Turing‐Turing and Turing‐Hopf bifurcations in a general diffusive Brusselator model

Abstract

AbstractA general reaction‐diffusion Brusselator model subject to homogeneous Neumann boundary conditions is investigated in this paper. First, the stability of the unique positive equilibrium is studied, and we identify the existence of the Hopf bifurcation. Then, occurrence conditions of the Turing instability, the Turing‐Turing, and the Turing‐Hopf bifurcations are established. To explore the spatiotemporal solutions resulting from the bifurcation, the amplitude equations of the Turing‐Turing and the Turing‐Hopf bifurcations are established via the method of multiple time scale. It is found that the model admits the nonconstant steady state, the mixed nonconstant steady state, the spatially homogeneous periodic solution, and the spatially nonhomogeneous periodic solution. As such, the nonhomogeneous spatiotemporal solutions appear in the model. In the end, numerical simulations verify the validity of the theoretical results.