Complex Oscillations and Limit Cycles in Autonomous Two-Component Incommensurate Fractional Dynamical Systems

MSC 2010: 26A33, 34D05, 37C25

In the paper, long-time behavior of solutions of autonomous two-component incommensurate fractional dynamical systems with derivatives in the Caputo sense is investigated. It is shown that both the characteristic times of the systems and the orders of fractional derivatives play an important role for the instability conditions and system dynamics. For these systems, stationary solutions can be unstable for wider range of parameters compared to ones in the systems with integer order derivatives. As an example, the incommensurate fractional FitzHugh-Nagumo model is considered. For this model, different kinds of limit cycles are obtained by the method of computer simulation. A common picture of non-linear dynamics in fractional dynamical systems with positive and negative feedbacks is presented.