Abstract
Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences.
Highlights
The domain of fractional calculus is regarded as an important tool in modelling various phenomena from different scientific and engineering fields [1,2,3,4,5,6]
Stability results for linear systems with Caputo fractional-order difference equations with variable order of convolution type have been studied in [12], along with the recurrence formulas for the solutions of linear initial value problems for the considered fractional operators
We explore the stability and instability properties for systems that consist of two fractional-order difference equations
Summary
The domain of fractional calculus is regarded as an important tool in modelling various phenomena from different scientific and engineering fields [1,2,3,4,5,6]. Stability results for linear systems with Caputo fractional-order difference equations with variable order of convolution type have been studied in [12], along with the recurrence formulas for the solutions of linear initial value problems for the considered fractional operators. The most common and effective method used in the study of the stability properties of linear discrete-time fractional order systems is considered to be the Z -transform method. When taking into consideration the fact that fractional derivatives are successfully approximated by fractional h-differences of similar types, a link can be established between the stability properties of fractional-order differential systems and their discrete-time counterparts, i.e., fractional-order systems of difference equations. Fractional-order membrane potential dynamics have proven their utility in reproducing the electrical activity of neurons that were experimentally observed, as they are able to introduce capacitive memory effects [22]. FitzHugh–Nagumo neuronal model is later investigated, as an application to the theoretical findings, followed by numerical simulations that reveal rich spiking behaviour
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