Interaction of Turing and Hopf modes in the superdiffusive Brusselator model

Long-wave stability of spatiotemporal patterns near a codimension-2 Turing–Hopf point of the one-dimensional superdiffusive Brusselator model is analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by ∂α/∂|ξ|α∂α/∂|ξ|α, 1<α<21<α<2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, αα, is a measure of the rate of superdiffusion, which, in general, can be different for each of the two components. We first find the basic (spatially homogeneous, time independent) solution and study its linear stability, determining both Turing and Hopf instabilities, as well as a point at which both instabilities occur simultaneously. We then employ a weakly nonlinear stability analysis 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: 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 strongly on the superdiffusion rates. Also, when compared to the regular model and depending on specific values of the orders of the operators, the effect of anomalous diffusion may change the stability characteristics of the special solutions.