Second Order Stochastic Lattice Dynamical Systems: Existence Of Random Attractors

Random attraction is an important dynamic behavior that has received the increasing research attention. In this paper, we investigate the existence of the global random attra- ctors of second order stochastic lattice dynamical systems:Firstly, we establish Hilbert space F = lλ2×l2 and prove the uniqueness and existe- nce of the solutions of second order stochastic lattice dynamical systems in the space F = lλ2×l2. Furthermore, by stochastic analysis, we show that the solutions of the equ- ation are dependent continuously on initial conditions. Then, we estimate the"ends"of solutions of the system by stochastic analysis and nonlinear analysis.Finally, we establish the existence of the global random attractor with the set of temp- ered bounded sets by proving the asymptotic compactness of the random dynamical sys- tems.In section 1, background of stochastic dynamical systems and the global random attr- actor is introduced in this section.In Section 2, we introduce some basic concepts related to stochastic dynamical systems and the global random attractor. Meanwhile, we present some notations and give a simple description of the system.In Section 3, some bounded conditions and assumptions of nonlinear terms are given, and the existence and uniqueness of solutions of system (1) are established.In Section 4, we prove the existence of an absorbing set.In Section 5, we establish the existence conditions for a global random attractor of system (1), and some concluding remarks are given in this section.