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Random attractor for second-order stochastic delay lattice sine-Gordon equation
Boundary Value Problems volume 2021, Article number: 14 (2021)
Abstract
In this paper, we prove the existence of random \(\mathcal{D}\)-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random \(\mathcal{D}\)-attractor.
1 Introduction
This paper deals with the following second-order stochastic delay lattice sine-Gordon equation:
where \(i\in \mathbb{Z}^{k}\) with \(\mathbb{Z}\) being the set of integers and \(k\in \mathbb{N}\) a fixed positive integer, \(u_{it}=u_{it}( \tau )=u_{i}(t+\tau )\) is the delay term with the interval of delay time \([-h,0]\), and \(u_{i0}=u_{i}(\tau )\), \(\dot{u}_{i0}=\dot{u_{i}}(\tau )\) is the initial data, A is a linear operator defined by (3.1), \(u=(u_{i})_{i\in \mathbb{Z}^{k}}\), \(g=(g_{i})_{i\in \mathbb{Z}^{k}}\in l^{2}\), α, β, h are positive constants, \(\epsilon =(\epsilon _{i})_{i\in \mathbb{Z}^{k}}\in l^{2}\), \(f_{i}\) is a smooth function satisfying some dissipative conditions (see the hypotheses \(H_{1}\)–\(H_{3}\) in Sect. 3), \(\{w_{i}(t):i\in \mathbb{Z}^{k}\}\) is independent two-sided real valued standard Wiener processes.
Lattice dynamical systems, whose the spatial structure has a discrete character, arise from a variety of applications such as electrical engineering [1], biology [2, 3], chemical reaction [4], and pattern formation [5]. As a matter of fact, systems in the process of evolution are always influenced by the external environment, those influence may be random or the time delay. If the system add the random or the time delay terms, it makes up for the defects of some deterministic systems, and explains new evolutionary rules. Many researchers have discussed broadly the deterministic models in [6–8]. Stochastic lattice equations, driven by additive independent white noise, were discussed for the first time in [9], and then intensive researched in [10–20]. Furthermore, a kind of stochastic delay lattice systems were considered in [21–24], and these interesting results have attracted wide attention of scholars.
However, to the best of our knowledge, there is little literature about the existence of random attractors for a second-order stochastic delay lattice sine-Gordon equation on \(\mathbb{Z}^{k}\). To this end, this paper is devoted to study this problem. The main ideas and methods used in the proofs are motivated by [21, 22, 24–27].
This paper is organized as follows. In Sect. 2, we introduce some basic concepts and propositions. In Sect. 3, we present some sufficient conditions for the existence of continuous stochastic dynamical systems. Section 4 is devoted to proving the existence of a random \(\mathcal{D}\)-attractor for a stochastic lattice sine-Gordon equation. In Sect. 5, we study the upper bound of the Kolmogorov ε-entropy for the random \(\mathcal{D}\)-attractor.
2 Preliminaries
In this section, we recall some basic concepts and propositions related to random attractors for stochastic dynamical systems (SDS), more details can be found in the literature [9, 28, 29].
Let \((X,\|\cdot \|_{X})\) be a separable Hilbert space, and \((\Omega ,\mathcal{F},P)\) a probability space.
Definition 2.1
Let \(\mathcal{D}\) be a collection of random subsets of X, stochastic process \(S\{(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) a continuous random dynamical system and \(\{B_{0}(\omega )\}_{\omega \in \Omega }\in \mathcal{D}\). Then \(\{B_{0}(\omega )\}_{\omega \in \Omega }\) is said to be a random absorbing set for S in \(\mathcal{D}\) if for every \(B\in \mathcal{D}\) for P-a.e. \(\omega \in \Omega \), there exists some \(t_{B}(\omega )>0\) such that
Definition 2.2
Let \(\mathcal{D}\) be a collection of random subsets of X. Then a random set \(\{\mathcal{A}(\omega )\}_{\omega \in \Omega } \) in X is said to be a \(\mathcal{D}\)-random attractor for S, if the following conditions are satisfied, for P-a.e. \(\omega \in \Omega \):
-
(1)
\(\{\mathcal{A}(\omega )\}\) is compact, and \(\omega \rightarrow d(x,\mathcal{A}(\omega ))\) is measurable for every \(x\in X\);
-
(2)
\(\{\mathcal{A}(\omega )\} \) is invariant;
-
(3)
\(\{\mathcal{A}(\omega )\}\) attracts every set in \(\mathcal{D}\), i.e., for all \(B=\{B(\omega )\}_{\omega \in \Omega }\in \mathcal{D}\),
$$ \lim_{t\rightarrow \infty }d \bigl(S(t,\theta _{-t} \omega )B( \theta _{-t}\omega ),\mathcal{A}(\omega ) \bigr)=0, $$
where d denotes the Hausdorff semi-metric.
