On the asymptotic stability of solutions of nonstationary difference systems with homogeneous right-hand sides

Difference equations are widely used for the modeling of dynamical systems whose states are measured at discrete instants of time, as well as for the approximate replacement of continuous mathematical models. In particular, most part of numerical methods for solving of ordinary differential equations are based on their replacement by difference ones. One of directions of investigations arising in applications of difference equations is associated with the stability analysis of their solutions. In the present paper, by the use of the Lyapunov functions method, sufficient conditions of the uniform asymptotic stability of solutions of homogeneous time-varying systems of difference equations are derived. To obtain these conditions, a Lyapunov function is used which is constructed on the basis of the corresponding function found for the averaged system of ordinary differential equations. In this paper, the discrete counterparts of results concerning to the stability of solutions of homogeneous time-varying systems of ordinary differential equations are obtained. Compared with known for this type of difference systems results, the established conditions provide the uniform asymptotic stability of solutions. Bibliogr. 29.