On the forward dynamical behavior of nonautonomous lattice dynamical systems

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.