Definition 2.3
Let \(B(\omega )\subset X\) be a random set. For any \(\varepsilon >0\), \(\omega \in \Omega \), let \(\mathcal{N}_{\varepsilon ,\omega }(B(\omega ),X)=\mathcal{N}_{ \varepsilon ,\omega }(B(\omega ))\) be the minimal number of deterministic open balls in X with radii ε that is necessary to cover \(B(\omega )\). The number \(\mathcal{K}_{\varepsilon }(B(\omega ))=\mathcal{K}_{\varepsilon }(B( \omega ),X) =\ln \mathcal{N}_{\varepsilon ,\omega }(B(\omega ))\) is called the Kolmogorov ε-entropy of \(B(\omega )\) in X.
Proposition 2.1
(See [30])
Let \(n\in \mathbb{N}\) and \(\Upsilon =\{x=(x_{i})_{|i|\leq n}:x_{i}\in \mathbb{R},|x_{i}|\leq r \}\subset \mathbb{R}^{2n+1}\) be a regular polyhedron. Then ϒ can be covered by \(\mathcal{N}_{\varepsilon }(\Upsilon )= ([2r\cdot \frac{1}{\varepsilon }\sqrt{2n+1}]+1 )^{2n+1}\) balls in \(\mathbb{R}^{2n+1}\) with radii \(\frac{\varepsilon }{2}\), where \([\cdot ]\) denotes the integer-valued function.
Proposition 2.2
(See [29])
If \(r(\omega )>0\) is tempered and \(r(\theta _{t}\omega )\) is continuous in t for P-a.e. \(\omega \in \Omega \), then
-
(1)
for any \(t\in \mathbb{R}\), \(r(\theta _{t}\omega )\) is tempered. Moreover, for any \(h>0\), \(\max_{\tau \in [-h,0]}r(\theta _{\tau }\omega )\) is also tempered;
-
(2)
for any \(\beta >0\) and P-a.e. \(\omega \in \Omega \), \(R(\omega )=\int _{-\infty }^{0}e^{\beta s}r(\theta _{s}\omega )\,ds< \infty \) is tempered, and \(R(\theta _{t}\omega )\) is also continuous in t.
Proposition 2.3
(See [9])
Suppose that \(B_{0}(\omega )\in \mathcal{D}(X)\) is a closed random absorbing set for the continuous \(\operatorname{SDS}\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\), and, for a.e. \(\omega \in \Omega \), each sequence \(x_{n} \in S(t_{n},\theta _{-t_{n}}\omega ) B_{0}(\theta _{-t_{n}} \omega )\) with \(t_{n}\rightarrow \infty \) has a convergent subsequence in X. Then the \(\operatorname{SDS}\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) has a unique \(\mathcal{D}\)-random attractor \(A(\omega )\), which is given by
3 Second-order stochastic delay lattice sine-Gordon equation
Denote \(l^{p}\) (\(p\geq 1\)) defined by
with the norm
In particular, \(l^{2}\) is a Hilbert space with the inner product \((\cdot ,\cdot )\) and norm \(\|\cdot \|\) given by
for any \(u=(u_{i})_{i\in \mathbb{Z}^{k}}\), \(v=(v_{i})_{i\in \mathbb{Z}^{k}} \in l^{2}\).
Define a linear operator in the following way:
Then \(B_{j}^{\ast }\) is the adjoint operator of \(B_{j}\), and
By using the above equalities, we have
For any \(u=(u_{i})_{i\in \mathbb{Z}^{k}}\), \(v=(v_{i})_{i\in \mathbb{Z}^{k}} \in l^{2}\), we define a new inner product and norm on \(l^{2}\) by
It is obvious that
Denote
Then the norms \(\|\cdot \|\) and \(\|\cdot \|_{\lambda }\) are equivalent.
