Stability of Systems of Fractional-Order Difference Equations and Applications to a Rulkov-Type Neuronal Model

Necessary and sufficient conditions for the asymptotic stability and instability of two-dimensional linear autonomous incommensurate systems of fractional-order Caputo difference equations are presented. Moreover, the occurrence of discrete Flip and Hopf bifurcations is also discussed, choosing the fractional orders as bifurcation parameters. The theoretical results are then applied to the investigation of the stability and instability properties of a fractional-order version of the Rulkov neuronal model. Numerical simulations are further presented to illustrate the theoretical findings, revealing complex bursting behavior in the fractional-order Rulkov model.