Abstract Turing–Hopf bifurcation of the diffusive Brusselator model with homogeneous Neumann boundary conditions is considered in this paper. By stability analysis, the conditions of the Turing instability are obtained and the critical values of the Turing–Hopf bifurcation are also given. In order to better understand the dynamics near the Turing–Hopf bifurcation, the amplitude equations are derived by the method of the multiple time scale. Through the analysis of amplitude equations, complex dynamics are found, such as nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous periodic solutions, near the Turing–Hopf bifurcation point. The results show that, compared with the codimension-one Turing instability or Hopf bifurcation, the codimension-two Turing–Hopf bifurcation can induce more complex patterns: spatially inhomogeneous periodic solutions, which could be used to explain the phenomenon of spatiotemporal resonance between activators and inhibitors of chemical reactions. For bifurcation illustration of Brusselator model, the neighbourhood of bifurcation point is divided into six regions and various bifurcation solutions corresponding to different regions are presented via numerical simulations, respectively, which verify the theoretical analysis.
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