On Approximation of Almost-Periodic Solutions for a Non-Linear Countable System of Differential Equations by Quasi-Periodic Solutions for Some Linear System

Abstract

It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood later, that the question of existence of such trajectories is closely related to existence of invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.