Long-Time Behavior for Second Order Lattice Dynamical Systems

Abstract

Many researchers examined the existence of global attractors for various types of first and second order lattice dynamical systems. Here we prove the existence of a global attractor for a new type of second order lattice dynamical systems in the Hilbert space l 2 × l 2 . For specific choices of the linear operators this system can be regraded as a spatial discretization of a continuous damped nonlinear Boussinesq equation on R m , m ≥ 1.

Second order non-autonomous lattice systems and their uniform attractors

“…Recently, The existence of global attractor, uniform attractor, pullback attractor, and random attractor for different types of autonomous, non-autonomous, and stochastic LDSs in standard and weighted spaces of infinite lattices have been carefully investigated [1,2,3,4,5,7,8,14,15,17,18,20,21,22,23,25,26].…”

“…Remark 2. The upper semicontinuity of global attractors for infinite-dimensional autonomous LDSs [2,5,25] and non-autonomous LDSs [14,20] have been studied.…”

Exponential Attractor for Lattice System of Nonlinear Boussinesq Equation

“…Lattice dynamical systems (LDSs) have a wide range of applications in many areas such as electrical engineering, chemical reaction theory, laser systems, material science, and biology [1,2]. In recent years, many works about the asymptotic behavior of LDSs have been done, which include the global attractor, see [3][4][5][6][7][8][9][10][11] and the references therein. However, the global attractor sometimes attracts orbits at a relatively slow speed and it might take an unexpected long time to be reached.…”

“…Assume that(4) and(14) hold. Then, the semigroup { ( )} ≥0 of (10) possesses an exponential attractorM on A = ⋃ ≥ 0 ( )O with ( ) M is compact; ( ) B ⊂ M ⊂ O,where B is the global attractor; ( ) M has a finite fractal dimension dim (M) ≤ 2 0 (2 * + 1) ln √ ( * ) + 1 + 1, where 0 is a constant and * and * are as in (48); and (iv) there exist two positive constants 1 and 2 such that dist( ( ) , M) ≤ 1 − 2 for all ∈ O, ≥ 0.…”

Attractors for first order lattice systems with almost periodic nonlinear part

“…Recently, The existence of global attractors, uniform attractors, pullback attractors, and random attractors for different types of autonomous, non-autonomous, 1242 AHMED Y. ABDALLAH and stochastic LDSs in standard and weighted spaces have been carefully investigated [1,2,3,4,5,6,9,12,13,25,26,31,34,36,37,38,39,40,41,43,44]. For first order LDSs, the existence of global attractors for autonomous systems [9,42,43] and the existence of uniform global attractors for non-autonomous systems [4,36] have been studied.…”