A singularity theory analysis of bifurcation problems with octahedral symmetry

Dynam. Stab. Sys. 1 (1986) 293-321

Ian Melbourne


We analyse bifurcation problems with octahedral symmetry using results from singularity theory. For nondegenerate bifurcation problems equivariant with respect to the standard action of the octahedral group on R3, we find three branches of symmetry-breaking bifurcation corresponding to the three maximal isotropy subgroups of the symmetry group with one-dimensional fixed-point subspaces. Locally, one of these branches is never asymptotically stable, but precisely one of the other branches is stable if and only if all three branches bifurcate supercritically.

A singularity theory classification of these nondegenerate bifurcation problems yields two normal forms. One of these normal forms is of topological codimension one, and so a nondegenerate bifurcation problem need not be generic. Also, each normal form comprises a modal family. Hence the singularity theory classification is more delicate than a topological analysis. In particular, the modal parameters partition the space of nondegenerate bifurcation problems first according to branching and stability considerations but do not stop there. We discuss geometric interpretations of the delicate features of the singularity theory analysis.

A recent result of Gaffney about high-order terms is used to simplify calculations. Further simplifications arise from a connectedness result. In particular, we do not have to consider an explicit change of coordinates.