Approximation of Attractors of Nonautonomous Lattice Dynamical Systems

TL;DR

This paper investigates finite-dimensional approximations of nonautonomous lattice dynamical systems, demonstrating their uniform dissipativity and the upper semi-continuous convergence of their attractors.

Contribution

It establishes the uniform dissipativity and convergence properties of finite-dimensional approximations for nonautonomous lattice dynamical systems.

Findings

Finite-dimensional approximations are uniformly dissipative.

Attractors of approximations converge upper semi-continuously.

Provides a rigorous foundation for approximating infinite systems.

Abstract

The aim of this paper is to study the finite-dimensional approximations of the nonautonomous lattice dynamical systems of the form $u_{i}'=\nu (u_{i-1}-2u_i+u_{i+1})-\lambda u_{i}+F(u_i)+f_{i}(t)\ (i\in \mathbb Z)\ (*)$. We show that the finite-dimensional approximations for (*) are uniformly dissipative. The upper semi-continuous convergence of the attractors of the finite-dimensional approximations is established.