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

A 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.

Automated summary

This paper investigates a general reaction-diffusion Brusselator model under homogeneous Neumann boundary conditions. The stability of the unique positive equilibrium is studied, and the existence of the Hopf bifurcation is identified. Occurrence conditions of the Turing instability, Turing-Turing, and Turing-Hopf bifurcations are established. The spatiotemporal solutions resulting from the bifurcation are explored, and the validity of the theoretical results is verified through numerical simulations.