On a problem of linearized stability for fractional difference equations

This paper discusses the problem of linearized stability for nonlinear fractional difference equations. Computational methods based on appropriate linearization theorem are standardly applied in bifurcation analysis of dynamical systems. However, in the case of fractional discrete systems, a theoretical background justifying its use is still missing. Therefore, the main goal of this paper is to fill in the gap. We consider a general autonomous system of fractional difference equations involving the backward Caputo fractional difference operator and prove that any equilibrium of this system is asymptotically stable if the zero solution of the corresponding linearized system is asymptotically stable. Moreover, these asymptotic stability conditions for equilibria of the system are described via location of all the characteristic roots in a specific area of the complex plane. In the planar case, these conditions are given even explicitly in terms of trace and determinant of the appropriate Jacobi matrix. The results are applied to a fractional predator-prey model and the fractional Lorenz model. Related experiments are supported by a numerical code that is appended as well