Stability and Dynamics of Complex Order Fractional Difference Equations

Abstract

We extend the definition of n-dimensional difference equations to complex order α ∈ 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 ∈ 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 ∈ R, the solutions can be complex and dynamics in one-dimension is richer than the case for α ∈ 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.