Research Article Synchronization of Coupled Stochastic Systems Driven by α-stable Lévy Noises

Volume 2013, Article ID 685798,10pages http://dx.doi.org/10.1155/2013/685798

Research Article

Synchronization of Coupled Stochastic Systems Driven by

𝛼

-Stable Lévy Noises

Anhui Gu

College of Science, Guilin University of Technology, Guilin,Guangxi 541004, China

Correspondence should be addressed to Anhui Gu; [email protected]

Received 9 October 2012; Revised 25 December 2012; Accepted 28 December 2012 Academic Editor: Weihai Zhang

Copyright © 2013 Anhui Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The synchronization of the solutions to coupled stochastic systems ofN-Marcus stochastic ordinary differential equations which are driven by𝛼-stable L´evy noises is investigated(𝑁 ∈N, 1 < 𝛼 < 2). We obtain the synchronization between two solutions and among different components of solutions under certain dissipative conditions. The synchronous phenomena persist no matter how large the intensity of the environment noises. These results generalize the work of two Marcus canonical equations in X. M. Liu et al.’ s (2010).

1. Introduction

The synchronization of coupled systems is a ubiquitous phenomenon in the biological and physical science and is also known to occur in abundant of social science contexts; see for example [1–6] and references therein. In the recent book of Strogatz [4], a number of its diversity of occurrence and an extensive list of references can be found. Let 𝑢(𝑡), 𝑣(𝑡) ∈ R𝑑 be two functions defined in

[𝑡0, ∞) (𝑡0 ∈ R)and𝑢(𝑡), 𝑣(𝑡)are said to be synchronized

if lim_{𝑡 → ∞}‖𝑢(𝑡) − 𝑣(𝑡)‖ = 0. Synchronization of deterministic coupled systems has been investigated both for autonomous systems and nonautonomous systems (see, e.g., [7–10]). For coupled systems of Ito stochastic differential equations witḧ various Gaussian noises (in the terms of Brownian motion), the synchronization of solutions has been considered in the papers Caraballo and Kloeden [11], Caraballo et al. [12], Caraballo et al. [13] and Chueshov and Schmalfuß [14]. In [15], Shen et al. showed the synchronization of solutions for more general systems with multiplicative noise. Recently, Liu et al. [16, 17] studied the synchronization phenomenon for coupled systems driven by non-Gaussian noises (in terms of L´evy motion) and the analogous results also hold for the general systems with additive L´evy noises [18].

A L´evy motion 𝐿_𝑡 is a non-Gaussian process with independent and stationary increments; that is, increments

Δ𝐿_𝑡= 𝐿_𝑡+Δ𝑡− 𝐿_𝑡are stationary and independent for any non overlapping time lagsΔ𝑡. Moreover, its sample paths are only continuous in probability, namely,P(|𝐿_𝑡− 𝐿_𝑡0| ≥ 𝜖) → 0as

𝑡 → 𝑡₀for any positive𝜖. With a suitable modification, these paths may be taken as c`adl`ag; that is, paths are continuous on the right and have limits on the left (see, e.g., [19,20]). As a special case of L´evy processes, the symmetric𝛼-stable L´evy motion plays an important role among stable processes just like Brownian motion among Gaussian processes. A stochastic process {𝐿_𝑡, 𝑡 ≥ 0} is called the 𝛼-stable L´evy motion if (1)𝐿₀ = 0a.e., (2)𝐿has independent increments, and (3)𝐿_𝑡−𝐿_𝑠∼S_𝛼((𝑡 − 𝑠)1/𝛼, 𝛽, 0)for0 ≤ 𝑠 < 𝑡 < ∞and for some0 < 𝛼 ≤ 2, −1 ≤ 𝛽 ≤ 1, where S_𝛼(𝜎, 𝛽, 𝜈)denotes the

𝛼-stable distribution with index of stability𝛼, scale parameter

𝜎, skewness parameter𝛽, and shift parameter𝜈; in particular,

S₂(𝜎, 0, 𝜇) = 𝑁(𝜇, 2𝜎2)denotes the Gaussian distribution. For more details on𝛼-stable distributions, we can refer to [21,22].

Let(Ω,F,P)be a probability space, whereΩ = 𝐷(R,R𝑑) of c`adl`ag functions with the Skorohod metric (see [23]) as the canonical sample space and denote byF := B(𝐷(R,R𝑑)) the Borel𝜎-algebra on Ω. Let𝜇_𝐿be the (L´evy) probability measure onFwhich is given by the distribution of a two-sided L´evy process with paths inΩ, that is,𝜔(𝑡) = 𝐿_𝑡(𝜔). Define𝜃 = (𝜃_𝑡, 𝑡 ∈ R)onΩthe shift by(𝜃_𝑡𝜔)(𝑠) := 𝜔(𝑡 +

𝑠) − 𝜔(𝑡). Then, the mapping(𝑡, 𝜔) → 𝜃_𝑡𝜔is continuous and measurable [24], and the (L´evy) probability measure is

𝜃-invariant, that is,𝜇_𝐿(𝜃−1_𝑡 (𝐴)) = 𝜇_𝐿(𝐴), for all𝐴 ∈ F; see [19] for more details.

