M. Field, M. Golubitsky and I.N. Stewart

Bifurcations on hemispheres

J. Nonlinear Science. 1 (1991) 201-223.

It is now well known that the number of parameters and symmetries of an equation affect the bifurcation structure of that equation. The bifurcation behavior of reaction diffusion equations on certain domains with certain boundary conditions is nongeneric in the sense that the bifurcation of steady states in these equations is not what would be expected if one considered only the number of parameters in the equations and the type of symmetries of the equations. This point was made previously in work by Fujii, Mimura and Nishiura [6] and Armbruster and Dangelmayr [1] who considered reaction diffusion equations on an interval with Neumann boundary conditions.

As was pointed out by Crawford et al. [5], the source of this o nongenericity is that reaction diffusion equations are invariant under translations and reflections of the domain and, depending on boundary conditions, may naturally and uniquely be extended to larger domains with larger symmetry groups. These extra symmetries are the source of the nongenericity. In this paper we consider in detail the steady-state bifurcations of reaction diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a natural O(2) symmetry, but may be extended to the full sphere where the natural symmetry group is O(3). We also determine alarge class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.