Let \(H=l^{2}_{\lambda }\times l^{2}\) be endowed with the inner product and norm
for \(\psi _{j}=(u^{(j)},v^{(j)})^{T}= ((u^{(j)}_{i},v^{(j)}_{i}) )^{T}_{i \in \mathbb{Z}^{k}}\in H\), \(j=1, 2\), \(\psi =(u,v)^{T}= ((u_{i},v_{i}) )^{T}_{i\in \mathbb{Z}^{k}}\in H\). In addition, the space \(H_{0}=C([-h,0],H)\) is endowed with \(\|\psi \|_{H_{0}}=\max_{\tau \in [-h,0]}\| \psi (\tau )\|_{H}\).
In the following, we consider the probability space \((\Omega ,\mathcal{F},P)\), where
\(\mathcal{F}\) is the Borel σ-algebra induced by the compact-open topology of Ω, and P the corresponding Wiener measure on \((\Omega ,\mathcal{F})\). We will identify ω with
Define the time shift by
Then \((\Omega ,\mathcal{F},P,(\theta _{t})_{t\in \mathbb{R}})\) is a metric dynamical system with the filtration
where \(\mathcal{F}^{t}_{s}=\sigma \{w(t_{2})-w(t_{1}):s\leq t_{1}\leq t_{2} \leq t\}\) is the smallest σ-algebra generated by \(w(t_{2})-w(t_{1})\) for all \(s\leq t_{1}\leq t_{2}\leq t\).
For convenience, we rewrite Eq. (1.1) as
where \(u=(u_{i})_{i\in \mathbb{ Z}^{k}}\), \(\dot{u}=(\dot{u}_{i})_{i\in \mathbb{ Z}^{k}}\), \(\ddot{u}=(\ddot{u}_{i})_{i\in \mathbb{ Z}^{k}}\), \(u_{0}=(u_{0i})_{i\in \mathbb{ Z}^{k}}\), \(\dot{u}_{0}=(\dot{u}_{0i})_{i\in \mathbb{ Z}^{k}}\), \(Au=(Au_{i})_{i\in \mathbb{ Z}^{k}}\), \(\lambda u=(\lambda u_{i})_{i\in \mathbb{ Z}^{k}}\), \(\beta \sin {u}=\beta (\sin {u_{i}})_{i\in \mathbb{ Z}^{k}}\), \(f(u_{t})=(f_{i}(u_{it}))_{i\in \mathbb{ Z}^{k}}\), \(g=(g_{i})_{i\in \mathbb{ Z}^{k}}\) and \(\dot{w}=(\epsilon _{i}\dot{w}_{i})_{i\in \mathbb{ Z}^{k}}\). Let \(\bar{v}=\dot{u}+\delta u\), where δ is a positive constant and satisfies
then Eq. (3.2) can be rewritten as
where \(\varphi =(u,\bar{v})^{T}\), \(\varphi _{t}=(u_{t},\bar{v}_{t})^{T}\), \(F(\varphi (t))=(0,f(u_{t})+g-\beta \sin {u})^{T}\), \(G=(0,\dot{w})^{T}\) and
Also, we make the following assumptions:
- (\(H_{1}\)):
-
\(f_{i}:C([-h,0];\mathbb{R})\rightarrow \mathbb{R}\) is continuous and \(f_{i}(0)=0\);
- (\(H_{2}\)):
-
\(|f_{i}(\xi )|\leq M_{0,i}+M_{1,i}\max_{\tau \in [-h,0]}|\xi (\tau )|\) for all \(\xi \in C([-h,0];\mathbb{R})\), where \(M_{r,i}\geq 0\), \((M_{r,i})_{i\in \mathbb{Z}^{k}}\in l^{2}\), \(M_{r}^{2}:=\sum_{i\in \mathbb{Z}^{k}}M_{r,i}^{2}\) (\(r=0,1\));
- (\(H_{3}\)):
-
for any bounded set \(Y\subset l^{2}\), there exists a constant \(L_{f}>0\), such that
$$ \bigl\Vert f(u)-f(v) \bigr\Vert \leq L_{f} \Vert u-v \Vert , \quad \forall u,v\in Y. $$
Lemma 3.1
Suppose (\(H_{1}\))–(\(H_{3}\)) hold. For any \(T>0\) and an initial data \(\varphi _{0}\in H_{0}\), there exists a unique solution \(\varphi _{t}\in L^{2}(\Omega ,C([0,T]);H)\) of Eq. (3.4) with \(\varphi _{t}(\cdot ,\varphi _{0})\in H_{0}\) for \(t\in [0,T]\) and \(\varphi _{0}(\cdot ,\varphi _{0})= \varphi _{0}\). Moreover, \(\varphi _{t}(\cdot ,\varphi _{0})\) depends continuously on the initial data \(\varphi _{0}\) for each \(\omega \in \Omega \).