Consider the following Marcus stochastic ordinary differ-ential equations (MSODEs) system driven by𝛼-stable L´evy noises inR𝑑:

𝑑𝑋(𝑗)_𝑡 = 𝑓(𝑗)(𝑋(𝑗)_𝑡 )𝑑𝑡 +∑𝑚

𝑖=1𝑐

(𝑗)

𝑖 𝑋(𝑗)𝑡 ⬦ 𝑑𝐿(𝑖)𝑡 , 𝑗 = 1, . . . , 𝑁,

(1) where𝑐_𝑖(𝑗) ∈R,𝐿(𝑖)_𝑡 are independent𝛼-stable L´evy noises on

(Ω,F,P),1 < 𝛼 < 2,⬦denotes the Marcus integral (see, e.g., [25]), and𝑓(𝑗), 𝑗 = 1, . . . , 𝑁are regular enough to ensure the existence and uniqueness of solutions and satisfy the one-sided dissipative Lipschitz condition

⟨𝑥₁− 𝑥₂, 𝑓(𝑗)(𝑥₁) − 𝑓(𝑗)(𝑥_{2)⟩ ≤ −𝐿󵄩󵄩󵄩󵄩𝑥1}− 𝑥₂󵄩󵄩󵄩󵄩2, 𝑗 = 1, . . . , 𝑁, (2)

onR𝑑for some𝐿 > 0. Set

𝑥(𝑗)(𝑡, 𝜔) = 𝑒−𝑂(𝑗)𝑡 𝑋(𝑗)

𝑡 (𝜔) , 𝑡 ∈R, 𝜔 ∈ Ω, 𝑗 = 1, . . . , 𝑁,

(3) where

𝑂(𝑗)_𝑡 := 𝑂(𝑗)_𝑡 (𝜔) =∑𝑚

𝑖=1𝑐

(𝑗)

𝑖 𝑒−𝑡∫

𝑡

−∞𝑒

𝑠_𝑑𝐿(𝑖)

𝑠 , 𝑗 = 1, . . . , 𝑁,

(4) are the stationary solutions of the Ornstein-Uhlenbeck stochastic differential equations

𝑑𝑂_𝑡(𝑗)= −𝑂(𝑗)_𝑡 𝑑𝑡 +∑𝑚

𝑖=1𝑐

(𝑗)

𝑖 ⬦ 𝑑𝐿(𝑖)𝑡 , 𝑗 = 1, . . . , 𝑁. (5)

Then system (1) can be translated into the following random ordinary differential equations (RODEs):

𝑑𝑥(𝑗)

𝑑𝑡₊ = 𝐹(𝑗)(𝑥(𝑗), 𝑂𝑡(𝑗))

:= 𝑒−𝑂(𝑗)𝑡 𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗)) + 𝑂(𝑗)

𝑡 𝑥(𝑗), 𝑗 = 1, . . . , 𝑁,

(6)

where𝑑𝑥(𝑗)/𝑑𝑡₊is right-hand derivative of𝑥(𝑗)at𝑡.

Now we consider the linear coupled RODEs of (6), as follows:

𝑑𝑥(𝑗)

𝑑𝑡₊ = 𝐹(𝑗)(𝑥(𝑗), 𝑂(𝑗)𝑡 ) + 𝜈(𝑥(𝑗−1)− 2𝑥(𝑗)+ 𝑥(𝑗+1)) ,

𝑗 = 1, . . . , 𝑁,

(7)

with the coupled coefficient𝜈 > 0, where𝑥(0) = 𝑥(𝑁) and

𝑥(𝑁+1) _{= 𝑥}(1)_{. Hence, (7) can be written as the following}

equivalent MSODEs:

𝑑𝑋_𝑡(𝑗)= 𝑓(𝑗)(𝑋(𝑗)_𝑡 )𝑑𝑡 + 𝜈(𝑒−Δ ̌𝑂(𝑗)𝑡 𝑋(𝑗−1)

𝑡 − 2𝑋(𝑗)𝑡 + 𝑒−Δ̂𝑂

(𝑗)

𝑡 𝑋(𝑗+1)

𝑡 ) 𝑑𝑡

+∑𝑚

𝑖=1

𝑐_𝑖(𝑗)𝑋(𝑗)_𝑡 ⬦ 𝑑𝐿(𝑖)_𝑡 , 𝑗 = 1, . . . , 𝑁,

(8)

whereΔ ̌𝑂(𝑗)_𝑡 = 𝑂(𝑗)_𝑡 − 𝑂_𝑡(𝑗−1), Δ̂𝑂(𝑗)_𝑡 = 𝑂(𝑗)_𝑡 − 𝑂_𝑡(𝑗+1), 𝑂_𝑡(0)=

𝑂(𝑁)

𝑡 , and𝑂(𝑁+1)𝑡 = 𝑂𝑡(1).

