Bifurcation characteristics of fractional reaction-diffusion systems

Anomalous behavior of many complex heterogeneous systems is known to be adequate modeled with the fractional differential equations and in particular with the fractional reaction-diffusion systems (FRDS). In this article, a generalized reaction-diffusion model in form of a system of nonlinear fractional partial differential equations is considered. It is shown that orders of the fractional derivatives contained in the FRDS are new bifurcation parameters that can change stability both of the spatially-homogeneous and of the spatially-nonhomogeneous stationary solutions. A general principle of linear stability for FRDS is formulated. The results of linear stability analysis are confirmed by computer simulations of some basic FRDS with classical nonlinearities. It is shown that stability of steady state solutions of FRDS and their evolution are mainly determined by the orders of the fractional derivatives and the eigenvalue spectrum of the linearized systems. Moreover, new types of spatiotemporal solutions a...