Stability Analysis of Higher Order Fractional Difference Equations

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 $N \in \mathbb{N}$.