For synchronization of solutions (in the sense of Carath´eodory [26]) to RODEs system (7), there are two cases: one for any two solutions and the other for components of solutions. When𝑁 = 2, Liu et al. [17] consider both types of synchronization. Under the one-sided dissipative Lipschitz condition (2), they firstly proved that synchronization of any two solutions occurs and the random dynamical system generated by the solution of (7)𝑁 = 2has a singleton sets random attractor, then they proved that the synchronization between any two components of solutions occurs as the cou-pled coefficient𝜈tends to infinity. The synchronization result implies that coupled dynamical systems share a dynamical feature in an asymptotic sense. Based on the work of [15,17], we consider the synchronization of solutions of (7) in the case of 𝑁 ≥ 3 and obtain the similar results. We show that the random dynamical system (RDS) generated by the solution of the coupled RODEs system (7) has a singleton sets random attractor which implies the synchronization of any two solutions of (7). Moreover, the singleton set random attractor determines a stationary stochastic solution of the equivalently coupled RODEs system (8). We also show that any two solutions of RODEs system (7) converge to a solution

𝑍(𝑡, 𝜔)of the averaged RODEs as follows:

𝑑Z

𝑑𝑡₊ = 1 𝑁

𝑁

∑

𝑗=1

𝑒−𝑂(𝑗)𝑡 𝑓(𝑗)(𝑒𝑂𝑡(𝑗)𝑍) + 1

𝑁

𝑁

∑

𝑗=1

𝑂(𝑗)_𝑡 𝑍, (9) as the coupling coefficient𝜈 → ∞.

When𝛼 = 2, we have the standard Brownian motion, which the Marcus integral reduces to the Stratonovich stochastic integral, and both types of the synchronization of system (8) have been considered in [15]. It is worth mentioning that the generalization is not trivial because new techniques similar to [15] are needed. We restrict here that

𝛼 ∈ (1, 2), only in this case, the solutions of the Ornstein-Uhlenbech equations based on 𝛼-stable L´evy noises are stationary, which is crucial to our purpose. When𝛼 ∈ (0, 1), dealing with such values of the parameter seems to be a new challenging for us.

The paper is organized as follows. InSection 2, we recall some basic facts on random dynamical systems, and then we give two lemmas which will be frequently used. InSection 3, we show the synchronization of two solutions to the coupled RODEs (7) and obtain the staionary stochastic solution to the equivalent MSODEs (8). In Section 4, we give the

synchronization of components of solutions to the coupled RODEs (7), which implies that the equivalent MSODEs (8) share the similarly synchronous phenomenon when driven by the same𝛼-stable L´evy noises.

2. Random Dynamical Systems and

Auxiliary Lemmas

We will frequently use the following results.

Lemma 1. There exists a{𝜃_𝑡}_𝑡∈R-invariant subsetΩ ∈ Fof

full measure for a.e.𝜔 ∈ Ω, and the sample paths𝜔(𝑡)of𝐿_𝑡

satisfy

lim

𝑡 → ±∞

𝜔(𝑡)

𝑡 = 0, 𝑡 ∈R. (10)

In addition, for 𝑗 = 1, . . . , 𝑁, there exist random variables

𝑂(𝑗)= 𝑂(𝑗)_𝑡 and𝑇_𝜔> 0such that

𝑂(𝑗)(𝜃_𝑡𝜔) = 𝑂(𝑗)_𝑡 (𝜔) , lim

𝑡 → ±∞

1 𝑡 ∫

𝑡

0𝑂 (𝜃𝑠𝜔)𝑑𝑠 = 0, 𝜔 ∈ Ω,

(11)

𝑒2 ∫𝜏𝑡𝑂(𝑗)𝑠 𝑑𝑠≤ 𝑒(𝐿/2)(𝑡−𝜏) _for − 𝜏, 𝑡 > 𝑇

𝜔. (12)

Proof. The equalities (10) and (11) can be found in [17, Lemma

2]. By (11), we have lim_{𝑡 → ∞}(1/𝑡)∫₀𝑡𝑂(𝑗)_𝑠 𝑑𝑠 = 0, then there exists𝑇_𝜔(1) > 0such that∫₀𝑡𝑂_𝑠(𝑗)𝑑𝑠 ≤ (𝐿/4)𝑡for𝑡 > 𝑇_𝜔(1). Similarly, lim_{𝜏 → −∞}(1/𝜏)∫_𝜏0𝑂_𝑠(𝑗)𝑑𝑠 = 0, which implies that there exists𝑇_𝜔(2) > 0 such that∫_𝜏0𝑂(𝑗)_𝑠 𝑑𝑠 ≤ −(𝐿/4)𝜏for

𝜏 < −𝑇_𝜔(2). Denoting 𝑇_𝜔 = max{𝑇_𝜔(1), 𝑇_𝜔(2)}, we have

2 ∫_𝜏𝑡𝑂_𝑠(𝑗)𝑑𝑠 ≤ (𝐿/2)(𝑡 − 𝜏)for−𝜏, 𝑡 > 𝑇_𝜔, which completes the proof.