Proof
Rewriting (3.4) as
By (\(H_{1}\)) and (\(H_{3}\)), we know that
and
where \(d_{1}=\max \{2\delta ^{2}+\frac{54k^{2}}{\lambda }+3\lambda + \frac{6\delta ^{2}(\delta -\alpha )^{2}}{\lambda }, 6(\delta -\alpha )^{2}+2(4k+ \lambda )\}\). Thus, F and D map the bounded sets into bounded sets. In this way, by the standard theory of differential equations, we find that there exists a unique local solution. Then calculations in blow shows that this local solution is actually global. Indeed, suppose the solutions \(\varphi ^{(1)}(t)\), \(\varphi ^{(2)}(t)\) of Eq. (3.4) with the initial data \(\varphi _{0}^{(1)}\), \(\varphi _{0}^{(2)}\in H_{0}\), respectively, we have
which implies that
Applying the Gronwall inequality, we have
from which we get
The proof is complete. □
Lemma 3.2
Suppose (\(H_{1}\))–(\(H_{3}\)) hold, Eq. (3.4) generates a continuous random dynamical system \(\varphi _{t}\) over \((\Omega ,\mathcal{F},P,(\theta _{t})_{t\in R})\).
Proof
The proof is similar to that of Theorem 3.2 in [9], so here it is omitted. □
4 Existence of random attractor
This section will be devoted to prove the existence of a \(\mathcal{D}\)-random attractor for \(\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) in \(H_{0}\). Firstly, we introduce an Ornstein–Uhlenbeck process in \(l^{2}\) on the metric dynamical system \((\Omega ,\mathcal{F},P,(\theta _{t})_{t\in R})\) given by the Wiener process:
where \(\alpha >0\), the above integral solves the following Itô equation:
In fact, there exists a \(\theta _{t}\)-invariant set \(\Omega '\subset \Omega \) such that
-
(i)
the mapping \(t\rightarrow z(\theta _{t}\omega )\) is continuous for P-a.s. \(\omega \in \Omega '\);
-
(ii)
the random variable \(\|z(\theta _{t}\omega )\|\) is tempered.
Denote
where \(\varphi (t)\) is the solution of Eq. (3.4). Then \(\psi (t)\) satisfies
where \(z(\theta _{t_{-}}\omega )=z(\theta _{t+\tau }\omega )\), \(\tau \in [-h,0]\), for any \(t\geq 0\), \(\psi _{t}=\varphi _{t}-(0,z(\theta _{t_{-}}\omega ))^{T}\), \(C(\psi (t),t, \omega )=(z(\theta _{t}\omega ),f(u_{t})+g-\beta \sin {u}+\delta z( \theta _{t}\omega ))^{T}\), and
Lemma 4.1
Suppose (\(H_{1}\))–(\(H_{3}\)) hold, and \(\delta >\frac{2M_{1}e^{\delta h}}{\sqrt{\lambda }} \), Then there exists a random absorbing set \(B_{0}(\omega )\in \mathcal{D}(H_{0})\) for \(\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\).
