Symmetric Spiral Patterns on Spheres

Abstract

Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction-diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate on the case where the system has an additional sign-change symmetry, in which case the resulting spiral patterns do not rotate. Spiral patterns arise through a mode interaction between spherical harmonics of degrees $\ell$ and $\ell+1$. Using the methods of equivariant bifurcation theory, possible symmetry types are determined for each $\ell$. For small values of $\ell$, the center manifold equations are constructed, and spiral solutions are found explicitly. Bifurcation diagrams are obtained showing how spiral states can appear at secondary bifurcations from primary solutions or at tertiary bifurcations. The results are consistent with numerical simulations of a model pattern-forming system.