Mathematical modeling of traveling autosolitons in fractional-order activator-inhibitor systems

In the article, basic properties of traveling spatially nonhomogeneous auto-wave solutions in nonlinear fractional-order reactiondiffusion systems are investigated. Such solutions, called autosolitons, arise in a stability region of the system and can coexist with the spatially homogeneous states. By a linear stability analysis and computer simulation, it is shown that the order of the fractional derivative can substantially change the properties of such auto-wave solutions and significantly enrich nonlinear system dynamics. The results of the linear stability analysis are confirmed by computer simulations of the generalized fractional van der Pol-FitzHugh-Nagumo model. A common picture of traveling auto-waves including series in time-fractional two-component activator-inhibitor systems is presented. The results obtained in the article for the distributed system have also been of interest for nonlinear dynamical systems described by fractional ordinary differential equations.