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 ${\mathrm{\Delta }}^{\alpha }x(t)+a{\mathrm{\Delta }}^{\beta }x(t+\alpha -\beta -1)=(b-1)x(t+\alpha -2)$. The stability results are derived, and we discuss the bifurcations for $0<\beta \le 1<\alpha \le 2$, $a>0$, $b\in \mathbb{C}$ 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 ${\mathrm{\Delta }}^{\alpha }x(t)=(c-1)x(t+\alpha -N)$ with $N-1<\alpha \le N$, where $N\in \mathbb{N}$.