Interaction of Turing and Hopf Modes in the Superdiffusive Brusselator Model Near a Codimension Two Bifurcation Point

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by partial derivative(alpha)/partial derivative vertical bar xi vertical bar(alpha), 1 < alpha < 2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, alpha, is a measure of the rate of superdiffusion, which, in general, can be different for each of the two components. A weakly nonlinear analysis is used to derive two coupled amplitude equations describing the slow time evolution of the Turing and Hopf modes. We seek special solutions of the amplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed mode solution, and analyze their stability to long-wave perturbations. We find that the stability criteria of all three solutions depend greatly on the rates of superdiffusion of the two components. In addition, the stability properties of the solutions to the anomalous diffusion model are different from those of the regular diffusion model. Numerical computations in a large spatial domain, using Fourier spectral methods in space and second order Runge-Kutta in time are used to confirm the analysis and also to find solutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime. Specifically, we find a large number of steady state patterns consisting of a localized region or regions of stationary stripes in a background of time periodic cellular motion, as well as patterns with a localized region or regions of time periodic cells in a background of stationary stripes. Each such pattern lies on a branch of such solutions, is stable and corresponds to a different initial condition. The patterns correspond to the phenomenon of pinning of the front between the stripes and the time periodic cellular motion. While in some cases it is difficult to isolate the effect of the diffusion exponents, we find characteristics in the spatiotemporal patterns for anomalous diffusion that we have not found for regular (Fickian) diffusion.