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Stability and Dynamics of Complex Order Fractional Difference Equations

We extend the definition of $n$-dimensional difference equations to complex order $α\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 $A =λ\in \mathbb {C}$, 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 $ λ\in \mathbb{R} $, the solutions can be complex and dynamics in one-dimension is richer than the case for $ α\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.