Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking

Abstract

Recently it has been observed, that perturbations of symmetric ODE’s can lead to highly nontrivial dynamics. In this paper we want to establish a similar result for certain nonlinear partial differential systems. Our results are applied to equations which are motivated from chemical reactions. In fact we show that the theory applies to the Brusselator on a sphere. To be more precise, we consider solutions of a semi-linear parabolic equation on the 2-sphere. When this equation has an axisymmetric equilibrium $u_\alpha$, the group orbit of $u_\alpha$ (under rotations) gives a whole (invariant) manifold $M$ of equilibria. Under generic conditions we have that, after perturbing our equation by a (small) $L\subset {{\textbf {O}(3)}}$-equivariant perturbation, $M$ persists as an invariant manifold $\widetilde M$. However, the flow on $\widetilde M$ is in general no longer trivial. Indeed, we find slow dynamics on $\widetilde M$ and, in the case $L=\mathbb {T}$ (the tetrahedral subgroup of ${{\textbf {O}(3)}}$), we observe heteroclinic cycles. In the application to chemical systems we would expect intermittent behaviour. However, for the Brusselator equations this phenomenon is not stable. In order to see it in a physically relevant situation we need to introduce further terms to get a higher codimension bifurcation.