Mathematical modeling of nonlinear dynamics in bistable reaction-diffusion systems with fractional derivatives

Abstract

We investigate the dynamics and conditions of emergence of complex autowave solutions in bistable reaction-diffusion systems with time fractional derivatives. It is shown that fractional reaction-diffusion systems have new properties in comparison with standard systems with derivatives of integer order. In particular, in bistable systems with fractional derivatives, we have found new types of autowave solutions, which cannot exist in standard reaction-diffusion systems. Results of the linear theory are substantiated by using computer simulation of a system with cubic nonlinearity, which enables us to simulate characteristic feedbacks and main types of autowave solutions. On the basis of the computational experiment, we show that the order and relation between time fractional derivatives change qualitatively the conditions of instability and nonlinear dynamics of bistable systems.