Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection

TL;DR

This paper analytically demonstrates the existence and stability of heteroclinic connections in a six-dimensional reversible system modeling orthogonal domain walls in convection, with implications for understanding complex fluid dynamics.

Contribution

It provides the first analytical proof of heteroclinic connections in a 6D reversible system related to convection, including stability and persistence under perturbations.

Findings

Existence of heteroclinic connection between equilibria

Transversal intersection of unstable and stable manifolds

Persistence of heteroclinic under reversible perturbations

Abstract

A six-dimensional reversible normal form system occurs in B{\'e}nard-Rayleigh convection between parallel planes, when we look for domain walls intersecting orthogonally (see Buffoni et al [1]). On the truncated system, we prove analytically the existence, local uniqueness, and analyticity in parameters, of a heteroclinic connection between two equilibria, each corresponding to a system of convective rolls. We prove that the 3-dimensional unstable manifold of one equilibrium, intersects transversally the 3-dimensional stable manifold of the other equilibrium, both manifolds lying on a 5-dimensional invariant manifold. We also study the linearized operator along the heteroclinic, allowing to prove (in [9]) the persistence under reversible perturbation, of the heteroclinic obtained in [1].