Diffusion‐driven codimension‐2 Turing–Hopf bifurcation in the general Brusselator model

Abstract

In this paper, spatiotemporal dynamics for a general reaction–diffusion system of Brusselator type under the homogeneous Neumann boundary condition is considered. It is shownthat the reaction–diffusion system has a unique steady state solution. For some suitable 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.