Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking
- Lauterbach, Reiner
- Maier-Paape, Stanislaus
2010 Mathematics Subject Classification
- 37G40 35B32 34C14 37G25
- axisymmetric equilibrium, group orbit, invariant manifold, higher codimension, bifurcation
We consider solutions of the semilinear parabolic equation (1.1) on the 2-Sphere. Assuming (1.1) has an axisymmetric equilibrium uα, the group orbit of uα gives a whole (invariant) manifold M of equilibria for (1.1). Under generic conditions we have that, after perturbing (1.1) by a (small) L ⊂ O(3)-equivariant perturbation, M persists as an invariant manifold ͠M slightly changed. However, the flow on ͠M is in general no longer trivial. Indeed, we find heteroclinic orbits on ͠M and, in case L = 𝕋 (the tetrahedral subgroup of O(3)), even heteroclinic cycles.