We consider the pattern defect in Bénard-Rayleigh convection which consists in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the $x$-coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit of a 6-dimensional reversible normal form system (see Buffoni et al [1]). We prove analytically on this 6-dimensional system (see [2]) the existence, local uniqueness, and analyticity in parameters, of the heteroclinic connection between two equilibria, each corresponding to a system of convective rolls. 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. This gives an analytic proof of the result obtained by a variational method in [1]. We study the linearized operator along this heteroclinic, and study the complete perturbed system given by the now 8-dimensional normal form (a phase is added for the limiting periodic equilibrium). We are then able (see [3]) to prove, via an adapted Lyapunov-Schmidt method, the persistence of the heteroclinic corresponding to the orthogonal wall, showing near the bifurcation, the existence of a one parameter family of walls, with a clear link between wave numbers at $\pm\infty$.
Keywords. reversible dynamical systems, invariant manifolds, bifurcations, heteroclinic connection, domain walls in convection
References
[1] B.Buffoni, M.Haragus, G.Iooss. Heteroclinic orbits for a system of
amplitude equations for orthogonal domain walls. J. Diff. Equ. (2023). (link)
[2] G.Iooss. Heteroclinic for a 6-dimensional reversible system related with or-
thogonal domain walls in convection. Preprint 2023 (submitted to JDDE).
[3] G.Iooss. Existence of orthogonal domain walls in Bénard-Rayleigh con-
vection. Preprint 2023 (submitted to JMFM).