Proof
Taking the inner product \((\cdot ,\cdot )_{H}\) on both sides of (4.1) with \(\psi (t)\), we get
Now, we estimate the terms of (4.2) one by one. We have
and
By (4.2)–(4.4) and (3.3), we have
which gives
One deduces by integrating the above inequality on [0,t] that
For fixed \(\tau \in [-h,0]\), we have, for all \(t\geq 0\),
This implies that
Let us introduce the following notations:
Thus, the inequality (4.9) can be rewritten as
Applying the Gronwall inequality to the above inequality yields
and hence
Replacing ω by \(\theta _{-t}\omega \) in (4.12), we have
Since \(\varphi _{t}(\cdot ,\omega ,\varphi _{0}(\omega ))=\psi _{t} ( \cdot ,\omega ,\varphi _{0}(\omega )-(0,z(\theta _{0_{-}}\omega ))^{T} ) +(0,z(\theta _{t_{-}}\omega ))^{T}\), it follows from (4.13) that
By assumption, \(B(\omega )\in \mathcal{D}(H_{0})\) is tempered. On the other hand, by Proposition 2.2, we know that \(\max_{\tau \in [-h,0]}\|z(\theta _{\tau } \omega )\|^{2}\), \(\max_{\tau \in [-h,0]}\|z( \theta _{-t+\tau }\omega )\|^{2}\), and \(\int _{-\infty }^{0}e^{(\delta -c_{3})s}\|z(\theta _{s}\omega )\|^{2}\,ds\) are also tempered. Thus, if \(\varphi _{0}(\theta _{-t}\omega )\in B(\theta _{-t}\omega )\), then there exists some \(T_{B}(\omega )>0\) such that, for all \(t\geq T_{B}(\omega )\),
that is, \(B_{0}(\omega )=\{\xi \in H_{0}:\|\xi \|_{H_{0}}\leq R_{0}(\omega )\}\) is a random absorbing set for \(\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\). The proof is complete. □
For convenience, we denote \(\|\psi \|_{H}^{2}=\sum_{i\in \mathbb{Z}^{k}}|\psi _{i}|_{H}^{2}\), where \(|\psi _{i}|_{H}^{2}=\sum_{j=1}^{k}(B_{j}u)_{i}^{2}+\lambda u_{i}^{2}+v_{i}^{2}\), for any \(\psi =(\psi _{i})_{i\in \mathbb{Z}^{k}}=((u_{i},v_{i}))^{T}_{i\in \mathbb{Z}^{k}}\).
Lemma 4.2
Suppose the conditions of Lemma 4.1hold, and \(\varphi _{0}(\omega )\in B_{0}(\omega )\). Then, for any \(\varepsilon >0\), there exist \(M(\varepsilon ,\omega )\in N\) and \(T(\varepsilon ,\omega )>0\) such that the solution \(\varphi (t,\omega ,\varphi _{0}(\omega ))\) of Eq. (3.4) satisfies
where \(i\in \mathbb{Z}^{k}\) and \(\|i\|=\max_{1\leq j\leq k}|i_{j}|\).
Proof
Define a smooth increasing function \(\eta (x)\in C(R_{+},[0,1])\bigcap C^{1}(R_{+},R_{+}) \) such that
and \(|\eta (x)'|\leq \eta _{0}\) (constant) for all \(x\in R_{+}\). Set \(\xi =(\xi _{i})_{i\in \mathbb{Z}^{k}}=(p,q)^{T}\) with \(\xi _{i}=(p_{i},q_{i})^{T}=(\eta (\frac{\|i\|}{M})u_{i},\eta ( \frac{\|i\|}{M})v_{i})^{T} \), where M is a fixed positive integer. Taking the inner product \((\cdot ,\cdot )_{H}\) of Eq. (4.1) with ξ, we have
Next, we estimate the terms of (4.17) one by one. Firstly,
and
Secondly,
and
Thirdly,
and
Combining (4.28)–(4.33), we have
By (4.17), (4.20), (4.27) and (4.34), we get
It follows from (3.3) that
where \(h_{1}=\frac{k\eta _{0}}{M}\max \{\frac{5+8\delta }{\lambda },4\}+ \frac{2k\eta _{0}}{\sqrt{\lambda }M}\), \(h_{2}= \frac{4k+4\lambda }{\delta }+3\delta \). One deduces by integrating the above inequality on [0,t] that
Note that \(\|\psi (\tau )\|_{H}\leq \|\psi _{0}\|_{H_{0}}\) for fixed \(\tau \in [-h,0]\), hence we get, for all \(t\geq 0\),
from which we get
where \(k_{1}=\frac{3}{\delta ^{2}}\sum_{i\in \mathbb{Z}^{k}}\eta ( \frac{\|i\|}{M})(M_{0i}^{2}+g_{i}^{2})\), \(k_{2}=h_{1}e^{\delta h}\), \(k_{3}= \frac{8k\eta _{0}}{M}e^{\delta h}\), \(k_{4}=h_{2}e^{\delta h}\). Using the Gronwall inequality yields
