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.