Transition from Rotating Waves to Modulated Rotating Waves on the Sphere

In this article, we study parameter-dependent systems of reaction-diffusion equations on the sphere, which are equivariant under the group SO(3) of all rigid rotations on a sphere. Two main types of spatial-temporal patterns that can appear in such systems are rotating waves (equilibrium in a co-rotating frame) and modulated rotating waves (periodic solution in a corotating frame). The transition from rotating waves to modulated rotating waves on spherical domains is explained via a supercritical Hopf bifurcation from a rotating wave, SO(3)-symmetry and finite-dimensional equivariant center manifold reduction. The Baker-Campbell-Haussdorff formula in the Lie algebra so(3) is used to get reduced differential equations on so(3), a formula for a primary frequency vector, as well as a formula for the periodic part associated to any modulated rotating wave obtained by a supercritical Hopf bifurcation from a rotating wave. As a consequence, there are three types of motions for the tips of these modulated rotating waves. In the resonant case (with two parameters), we obtain that the primary frequency vectors of a branch of these modulated rotating waves are generically orthogonal to the frequency vector of the initial rotating wave undergoing Hopf bifurcation.