This implies that
Replacing ω with \(\theta _{-t}\omega \), we have
Next, we estimate the terms of (4.42). Note that \(\|\varphi _{0}(\theta _{-t}\omega )\|^{2}\), \(\max_{\tau \in [-h,0]} \|z(\theta _{-t+\tau }\omega )\|^{2}\) are tempered, we find that, for every \(\varepsilon >0\), there exists some \(T_{1}(\varepsilon ,\omega )>0\) such that, for all \(t\geq T_{1}(\varepsilon ,\omega )\),
Since \(M_{0}, g\in l^{2}\), there exists \(M_{1}(\varepsilon ,\omega )>0\) such that
In the light of \(\int _{-\infty }^{0}e^{(\delta -c_{3})s}\|z(\theta _{s}\omega )\|^{2}\,ds< \infty \) and the Lebesgue theorem of dominated convergence, there exists some \(M_{2}(\varepsilon ,\omega )>0\) such that, for all \(M\geq M_{2}(\varepsilon ,\omega )\),
Moreover, it follows from (4.13) that
which shows that there exist \(T_{2}(\varepsilon ,\omega )>0\) and \(M_{3}(\varepsilon ,\omega )>0\) such that
We set
we have
from which we get
where \(M(\varepsilon ,\omega )=\max \{M_{4}(\varepsilon ,\omega ),M_{5}( \varepsilon ,\omega )\}\), and \(M_{5}(\varepsilon ,\omega )\in N\) leads to
The proof is complete. □
Theorem 4.1
Suppose that Lemma 4.1holds, then the \(\operatorname{SDS}\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) over \((\Omega ,F,P,(\theta _{t})_{t\in R})\) defined by Eq. (3.4) has a unique \(\mathcal{D}\)-random attractor \(\mathcal{A}(\omega )\).
Proof
In the light of Proposition 2.3 and Lemma 4.1, it suffices to prove that, for a.e. \(\omega \in \Omega \), each sequence \(\varphi _{t_{n}}(\cdot ,\theta _{-t_{n}}\omega ,\varphi _{0}(\theta _{-t_{n}} \omega ))=S(t_{n},\theta _{-t_{n}}\omega )\varphi _{0} (\theta _{-t_{n}} \omega )\) has a convergent subsequence in \(H_{0}\) as \(t_{n}\rightarrow \infty \) and \(\varphi _{0}(\theta _{-t_{n}}\omega )\in B_{0}(\theta _{-t_{n}} \omega )\). By (4.14) we have
where \(C>0\) is a given constant. For a fixed \(\tau \in [-h,0]\), we can find a subsequence \(\{\varphi ({t_{n}}+\tau ,\theta _{-t_{n}}\omega ,\varphi _{0}( \theta _{-t_{n}}\omega ))\}\) and \(\mu (\tau )\in H\) such that
Next, we prove the above convergence is also strong. We know, for any \(\varepsilon >0\), there exist \(N(\varepsilon ,\omega )\) and \(\hat{M}(\varepsilon ,\omega )\) such that
and for \(n\geq N(\varepsilon ,\omega )\) we have
This shows that, for any \(\tau \in [-h,0]\), \(\varphi ({t_{n}}+\tau ,\theta _{-t_{n}}\omega ,\varphi _{0} (\theta _{-t_{n}} \omega ))\rightarrow \mu (\tau )\) is strong in H as \(n\rightarrow \infty \). In addition, making use of the integral representation of solutions, we obtain for any \(\tau _{1},\tau _{2}\in [-h,0]\)
It is obvious that D is a linear operator from H into itself and the operator C is bounded. Hence we have
and
Hence
which is the required equicontinuity. In view of the Ascoli–Arzelá theorem, we conclude that there exists a subsequence \(\{\varphi _{t_{n_{k}}}(\cdot ,\theta _{-t_{n_{k}}}\omega ,\varphi _{0}( \theta _{-t_{n_{k}}}\omega ))\}\) of \(\{\varphi _{t_{n}}(\cdot ,\theta _{-t_{n}}\omega ,\varphi _{0}( \theta _{-t_{n}}\omega ))\}\) such that
The proof is complete. □
5 An upper bound of the Kolmogorov ε-entropy
In this section, we study the upper bound of the Kolmogorov ε-entropy of the global random \(\mathcal{D}\)-attractor \(\mathcal{A}(\omega )\) given by Theorem 4.1.
