Stability Analysis of Higher Order Fractional Difference Equations

TL;DR

This paper investigates the stability properties of higher-order fractional difference equations, providing new stability criteria, bifurcation analysis, and extending results to nonlinear cases, with applications to complex systems modeling.

Contribution

It introduces new stability analysis methods for higher-order fractional difference equations, including bifurcation discussion and nonlinear extensions.

Findings

Derived stability conditions for linear fractional difference equations.

Analyzed bifurcation phenomena in fractional systems.

Extended stability results to nonlinear fractional difference equations.

Abstract

Fractional difference equations provide a flexible mathematical framework for modeling complex systems with memory, hereditary, and non-local effects. In this work, we study the stability of higher-order two-term fractional linear difference equations $\Delta^{\alpha} x(t) + a \, \Delta^{\beta} x(t+\alpha-\beta-1) =(b-1)x(t+\alpha-2)$. The stability results are derived, and we discuss the bifurcations for $0<\beta \leq 1 < \alpha \leq 2$, $a>0$, $b \in \mathbb{C}$ or $b \in \mathbb{R}$ with examples. We extend this to the stability of an equilibrium point of a nonlinear higher-order fractional difference equation. Moreover, we study the stability of higher-order one-term linear fractional difference equations $\Delta^{\alpha} x(t) = (c-1) x(t+\alpha-N)$ with $N-1<\alpha \leq N$, where $N \in \mathbb{N}$.