional space, introducing new analytical methods to handle high-dimensional cycles and networks, with insights applicable to Boussinesq convection.

Contribution

It presents a novel analysis of heteroclinic cycles of a previously unstudied type in high dimensions, developing new geometric tools for stability analysis.

Findings

Identified stability conditions for the heteroclinic network

Developed new geometric analysis methods for return maps

Illustrated the analysis with a case from Boussinesq convection

Abstract

This article is concerned with three heteroclinic cycles forming a heteroclinic network in ${\mathbb R}^6$. The stability of the cycles and of the network are studied. The cycles are of a type that has not been studied before, and provide an illustration for the difficulties arising in dealing with cycles and networks in high dimension. In order to obtain information on the stability for the present network and cycles, in addition to the information on eigenvalues and transition matrices, it is necessary to perform a detailed geometric analysis of return maps. Some general results and tools for this type of analysis are also developed here.