Lemma 2 (Gronwall type inequality). Suppose that𝐷(𝑡)is an

𝑛 × 𝑛matrix andΦ(𝑡)andΨ(𝑡)are𝑛-dimensional vectors on

[𝑇₀, 𝑇] (𝑇 ≥ 𝑇₀, 𝑇, 𝑇₀ ∈R)which are sufficiently regular. If the following inequality holds in the componentwise sense:

𝑑

𝑑𝑡+Φ(𝑡) ≤ 𝐷(𝑡)Φ(𝑡) + Ψ(𝑡) , 𝑡 ≥ 𝑇0, (13)

where(𝑑/𝑑𝑡₊)Φ(𝑡) := lim_ℎ↓0+((Φ(𝑡 + ℎ) − Φ(𝑡))/ℎ)is

right-hand derivative ofΦ(𝑡), then

Φ(𝑡) ≤exp(∫

𝑡

𝑇0𝐷(𝑠)𝑑𝑠)Φ(𝑇0)

+ ∫𝑡

𝑇0

exp(∫𝑡

𝜏𝐷(𝑠)𝑑𝑠)Ψ(𝜏)𝑑𝜏, 𝑡 ≥ 𝑇0.

(14)

Proof. See Lemma 2.8 in [27] and the proof of lemma 2.2 in

[15].

Proposition 3 (Random attractor for c`adl`ag RDS (see [16])).

Let(𝜃, 𝜑)be an RDS onΩ×R𝑑and let𝜑be continuous in space

but c`al`ag in time. If there exists a familyB= {B(𝜔), 𝜔 ∈ Ω}

of nonempty measurable compact subsetsB(𝜔)ofR𝑑and a

𝑇_𝐵,𝜔≥ 0such that

𝜑(𝑡, 𝜃−𝑡𝜔, 𝐵(𝜃−𝑡𝜔)) ⊂B(𝜔) , ∀𝑡 ≥ 𝑇𝐵,𝜔, (15)

for all families 𝐵 = {𝐵(𝜔), 𝜔 ∈ Ω} in a given attracting

universe. then, the RDS(𝜃, 𝜑)has a random attractor A =

{A(𝜔), 𝜔 ∈ Ω}with the component subsets defined for each

𝜔 ∈ Ωby

A(𝜔) = ⋂

𝑠>0

⋃

𝑡≥𝑠

𝜑(𝑡, 𝜃_−𝑡𝜔, 𝐵(𝜃_−𝑡𝜔)). ₍₁₆₎

Furthermore, if the random attractor consist of singleton sets,

that is,A(𝜔) = {𝑋∗(𝜔)}for some random variable𝑋∗, then

𝑋∗

𝑡(𝜔) = 𝑋∗𝑡(𝜃𝑡𝜔)is a stationary stochastic process.

3. Synchronization of Two Solutions

Consider the coupled system (7) with the following initial data:

𝑥(𝑗)(0, 𝜔) = 𝑥(𝑗)₀ (𝜔) ∈R𝑑, 𝜔 ∈ Ω, 𝑗 = 1, . . . , 𝑁. (17) For asymptotic behavior of the difference between two solutions of RODEs system (7) with initial data (17) (omitting to RODEs system (7) for brevity), we get the following:

Lemma 4. For any two solutions (𝑥(1)₁ (𝑡), 𝑥(2)₁ (𝑡)

, . . . , 𝑥(𝑁)₁ (𝑡))T and (𝑥(1)₂ (𝑡), 𝑥(2)₂ (𝑡), . . . , 𝑥(𝑁)₂ (𝑡))T of RODEs

system(7),

lim

𝑡 → ∞󵄩󵄩󵄩󵄩󵄩󵄩𝑥

(𝑗)

1 (𝑡) − 𝑥(𝑗)2 (𝑡)󵄩󵄩󵄩󵄩󵄩󵄩 = 0, 𝑗 = 1, . . . , 𝑁, (18)

that is, all solutions of the coupled RODEs system(7)converge

pathwise to each other as time𝑡tends to infinity.

Proof. By the dissipative Lipschitz condition (2), for 𝑗 =

1, . . . , 𝑁, we have

𝑑

𝑑𝑡₊󵄩󵄩󵄩󵄩󵄩󵄩𝑥1(𝑗)(𝑡) − 𝑥(𝑗)2 (𝑡)󵄩󵄩󵄩󵄩󵄩󵄩 2

= 2 ⟨𝑥(𝑗)₁ (𝑡) − 𝑥(𝑗)₂ (𝑡) , 𝑑 𝑑𝑡𝑥

(𝑗)

1 (𝑡) −_𝑑𝑡𝑑𝑥(𝑗)2 (𝑡)⟩

= 2𝑒−𝑂(𝑗)𝑡 ⟨𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗)

1 ) − 𝑓(𝑗)(𝑒𝑂 (𝑗)

𝑡 𝑥(𝑗)

2 ) , 𝑥(𝑗)1 (𝑡)

− 𝑥(𝑗)₂ (𝑡)⟩ × (2𝑒𝑂𝑡(𝑗)− 4𝜈)󵄩󵄩󵄩󵄩

󵄩󵄩𝑥(𝑗)1 (𝑡) − 𝑥(𝑗)2 (𝑡)󵄩󵄩󵄩󵄩󵄩󵄩 2

+ 2𝜈 ⟨𝑥(𝑗−1)₁ (𝑡) − 𝑥(𝑗−1)₂ (𝑡) , 𝑥(𝑗)₁ (𝑡) − 𝑥(𝑗)₂ (𝑡)⟩ + 2𝜈 ⟨𝑥(𝑗+1)₁ (𝑡) − 𝑥(𝑗+1)₂ (𝑡) , 𝑥(𝑗)₁ (𝑡) − 𝑥(𝑗)₂ (𝑡)⟩ ≤ (2𝑒𝑂(𝑗)𝑡 − 2𝐿 − 2𝜈)󵄩󵄩󵄩󵄩

