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