Hopf Bifurcation and Self-Organization Pattern of a Modified Brusselator Model

Abstract

This paper reports the Hopf bifurcation and self-organization pattern of a modified Brusselator model. The model is a non-standard Brusselator model, it involves the nonlinear restraint term. For the non-diffusive model, we give the types of unique positive equilibrium. It is found that the unique positive equilibrium may be focus, node, or center and we establish their stability, respectively. Especially, there exists the spatial homogeneous Hopf bifurcation when the equilibrium is a center. The first Lyapunov number technique is applied to perform the direction of the spatial homogeneous Hopf bifurcation. In the sequel, the occurrence conditions of the Turing instability and the spatial inhomogeneous Hopf bifurcation are given for the diffusive model. Moreover, by using the normal form theory, we show that the Hopf bifurcation is supercritical or subcritical. Finally, the self-organization patterns induced by the Turing instability and periodic solutions resulting from the Hopf bifurcation are displayed by employing numerical simulations. Our theoretical predictions and numerical results reveal that the modified Brusselator model enjoys the temporal period oscillation and spatial oscillation due to the Hopf bifurcation and Turing instability, respectively. These results may help us to figure out the spatio-temporal dynamics of such modified Brusselator model.