On the Structure of the Global Attractor for Non-autonomous Difference Equations with Weak Convergence

TL;DR

This paper investigates the detailed structure of global attractors in non-autonomous difference equations with recurrent coefficients, focusing on weak convergent systems and their almost periodic solutions within a dynamical systems framework.

Contribution

It provides a new analysis of the global attractor structure for weak convergent non-autonomous difference systems, extending previous results to almost periodic and recurrent solutions.

Findings

Characterization of global attractors for weak convergent systems

Existence of almost periodic solutions in these systems

Application of cocycle theory to analyze solution behavior

Abstract

The aim of this paper is to describe the structure of global attractors for non-autonomous difference systems of equations with recurrent (in particular, almost periodic) coefficients. We consider a special class of this type of systems (the so--called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers to study the almost periodic (almost automorphic, recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent) solutions of difference equations.