Theorem 5.1
Under the same conditions of Theorem 4.1, for a.e. \(\omega \in \Omega \),
where \(\hat{M}(\varepsilon ,\omega )\doteq \hat{M}( \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon , \omega ,B_{0})\) is the minimal positive integer such that
Proof
By Lemma 4.1, we obtain \(\mathcal{A}(\omega )=\varphi _{t}(t,\theta _{-t}\omega )\mathcal{A}( \theta _{-t}\omega )\subset B_{0}(\omega )\) for \(t>T(\omega ,\mathcal{A})\). Thus, for any \(\varepsilon >0\) and \(\varphi _{t}=(\varphi _{it})_{i\in \mathbb{Z}^{k}}= ((u_{it},v_{it})_{i \in \mathbb{Z}^{k}} )^{T} =\varphi _{t}(t,\theta _{-t}\omega ) \varphi _{0}(\theta _{-t}\omega )\in \mathcal{A}\), where \(\varphi _{0}(\omega )\in \mathcal{A}(\omega )\subset B_{0}(\omega )\), and by Lemma 4.2 there exists some \(\hat{M}(\varepsilon ,\omega )\doteq \hat{M}( \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon , \omega , B_{0})\in \mathbb{N}\) such that (5.2) holds. Next, we decompose φ into two parts as
where
and
this implies that
Consider the regular polyhedron
by Lemma 2.1 we see that \(\Upsilon _{1}\) can be covered by
balls in \(\mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\) with radii \(\frac{\varepsilon }{4k+\lambda } \). Next, we study the other regular polyhedron
it can be covered by
balls in \(\mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\) with radii \(\frac{\varepsilon }{4k+\lambda } \). Therefore, the polyhedron
can be covered by
balls in \(\mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\times \mathbb{R}^{2 \hat{M}(\varepsilon ,\omega )+1}\) with radii \(\frac{\sqrt{2}\varepsilon }{4k+\lambda }\). Let the centers of those balls be
where \(\iota =1,2,\ldots ,\mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\). We choose
Then there exists some ι (\(1\leq \iota \leq \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\)) such that
Thus, for any \(\varphi \in \mathcal{A}(\omega )\subset \mathcal{B}_{0}(\omega )\), there exists some ι (\(1\leq \iota \leq \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\)) such that
which means that the global random \(\mathcal{D}\)-attractor \(\mathcal{A}(\omega )\subset H\) can be covered by \(\mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\) balls centered at \(\hat{\varphi }_{\iota t}\), \(\iota =1,2,\ldots , \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\), with radii ε. The proof is complete. □
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References
Carrol, T.L., Pecora, L.M.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
Chua, L.O., Roska, T.: The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156 (1993)
Winalow, R.L., Kimball, A.L., Varghese, A.: Simulating cartidiac sinus and atrial network dynamics on connection machine. Physica D 64, 281–298 (1993)
Kapral, R.: Discrete models for chemically reacting systems. J. Math. Chem. 6, 113–163 (1991)
Malletparet, J., Chow, S.: Pattern formation and spatial chaos in lattice dynamical systems. I. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 42(10), 746–751 (2002)
Cheng, C., Feng, Z., Su, Y.: Global stability of traveling wave fronts for a reaction–diffusion system with a quiescent stage on a one-dimensional spatial lattice. Appl. Anal. 97, 2920–2940 (2018)
Guo, J., Wu, C.: The existence of traveling wave solutions for a bistable three-component lattice dynamical system. J. Differ. Equ. 260, 1445–1455 (2016)
Wu, C.: A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems. J. Dyn. Differ. Equ. 28, 317–338 (2016)
Bates, P.W., Lisei, H., Lu, K.: Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 6, 1–21 (2006)