󵄩󵄩𝑥(𝑗)1 (𝑡) − 𝑥(𝑗)2 (𝑡)󵄩󵄩󵄩󵄩󵄩󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩_󵄩󵄩𝑥(𝑗−1)₁ (𝑡) − 𝑥(𝑗−1)₂ (𝑡)󵄩󵄩󵄩󵄩_󵄩󵄩2+ 𝜈󵄩󵄩󵄩󵄩_󵄩󵄩𝑥(𝑗+1)₁ (𝑡) − 𝑥(𝑗+1)₂ (𝑡)󵄩󵄩󵄩󵄩_󵄩󵄩2.

Define for𝑡 ∈R,

x(𝑡) = (󵄩󵄩󵄩󵄩_󵄩𝑥(1)₁ (𝑡) − 𝑥(1)₂ (𝑡)󵄩󵄩󵄩󵄩_󵄩2_{, 󵄩󵄩󵄩󵄩󵄩𝑥}(2)₁ (𝑡) − 𝑥(2)₂ (𝑡)󵄩󵄩󵄩󵄩_󵄩2, . . . , 󵄩󵄩󵄩󵄩

󵄩𝑥(𝑁)1 (𝑡) − 𝑥(𝑁)2 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

)T,

𝐷_𝜈(𝑡) =(((₍ (

𝜆(1)

𝜈 (𝑡) 𝜈 0 ⋅ ⋅ ⋅ 𝜈

𝜈 𝜆(2)

𝜈 (𝑡) 𝜈 0 ⋅ ⋅ ⋅

0 𝜈 𝜆(3)_𝜈 (𝑡) . .. . ..

..

. . .. . .. ... 𝜈

𝜈 ⋅ ⋅ ⋅ 0 𝜈 𝜆(𝑁)

𝜈 (𝑡)

) ) ) )

)𝑁×𝑁

,

(20) where𝜆(𝑗)_𝜈 (𝑡) = 2𝑒𝑂(𝑗)𝑡 − 2𝐿 − 2𝜈, 𝑗 = 1, . . . , 𝑁_{. Thus, the}

differential inequalities can be written as a simple form

̇

x(𝑡) ≤ 𝐷_𝜈(𝑡)x(𝑡) , −componentwise. (21) ByLemma 2, it yields from (21) that

x(𝑡) ≤ exp(∫

𝑡

0𝐷𝜈(𝑡)𝑑𝑠)x(0) , −componentwise. (22)

By [15, Lemma 3.2], we know that for 𝑡 ≥ 𝑇_𝜔 defined in

Lemma 1, and𝜈 > 0,

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩exp(∫

𝑡

0𝐷𝜈(𝑡)𝑑𝑠)x(0)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤ 𝑒

−𝐿𝑡_{‖x(0)‖ , (23)}

which leads to lim

𝑡 → ∞󵄩󵄩󵄩󵄩󵄩󵄩𝑥

(𝑗)

1 (𝑡) − 𝑥(𝑗)2 (𝑡)󵄩󵄩󵄩󵄩󵄩󵄩 = 0, 𝑗 = 1, . . . , 𝑁, (24)

and completes the proof.

Now, we use the theory of random dynamical systems which are generated by stochastic differential equations driven by𝛼-stable L´evy noise to find what the solutions of (7) will converge to. Obviously by condition (2) and [16, Lemma 4], we know that the solution

𝜑(𝑡, 𝜔) = (𝑥(1)(𝑡, 𝜔), 𝑥(2)(𝑡, 𝜔) , . . . , 𝑥(𝑁)(𝑡, 𝜔))T, 𝜔 ∈ Ω,

(25) of system (7) generates a c`adl`ag RDS over(Ω,F,P, (𝜃_𝑡)_𝑡∈R) with state spaceΩ ×R𝑁𝑑.

Then, we have the result for this RDS𝜑.

Theorem 5. Under the dissipative condition of (2), the RDS

𝜑(𝑡, 𝜔), 𝑡 ∈ R, 𝜔 ∈ Ω, has a singleton sets random attractor given by

A_𝜈(𝜔) = (𝑥(1)_𝜈 (𝜔), 𝑥(2)_𝜈 (𝜔), . . . , 𝑥(𝑁)_𝜈 (𝜔))T, (26)

which implies the synchronization of any two solutions of

system(7). Furthermore,

(𝑥(1)_𝜈 (𝜃_𝑡𝜔) 𝑒𝑂(1)𝑡 (𝜔), 𝑥(2)

𝜈 (𝜃𝑡𝜔) 𝑒𝑂

(2) 𝑡 (𝜔), . . . ,

𝑥(𝑁)_𝜈 (𝜃_𝑡𝜔)𝑒𝑂(𝑁)𝑡 (𝜔))T

(27)

is the stationary stochastic solution of the equivalent coupled

MSODEs(8).

