Conditions of convergence of some classes of nonlinear difference systems

One important problem of the qualitative theory of differential and difference equations is to determine the conditions of existence and stability of forced stationary oscillations, which arise in nonlinear systems under external disturbances. Of great practical interest is the situation when these stationary modes are globally asymptotically stable. Such a phenomenon is called convergence. The theorem on the conditions for almost periodic convergence for differential equations systems was proved by V. I. Zubov. These conditions are formulated in terms of the existence of Lyapunov’s functions with certain properties. N. N. Ataeva extended Zubov’s theorem to the difference systems. However, classes of the difference systems were not specified for which the theorem gives conditions of convergence. At the same time it should be noted that so far, there are no general constructive methods for constructing Lyapunov’s functions that satisfy the requirements of the Zubov’s theorem. A method for constructing Lyapunov’s functions that satisfy the requirements of Zubov’s theorem was proposed for some types of systems of differential equations. In this work sufficient conditions for convergence of the difference systems are obtained by the use of the Lyapunov’s direct method and the discrete analogue of Zubov’s theorem.