Stability and Dynamics of Complex Order Fractional Difference Equations

TL;DR

This paper extends difference equations to complex orders, analyzing stability conditions for linear and nonlinear systems, revealing complex dynamics and stability boundaries in the complex plane.

Contribution

It introduces the definition of complex order difference equations and derives stability conditions for both linear and nonlinear systems, including boundary curves and stability regions.

Findings

Stability regions are enclosed by boundary curves in the complex plane.

No stable region exists if the boundary curve is self-intersecting.

One-dimensional solutions exhibit richer dynamics when solutions are complex.

Abstract

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.