Proof. For𝑗 = 1, . . . , 𝑁, we have

𝑑

𝑑𝑡󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)(𝑡)󵄩󵄩󵄩󵄩󵄩

2

= 2 ⟨𝑥(𝑗)(𝑡) ,_𝑑𝑡𝑑𝑥(𝑗)(𝑡)⟩

= 2 ⟨𝑒−𝑂(𝑗)𝑡 𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗)(𝑡)), 𝑥(𝑗)(𝑡)⟩

+ 2 ⟨𝑒𝑂(𝑗)𝑡 𝑥(𝑗)(𝑡), 𝑥(𝑗)(𝑡)⟩ − 4𝜈󵄩󵄩󵄩󵄩

󵄩𝑥(𝑗)(𝑡)󵄩󵄩󵄩󵄩󵄩2 + 2𝜈 ⟨𝑥(𝑗)(𝑡), 𝑥(𝑗−1)(𝑡)⟩ + 2𝜈 ⟨𝑥(𝑗)(𝑡) , 𝑥(𝑗+1)(𝑡)⟩ ≤ (2𝑒𝑂𝑡(𝑗)− 2𝐿 − 2𝜈) 󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)(𝑡)󵄩󵄩󵄩󵄩

󵄩2+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗−1)(𝑡)󵄩󵄩󵄩󵄩󵄩2 + 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗+1)(𝑡)󵄩󵄩󵄩󵄩_󵄩2_{+ 2 󵄩󵄩󵄩󵄩󵄩𝑥}(𝑗)(𝑡)_󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩𝑓(𝑗)(0)󵄩󵄩󵄩󵄩_{󵄩 𝑒}−𝑂(𝑗)𝑡

≤ (2𝑒𝑂𝑡(𝑗)− 𝐿 − 2𝜈) 󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)_(𝑡)󵄩󵄩󵄩󵄩

󵄩2+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗−1)(𝑡)󵄩󵄩󵄩󵄩󵄩2 + 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗+1)(𝑡)󵄩󵄩󵄩󵄩_󵄩2+𝑒−2𝑂

(𝑗) 𝑡

𝐿 󵄩󵄩󵄩󵄩󵄩𝑓(𝑗)(0)󵄩󵄩󵄩󵄩󵄩

2

.

(28) Analogous to (21), we get

̇

y(𝑡) ≤ ̃𝐷_𝜈(𝑡)y(𝑡) +g(𝑡) , (29) where𝑡 ∈R,

y(𝑡) = (󵄩󵄩󵄩󵄩_󵄩𝑥(1)(𝑡)󵄩󵄩󵄩󵄩_󵄩2_{, 󵄩󵄩󵄩󵄩󵄩𝑥}(2)(𝑡)󵄩󵄩󵄩󵄩_󵄩2_{, . . . , 󵄩󵄩󵄩󵄩󵄩𝑥}(𝑁)(𝑡)󵄩󵄩󵄩󵄩_󵄩2)T,

g(𝑡) = (𝑒−2𝑂

(1) 𝑡

𝐿 󵄩󵄩󵄩󵄩󵄩𝑓(1)(0)󵄩󵄩󵄩󵄩󵄩

2

,𝑒−2𝑂

(2) 𝑡

𝐿 󵄩󵄩󵄩󵄩󵄩𝑓(2)(0)󵄩󵄩󵄩󵄩󵄩

2

, . . . , 𝑒−2𝑂(𝑁)

𝑡

𝐿 󵄩󵄩󵄩󵄩󵄩𝑓(𝑁)(0)󵄩󵄩󵄩󵄩󵄩

2

) T

,

̃

𝐷_𝜈(𝑡) =(((₍ (

̃𝜆(1)

𝜈 (𝑡) 𝜈 0 ⋅ ⋅ ⋅ 𝜈

𝜈 ̃𝜆(2)_𝜈 (𝑡) 𝜈 0 ⋅ ⋅ ⋅

0 𝜈 ̃𝜆(3)

𝜈 (𝑡) . .. ...

..

. . .. . .. ... 𝜈

𝜈 ⋅ ⋅ ⋅ 0 𝜈 ̃𝜆(𝑁)

𝜈 (𝑡)

) ) ) )

)𝑁×𝑁

,

(30) where ̃𝜆(𝑗)_𝜈 (𝑡) = 2𝑒𝑂(𝑗)𝑡 − 𝐿 − 2𝜈, 𝑗 = 1, . . . , 𝑁_{. Then by}

Lemma 2,

y(𝑡) ≤exp(∫

𝑡 𝑡0

̃

𝐷_𝜈(𝑡)𝑑𝑠)y(𝑡₀) + ∫𝑡

𝑡0exp(∫

𝑡 𝜏

̃ 𝐷_𝜈(𝑡)𝑑𝑠) ×g(𝜏)𝑑𝜏, 𝑡 ≥ 𝑡0.

By (23), we have

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩exp(∫

𝑡

𝑡0

̃

𝐷_𝜈(𝑡)𝑑𝑠)y(𝑡₀)󵄩󵄩󵄩󵄩_{󵄩󵄩󵄩󵄩}

󵄩≤ exp(− 𝐿

2(𝑡 − 𝑡0))

× 󵄩󵄩󵄩󵄩y(𝑡_{0)󵄩󵄩󵄩󵄩 , 𝑡 ≥ 𝑡0}.

