Multiple bifurcations in a reaction-diffusion problem

We study multiple bifurcations in a system of reaction-diffusion equations defined on the unit square. First, we investigate linear stability of the system at the uniform steady-state solution. Then we discuss necessary conditions for mode interactions. In particular, we show that a corank-four bifurcation point of the system can be easily generated by the mode interactions of two double-bifurcation points. We also study the possibilities of preserving the multiplicity of bifurcation points in the discrete system. A continuation-BCG algorithm proposed by Chien et al. is exploited to trace the solution branches. Finally, sample numerical results are reported.