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

TL;DR

This paper derives stability conditions for two-component incommensurate fractional-order systems and applies these to analyze a fractional FitzHugh-Nagumo neuronal model, revealing complex dynamics and bifurcations through theoretical and numerical methods.

Contribution

It generalizes stability criteria for incommensurate fractional systems and applies them to a neuronal model, exploring bifurcations and spiking behavior.

Findings

Derived necessary and sufficient stability conditions for incommensurate fractional systems.

Identified conditions for Hopf bifurcations in the fractional FitzHugh-Nagumo model.

Numerical simulations show rich spiking dynamics differing from classical models.

Abstract

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.