Inhomogeneous oscillatory solutions in fractional reaction–diffusion systems and their computer modeling

Abstract

We investigate inhomogeneous oscillatory instability conditions in fractional reaction–diffusion systems. It is shown that non-linear stationary solutions emerge as a result of Turing instability becoming unstable according to oscillatory perturbation and transform to inhomogeneous oscillatory structures. It is also shown that with the certain value of the fractional derivatives index, a new type of instability takes place and the system becomes unstable towards perturbations of finite wave number. As a result, oscillatory perturbations with this wave number become unstable and lead to non-linear oscillations which result in spatial oscillatory structure formation. Computer simulation of a Bonhoeffer–van der Pol type reaction–diffusion systems with fractional time derivatives is performed.