Bifurcation of Mixed Mode Reaction-Diffusion Patterns in Spherical Caps

We study pattern formation in a chemical reaction-diffusion system of partial differential equations in spherical cap domains. For certain critical values of parameters corresponding to cap flatness, cap radius, and chemical precursor concentrations, the unpatterned solution is unstable to two different linear normal modes. We use center manifold and normal form reductions to analyze the existence and stability of pure and mixed modes of nonlinear patterned solutions of the reaction-diffusion system, for parameters with two cases of critical values. In one case, the system reduces to a well known example of mode interaction. In the other case, the mode interaction is new, due to very small quadratic terms in the normal form.