(32)

Define

𝜌_𝜈(𝜔) := ∫0

−∞exp(∫

0

𝜏 𝐷̃𝜈(𝑠)𝑑𝑠)g(𝜏) 𝑑𝜏, (33)

𝑅2_{𝜈(𝜔) = 1 + 󵄩󵄩󵄩󵄩𝜌}𝜈(𝜔)󵄩󵄩󵄩󵄩2, (34)

and letB_𝜈be a random ball inR𝑁𝑑centered at the origin with radius 𝑅_𝜈(𝜔). Obviously, the infinite integrals on the right hand side of (33) and (34) are well defined byLemma 1.

For a given attracting universe of tempered random bounded setsD, that is, for any 𝜔 ∈ Ω, 𝐵 ∈ D, and all

𝛾 > 0, we have lim_{𝑡 → ∞}𝑒−𝛾𝑡sup_{𝑥∈𝐵(𝜃−𝑡𝜔)}‖𝑥‖ = 0. Note that for all𝛾 > 0, if lim_{𝑡 → ∞}𝑒−𝛾𝑡‖y(𝑡₀)‖ = 0, then

𝑁

∑

𝑗=1󵄩󵄩󵄩󵄩󵄩𝑥

(𝑗)₍₀₎󵄩󵄩󵄩󵄩

󵄩2< 𝑅2𝜈(𝜔) as 𝑡0󳨀→ −∞, (35)

which implies that the closed random ballB_𝜈(𝜔)is a pullback absorbing set at𝑡 = 0of the c`adl`ag RDS𝜑(𝑡, 𝜔); that is

𝜑 (𝑡, 𝜃_−𝑡𝜔)𝐵 (𝜃_−𝑡𝜔) ⊂B_𝜈(𝜔) , ∀𝑡 ≥ 𝑡_B𝜈(𝜔) , (36)

in the attracting universe D. Hence by Proposition 3, the coupled system has a random attractorA_𝜈 = {A_𝜈(𝜔), 𝜔 ∈

Ω}withA_𝜈(𝜔) ⊂ B_𝜈satisfying thatA_𝜈(𝜔)is compact,𝜑 -invariant, that is,𝜑(𝑡, 𝜔)A_𝜈(𝜔) = A_𝜈(𝜃_𝑡𝜔)for all 𝑡 ≥ 0,

𝜔 ∈ Ω, and attracting inD, that is, for all𝐵 ∈D,

𝐻_𝑑∗(𝜑(𝑡, 𝜃_−𝑡𝜔, 𝐵(𝜃_−𝑡𝜔)),A_𝜈(𝜔)) 󳨀→ 0 as𝑡 󳨀→ ∞, (37) where 𝐻_𝑑∗ is the Hausdorff semidistance on R𝑁𝑑. By

Lemma 4, all solutions of (7) converge pathwise to each other; therefor,A_𝜈(𝜔)consists of singleton sets, that is,

A_𝜈(𝜔) = (𝑥(1)_𝜈 (𝜔), 𝑥(2)_𝜈 (𝜔), . . . , 𝑥(𝑁)_𝜈 (𝜔))T. (38) We transform the coupled RODEs (7) back to the coupled MSODEs (8), the corresponding pathwise singleton sets attractor is then equal to

(𝑥(1)_𝜈 (𝜃_𝑡𝜔)𝑒𝑂(1)𝑡 (𝜔), 𝑥(2)

𝜈 (𝜃𝑡𝜔) 𝑒𝑂

(2) 𝑡 (𝜔), . . . ,

𝑥(𝑁)_𝜈 (𝜃_𝑡𝜔)𝑒𝑂(𝑁)𝑡 (𝜔))T,

(39)

which is exactly a stationary stochastic solution of the coupled

MSODEs (8) because the Ornstein-Uhlenbeck process is stationary (see [17]).

4. Synchronization of

the Components of Solutions

It is known in Section 3 that all solutions of the coupled RODEs system (7) converge pathwise to each other in the future for a fixed positive coupling coefficient 𝜈. Here, we would like to discuss what will happen to solutions of the coupled RODEs system (7) as𝜈 → ∞. First, we will give some lemmas which play an important role in this section.

Similar to [15, Section 4], we can set up the following estimations. Suppose that (𝑥(1)_𝜈 (𝑡), 𝑥(2)_𝜈 (𝑡), . . . , 𝑥_𝜈(𝑁)(𝑡))T is a solution of the coupled RODEs system (7). For any two dif-ferent components𝑥(𝑗)_𝜈 (𝑡), 𝑥(𝑘)_𝜈 (𝑡)of the solution for all𝑗, 𝑘 ∈

{1, 2, . . . , 𝑁},

𝑑𝑗,𝑘

𝜈 (𝑡) = 2⟨𝑥(𝑗)𝜈 (𝑡) − 𝑥(𝑘)𝜈 (𝑡), 𝐹(𝑗)(𝑥(𝑗)𝜈 , 𝑂(𝑗)𝑡 )

