Dynamics of systems on infinite lattices

Abstract

The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l 2 space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.

Notes on the Global Attractors for Semigroup

“…In this section, firstly we recall some basic definitions in [1,7], then we show that the two necessary and sufficient conditions for the existence of global attractors for semigroups are equivalent directly. Definition 3.1 Let M be a complete metric space.…”

“…Furthermore, in [7], the author introduce the concept of asymptotically null and show that a lattice system has a global attractor if and only if:…”

“…Our main motivation of this paper is to prove that asymptotically compact   -limit compact, and then we prove that the conditions in [1,7] are equivalent directly in…”

Attractors for non-autonomous retarded lattice dynamical systems

“…Usually, the models under consideration are obtained by a spatial discretization of a parabolic or a hyperbolic equation (see e.g. [1], [2], [4], [5], [8], [11] [12], [15], [16], [19], [20], [22], [23], [26], [28], [29]). …”

Pullback Exponential Attractor for Second Order Nonautonomous Lattice System

“…In recent years, global attractors, uniform attractors, pullback attractors (or kernel sections), and random attractor for autonomous, nonautonomous, and stochastic LDSs have been studied; see [3][4][5][6][7][8][9][10][11][12]. However, these attractors sometimes attract orbits at a relatively slow speed, so that it might take an unexpected long time to be reached.…”