Xiang, X., Zhou, S.: Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term. J. Differ. Equ. Appl. 22, 235–252 (2016)
Wang, R., Wang, B.: Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise. Stoch. Process. Appl. 130, 7431–7462 (2020)
Zhao, M., Zhou, S., Sheng, F.: Random attractor for nonautonomous stochastic Boussinesq lattice equations with additive white noises. Acta Math. Sci. Ser. A 38, 924–940 (2018)
Zhou, S.: Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. J. Differ. Equ. 263, 2247–2279 (2017)
Han, Z., Zhou, S.: Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh–Nagumo lattice systems with quasi-periodic forces and multiplicative noise. Stoch. Dyn. 20, 2050036 (2020)
Su, H., Zhou, S., Wu, L.: Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise. Stoch. Dyn. 19, 1950044 (2019)
Su, H., Zhou, S., Wu, L.: Random exponential attractor for second order non-autonomous stochastic lattice dynamical systems with multiplicative white noise in weighted spaces. Adv. Differ. Equ. 2019, 45 (2019)
Zhang, Y., Lin, Z.: Long term behavior for a class of stochastic delay lattice systems in \(X_{\rho}\) space. Discrete Dyn. Nat. Soc. 2020, Article ID 5405947 (2020)
She, L., Wang, R.: Regularity, forward-compactness and measurability of attractors for non-autonomous stochastic lattice systems. J. Math. Anal. Appl. 479, 2007–2031 (2019)
Yan, W., Ji, S., Li, Y.: Random attractors for stochastic discrete Klein–Gordon–Schrödinger equations. Phys. Lett. A 373(14), 1268–1275 (2009)
Li, D., Wang, B., Wang, X.: Limiting behavior of non-autonomous stochastic reaction–diffusion equations on thin domains. J. Differ. Equ. 262, 1575–1602 (2017)
Yan, W., Li, Y., Ji, S.: Random attractors for first order stochastic retarded lattice dynamical systems. J. Math. Phys. 51, 032702 (2010)
Xu, L., Yan, W.: Stochastic FitzHugh–Nagumo systems with delay. Taiwan. J. Math. 16, 1079–1103 (2012)
Yan, Y., Li, X.: Long time behavior for the stochastic parabolic-wave systems with delay on infinite lattice. Nonlinear Anal. 197, 111866 (2020)
Zhang, C., Zhao, L.: The attractors for 2nd-order stochastic delay lattice systems. Discrete Contin. Dyn. Syst. 37, 575–590 (2016)
Han, X.: Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise. J. Math. Anal. Appl. 376, 481–493 (2011)
Wang, X., Li, S., Xu, D.: Random attractors for second-order stochastic lattice dynamical systems. Nonlinear Anal. 72, 483–494 (2010)
Zhao, C., Zhou, S.: Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications. J. Math. Anal. Appl. 354, 78–95 (2009)
Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)
Ding, X., Jiang, J.: Random attractors for stochastic retarded lattice dynamical systems. Abstr. Appl. Anal. 2012, Article ID 409282 (2012)
Lorentz, G., Golitschek, M.V., Makovoz, Y.: Constructive Approximation: Advanced Problems. Springer, Berlin (1996)
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The authors would like to thank anonymous referees and editors for their valuable comments and constructive suggestions.
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This work is supported by the Science and Technology Foundation of Guizhou Province ([2020]1Y007), the Natural Science Foundation of Education of Guizhou Province (KY[2019]139, KY[2019]143)) and School level Foundation of Liupanshui Normal University (LPSSYKJTD201907).
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Li, X., She, L. & Shan, Z. Random attractor for second-order stochastic delay lattice sine-Gordon equation. Bound Value Probl 2021, 14 (2021). https://doi.org/10.1186/s13661-021-01489-7
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DOI: https://doi.org/10.1186/s13661-021-01489-7