Abstract

The spatiotemporal dynamics for general reaction-diffusion systems of Brusselator type under the homogeneous Neumann boundary condition is considered. It is shown that the reaction-diffusion system has a unique steady state solution. For some suitable ranges of the parameters, we prove that the steady state solution can be a codimension-2 Turing-Hopf point. To understand the spatiotemporal dynamics in the vicinity of the Turing-Hopf bifurcation point, we calculate and analyze the normal form on the center manifold by analytical methods. A wealth of complex spatiotemporal dynamics near the degenerate point are obtained. It is proved that the system undergoes a codimension-2 Turing-Hopf bifurcation. Moreover, several numerical simulations are carried out to illustrate the validity of our theoretical results.