Stability of a heteroclinic network and its cycles: a case study 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.