CMMSE: New Properties of Auto‐Wave Solutions in Activator‐Inhibitor Reaction‐Diffusion Systems With Fractional Derivatives

ABSTRACT In this article, we analyze new properties of auto‐wave solutions in fractional reaction‐diffusion systems. These new properties arise due to a change in fractional derivative order and do not occur in systems with classical derivatives. It is shown that the stability of steady‐state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. It is also demonstrated that the basic properties of auto‐wave solutions in fractional‐order systems can essentially differ from those in standard systems. The results of the linear stability analysis are confirmed by computer simulations of the generalized fractional van der Pol–FitzHugh–Nagumo mathematical model. A common picture of possible instabilities and auto‐wave solutions in time‐fractional two‐component activator‐inhibitor systems is presented.