The attractors for 2nd-order stochastic delay lattice systems

This paper deals with the long-time dynamical behavior of a classof 2nd-order stochastic delay lattice systems. It is shown under thedissipative and sublinear growth conditions that such a systempossesses a compact global random attractor within the set oftempered random bounded sets. A numerical example is given toillustrate the obtained theoretical result.


1.
Introduction. The purpose of this paper is to study the long-time dynamical behavior for the following 2nd-order stochastic delay lattice differential systems with additive white noise: where i ∈ Z with Z being the integer set, u it = u i (t + θ) (θ ∈ [−h, 0]), α, λ and h are positive constants, a = (a i ) i∈Z ∈ l 2 , g = (g i ) i∈Z ∈ l 2 , each f i is a smooth function satisfying some dissipative condition, and w i (t) denotes an independent two-side Brownian motion. The various dynamical properties of deterministic lattice systems have caused wide concern in the past decades, and hence some significant related results have been presented in many references (see e.g., [4,18,19,30,31,32]). Comparing with the deterministic models, it was found that stochastic models can describe the actual physical phenomena more completely. Hence, recently, the researchers started to work for the investigation of stochastic lattice systems. Since Bates et al. [3] initiated the study of stochastic lattice systems, there is by now a rather comprehensive mathematical literature on the existence of random attractors for stochastic lattice systems. For example, the first order lattice systems with an additive noise were considered in [2,7,13,16]. The existence results of random attractors of second-order lattice systems with additive noise were presented in [20, 576 CHENGJIAN ZHANG AND LU ZHAO 25]. As to the case of the multiplicative noise, we refer the readers to [6,8,12,13,14] and the references therein.
On account of non-instant transmission or memory processes, the external force term may contain some genetic or memory features. In this direction, the study of delay systems has an important theoretical and practical significance, and hence a number of important results have been reported, seeing e.g. [5,9,10,11,15,17,21,22,14,23,24,26,27,29]. Especially, the existence of global attractors for deterministic delay lattice systems was analyzed in [26]. A kind of deterministic delay lattice systems with general external forces was studied in [9,24]. Stochastic Fitzhugh-Nagumo systems with delay were considered in [21]. Attractors for 1storder stochastic delay lattice systems were investigated in [11,22]. To the best of our knowledge, there are very few results on the existence of attractors for 2nd-order stochastic delay lattice systems.
In the present paper, motivated by the idea in [22,24], we will deal with the long-time dynamical behavior of stochastic delay lattice system (1.1). To this end, we first introduce a new norm in the phase space and derive an existence criterion of solution for (1.1). Then, we transform (1.1) into a stochastic functional differential equation with no noise and study the stochastic dynamical properties of (1.1). Especially, it is shown under the dissipative and sublinear growth conditions that such a system possesses a compact global random attractor within the set of tempered random bounded sets. In the end, a numerical example is given to illustrate the obtained theoretical result.

2.
Preliminaries. In this section, for the subsequent analysis, we recall some concepts and the related results on stochastic dynamical system (SDS). These background knowledge also can be found in the references [1,3]. Let (X, · X ) be a sparable Banach space with Borel σ-algebra B(X) and (Ω, F, P ) be a probability space. Then the following concepts and conclusions can be stated. Definition 2.1. (Ω, F, P, (ϑ t ) t∈R ) is called a metric dynamical system if ϑ : R×Ω → Ω is B(R × F, F) measurable, ϑ 0 = Id Ω , ϑ t+s = ϑ t • ϑ s for all t, s ∈ R, and ϑ t P = P for all t ∈ R.
In what follows, D(X) denotes the set of all tempered random sets of X.
For the function f , we make the following assumptions: An existence and uniqueness theorem of the solution of system (3.1) can be stated as follows.
Proof. Rewriting (3.1) as then, by (I) and (III), we have for any ϕ t ∈ H 0 that Thus F maps the bounded sets of H 0 into the bounded sets of H. Similarly, we have that In this way, the existence and uniqueness of the local solution of (3.1) can be proved by the standard technique. On the other hand, as shown in Lemma 4.1, we can find that ϕ t H0 is bounded in H 0 . Therefore, the local solution ϕ(t) of (3.1) is global.
Next, we assume ϕ In terms of Gronwall inequality, it holds that Therefore, the theorem is proven.
In order to obtain the existence of a random attractor, we need to use an estimate of the tails of solutions.
This completes the proof.
5. Numerical illustration. In this section, we present a numerical example to support our conclusions. Especially, taking using of several numerical experiments, we will test the influence of initial conditions to the system. Consider the 2nd-order stochastic delay lattice differential equation: u i + 4u i − (u i−1 − 2u i + u i+1 ) + 8u i + f i (u it ) +ẇ i (t) = 0, t > 0, i ∈ Z, (5.1) where k = h/2m denotes the computational stepsize in time, m is a given positive integer, u n i is an approximation to u i (t n ), and ∆ n i = w i (t n+1 ) − w i (t n ), which is independent N (0, k)-distributed Gaussian random variable.
Setting i = 1, 2, . . . , 100 and m = 100 and then using the numerical scheme (5.2) to solve the equations (5.1) with different initial functions: u i (t) = ∂ ∂t u i (t) = exp(t) cos( iπ 50 ) and u i (t) = ∂ ∂t u i (t) = exp(t) sin( iπ 50 ) (t ∈ [−h, 0]), respectively, we can obtain the corresponding numerical solutions. The 3D figures of the numerical solutions are plotted in Figure 1 and 2, respectively. And a detailed information of the numerical solutions at time t = 10 is shown in Figure 3. From the numerical results, we can find that the numerical solutions with different initial conditions arrive at the same stable state as the time increases, which implies the stochastic delay lattice systems have a D-random attractor. Hence, the previous theoretical results are further verified by the above numerical experiments.