Stability and dynamics of complex order fractional difference equations

We extend the definition of n-dimensional difference equations to complex order. We investigate the stability of linear systems defined by an n-dimensional matrix and derive the conditions for the stability of zero solution of linear systems. For the one-dimensional case, 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 the real eigenvalues, the solutions can be complex and dynamics in one-dimension is richer than the case for real order. 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.

Automated summary

This article extends the definition of n-dimensional difference equations to complex order and investigates the stability of linear systems defined by an n-dimensional matrix. It derives the conditions for the stability of the zero solution of linear systems. For the one-dimensional case, it finds that the stability region, if any, is enclosed by a boundary curve and obtains a parametric equation for it. Furthermore, it observes that there is no stable region if this parametric curve is self-intersecting. The article also highlights that even for real eigenvalues, the solutions can be complex, and the dynamics in one dimension are richer than the case for real order. These findings can be extended to n-dimensions. For nonlinear systems, it concludes that the stability of the linearized system determines the stability of the equilibrium point.