Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model

For two-dimensional autonomous linear incommensurate fractional-order dynamical systems with Caputo derivatives of different orders, necessary and sufficient conditions are obtained for the asymptotic stability and instability of the null solution. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional orders of the Caputo derivatives, leading to a generalization of the well known Routh-Hurwitz conditions. These theoretical results are then used to investigate the stability properties of a two-dimensional fractional-order FitzHugh-Nagumo neuronal model. The occurrence of Hopf bifurcations is also discussed. Numerical simulations are provided with the aim of exemplifying the theoretical results, revealing rich spiking behavior, in comparison with the classical integer-order FitzHugh-Nagumo model.