Attractors for Lattice Dynamical Systems

Abstract

We study the asymptotic behavior of solutions for lattice dynamical systems. We first prove asymptotic compactness and then establish the existence of global attractors. The upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.

Attractors for stochastic lattice dynamical systems with a multiplicative noise

“…The chaotic properties of solutions for such systems have been investigated by [11] and [14,40,13,22]. In the absence of the white noise, the existence of a global attractor for lattice differential equation (1.1) was established in [6].…”

Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems

“…"Tail ends" estimate method is usually used to get asymptotic compactness of autonomous infinite-dimensional lattice, and by this the existence of global compact attractor is obtained; see [15][16][17] . Authors in 18, 19 also prove that the uniform smallness of solutions of autonomous infinite lattice systems for large space and time variables is sufficient and necessary conditions for asymptotic compactness of it.…”

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]). …”