Stability and Dynamics of Complex Order Fractional Difference Equations

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.

21 pages, 17 figures

References in corpus (4)

• Discrete-Time Fractional Variational Problems
• Fractional $h$-difference equations arising from the calculus of variations
• Emergence of Order in Dynamical Phases in Coupled Fractional Gauss Map
• A Simplification in the proof presented for non existence of periodic solutions in time invariant fractional order systems

Cited by in corpus (1)

• Stability of Fixed Points in Generalized Fractional Maps of the Orders $0<\alpha <1$