Stability Analysis of Higher Order 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 $Δ^α x(t) + a \, Δ^β x(t+α-β-1) =(b-1)x(t+α-2)$. The stability results are derived, and we discuss the bifurcations for $0<β\leq 1 < α\leq 2$, $a>0$, $b \in \mathbb{C}$ 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 $Δ^α x(t) = (c-1) x(t+α-N)$ with $N-1<α\leq N$, where $N \in \mathbb{N}$.

Figures (12)

• Figure 1: The stability regions for $\alpha=1.9$ and $\beta=0.2$ and particular values of $a$, namely $2$ in (a), $3.24901$ in (b), $3.5$ in (c) and $3.79051$ in (d) respectively. In figures, S: stable, U: unstable.
• Figure 2: Behavior of the solutions of the system (\ref{['1']}) for $\alpha=1.9$, $\beta=0.2$ and different values of $a$ and $b$.
• Figure 3: The stability regions of system (\ref{['1']}) with blue (left boundary) and red (right boundary) for various values of $\alpha$ and $\beta$ in $ab-$plane. In figures, S: stable.
• Figure 4: The solutions of system (\ref{['1']}) for different values of $b$ within and outside the stable region.
• Figure 5: The solutions of system (\ref{['nonlinear']}) for different values of $\mu$ and $a$.

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 3.1
• Example 3.2
• Example 3.3
• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4