Review of the works of V. N. Shchennikova on the study of the convergence of nonlinear almost periodic systems by the comparison method

Abstract

The article provides an overview of the studies of V. N. Shchennikov on the problems of almost periodic convergence of nonlinear differential equations' systems. The problem of convergence established by linear or homogeneous approximation is considered. The conditions for convergence of complex systems are given, that are obtained by constructing Lyapunov vector functions and using the comparison method. It should be noted that in the course of the proof constructive estimates are made for the values of small parameters and interconnection functions. The dimensions of the region in which the limiting almost periodic mode is located are also specified. As an application, the problem of convergence in an electric circuit modeled by a second-order nonlinear differential equation with a small parameter is considered. In conclusion, possible applications and unsolved problems for new directions of research, on which V. N. Shchennikov worked in recent years, are discussed.