On the forward dynamical behavior of nonautonomous lattice dynamical systems

TL;DR

This paper investigates the forward dynamical behavior of nonautonomous lattice systems, constructing invariant and attracting sets, and applies these results to the discrete Gray-Scott model.

Contribution

It introduces a method to construct invariant and exponentially attracting sets for nonautonomous lattice systems, with applications to specific models.

Findings

Constructed a family of forward invariant sets near the global attractor.

Established conditions for uniform exponential attraction of bounded sets.

Applied theoretical results to the discrete Gray-Scott model.

Abstract

In this article, we study the forward dynamical behavior of nonautonomous lattice systems. We first construct a family of sets $\{\mathcal{A}_\varepsilon(\sigma)\}_{\sigma\in \Sigma}$ in arbitrary small neighborhood of a global attractor of the skew-product flow generated by a general nonautonomous lattice system, which is forward invariant and uniformly forward attracts any bounded subset of the phase space. Moreover, under some suitable conditions, we further construct a family of sets $\{\mathcal{B}_\varepsilon(\sigma)\}_{\sigma\in \Sigma}$ such that it uniformly forward exponentially attracts bounded subsets of the phase space. As an application, we study the discrete Gray-Scott model in detail and illustrate how to apply our abstract results to some concrete lattice system.