Analysis of the maps with variable fractional order

TL;DR

This paper investigates the stability of linear variable order difference equations with periodic order functions, providing general procedures and exact results for specific periods, highlighting how the order influences stability.

Contribution

It introduces a method to analyze stability of variable order difference equations with periodic orders, including exact solutions for periods 2 and 3, and insights into stability determination.

Findings

For T=2, lower order determines stability.

Numerical simulations suggest mean value approximates stability for odd T.

General procedure applicable for arbitrary T.

Abstract

Fractional order differential and difference equations are used to model systems with memory. Variable order fractional equations are proposed to model systems where the memory changes in time. We investigate stability conditions for linear variable order difference equations where the order is periodic function with period $T$. We give a general procedure for arbitrary $T$ and for $T=2$ and $T=3$, we give exact results. For $T=2$, we find that the lower order determines the stability of the equations. For odd $T$, numerical simulations indicate that we can approximately determine the stability of equations from the mean value of the variables.