Hopf bifurcation computation for two-component reaction-diffusion equations

Inspired by the numerical evaluation of Hopf bifurcation formulae for the Brusselator reaction-diffusion model, the spatially discretized two-component reaction-diffusion equations are proposed. Meanwhile, three kinds of spatially discretized boundary conditions are presented. Then the local stability of the constant steady state of the equations subject to one of the three kinds of boundary conditions is analyzed in a unified form, by employing the matrix form of the decoupling method. Hence the occurrence conditions for Hopf bifurcation are derived. At the same time, the expressions of the spatial eigenvalues and eigenvectors are given based on the theory of discrete Fourier transform. Further, the properties of the bifurcating periodic solutions are studied. It is shown that the effects of boundary conditions essentially arise from the different expressions of the spatial eigenvalues and eigenvectors. As an example, the derived results are applied to analyze the dynamics of the spatially discretized Brusselator model, both theoretically and numerically. For this kind of equations with m components (m ≥ 3) in n-dimensional space (n ≥ 2), the methods are also valid.