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A singularity theory analysis of bifurcation problems with octahedral symmetry

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

** Ian Melbourne **

** Abstract **

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
**R**^{3},
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.