A Hopf bifurcation with Robin boundary conditions

Abstract

We investigate Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations. By performing a smooth homotopy of boundary conditions from Neumann to Dirichlet type, we observe the creation of branches of periodic solutions with submaximal symmetry in codimension two bifurcations, although we do not fully calculate the branching behaviour. We also note that mode interactions behave generically on varying the boundary conditions. The investigation is performed using a numerical Liapunov-Schmidt reduction technique of Ashwin, Bohmer, and Mei (1994) and an analysis of Swift (1988).