Stability and Dynamics of Complex Order Fractional Difference Equations

We extend the definition of $n$-dimensional difference equations to complex order $\alpha \in \mathbb{C}$. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of equilibrium points for linear systems. For the one-dimensional case where $A=\lambda \in \mathbb{C}$, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for $\lambda \in \mathbb{R}$, the solutions can be complex and dynamics in one-dimension is richer than the case for $\alpha \in \mathbb{R}$. These results can be extended to $n$-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.