Hopf bifurcation on hemispheres

Field et al (1991 J. Nonlinear Sci. 1 201–23) consider the steady-state bifurcations of reaction–diffusion equations defined on the hemisphere with Neumann boundary conditions on the equator. We consider Hopf bifurcations for these equations. We show the effect of the hidden symmetries on spherical domains for the type of Hopf bifurcations that can occur. We obtain periodic solutions for the hemisphere problem by extending the problem to the sphere and then finding periodic solutions with spherical spatial symmetries containing the reflection across the equator. The equations on the hemisphere have O(2)-symmetry and the equations on the sphere have spherical symmetry. We find orbits of periodic solutions for the sphere problem containing multiple periodic solutions that restrict to periodic solutions of the Neumann boundary value problem on the hemisphere lying on different O(2)-orbits.