− 𝐹(𝑘)(𝑥(𝑘)_𝜈 , 𝑂(𝑘)_𝑡 )⟩

= 2⟨𝑥(𝑗)_𝜈 (𝑡) − 𝑥_𝜈(𝑘)(𝑡), 𝑒−𝑂𝑡(𝑗)𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗)

𝜈 )

− 𝑒−𝑂𝑡(𝑘)_𝑓(𝑘)_(𝑒𝑂(𝑘)𝑡 _𝑥(𝑘)

𝜈 )⟩

+ 2⟨𝑥_𝜈(𝑗)(𝑡) − 𝑥_𝜈(𝑘)(𝑡) , 𝑂_𝑡(𝑗)𝑥(𝑗)_𝜈 (𝑡) − 𝑂(𝑘)_𝑡 𝑥(𝑘)_𝜈 (𝑡)⟩ ≤ 2󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)𝜈 (𝑡) − 𝑥(𝑘)𝜈 (𝑡)󵄩󵄩󵄩󵄩_{󵄩 (𝑒}−𝑂

(𝑗)

𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩󵄩𝑓(𝑗)(𝑒𝑂𝑡(𝑗)𝑥(𝑗) 𝜈 (𝑡))󵄩󵄩󵄩󵄩󵄩󵄩󵄩

+󵄨󵄨󵄨󵄨_󵄨󵄨𝑂(𝑗)_𝑡 󵄨󵄨󵄨󵄨_󵄨󵄨󵄩󵄩󵄩󵄩_󵄩𝑥(𝑗)_𝜈 (𝑡)󵄩󵄩󵄩󵄩_{󵄩 )} + 2󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)

𝜈 (𝑡) − 𝑥(𝑘)𝜈 (𝑡)󵄩󵄩󵄩󵄩_{󵄩 (𝑒}−𝑂 (𝑘)

𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩𝑓(𝑘)(𝑒𝑂𝑡(𝑘)𝑥(𝑘) 𝜈 (𝑡))󵄩󵄩󵄩󵄩󵄩󵄩

+ 󵄨󵄨󵄨󵄨󵄨𝑂(𝑘)𝑡 󵄨󵄨󵄨󵄨_󵄨󵄩󵄩󵄩󵄩_󵄩𝑥(𝑘)𝜈 (𝑡)󵄩󵄩󵄩󵄩_{󵄩 ) ,}

(40)

thus, for fixed󰜚 > 0, we have

− 󰜚𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)𝜈 (𝑡) − 𝑥(𝑘)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝑑𝑘,𝑗_𝜈 (𝑡) ≤ 1

𝜈( 4 󰜚𝑒−2𝑂

(𝑗)

𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩󵄩𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗) 𝜈 (𝑡))󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

+4𝜈2 󰜚 󵄨󵄨󵄨󵄨󵄨󵄨𝑂𝑡(𝑗)󵄨󵄨󵄨󵄨󵄨󵄨

2

󵄩󵄩󵄩󵄩 󵄩𝑥(𝑗)𝜈 (𝑡)󵄩󵄩󵄩󵄩󵄩

2

) +_𝜈1(4_󰜚𝑒−2𝑂(𝑘)𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩𝑓(𝑘)(𝑒𝑂(𝑘)𝑡 𝑥(𝑘) 𝜈 (𝑡))󵄩󵄩󵄩󵄩󵄩󵄩

2

+4𝜈_󰜚2󵄨󵄨󵄨󵄨_󵄨𝑂(𝑘)_{𝑡 󵄨}󵄨󵄨󵄨󵄨2󵄩󵄩󵄩󵄩_󵄩𝑥(𝑘)_{𝜈 (𝑡)󵄩󵄩󵄩󵄩󵄩}2) .

Let

𝐶𝑗,𝑘,󰜚_𝑇1,𝑇2(𝜈, 𝜔)

= 4

󰜚_𝑡∈[sup_𝑇1,𝑇2][(𝑒−2𝑂

(𝑗)

𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩󵄩𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗) 𝜈 (𝑡))󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

+󵄨󵄨󵄨󵄨_{󵄨󵄨𝑂𝑡}(𝑗)󵄨󵄨󵄨󵄨_󵄨󵄨2󵄩󵄩󵄩󵄩_󵄩𝑥𝜈(𝑗)(𝑡)󵄩󵄩󵄩󵄩_󵄩2) + (𝑒−2𝑂(𝑘)𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩𝑓(𝑘)(𝑒𝑂(𝑘)𝑡 𝑥(𝑘) 𝜈 (𝑡))󵄩󵄩󵄩󵄩󵄩󵄩

2

+ 󵄨󵄨󵄨󵄨󵄨𝑂𝑡(𝑘)󵄨󵄨󵄨󵄨_󵄨

2󵄩󵄩󵄩󵄩

󵄩𝑥(𝑘)𝜈 (𝑡)󵄩󵄩󵄩󵄩󵄩 2

)]

(42)

in any bounded interval [𝑇₁, 𝑇₂]. Note that 𝜌_𝜈(𝜔) in (33) satisfies

𝑑

𝑑𝜈󵄩󵄩󵄩󵄩𝜌𝜈(𝜔)󵄩󵄩󵄩󵄩

2_{= 2⟨𝜌}

𝜈(𝜔),_𝑑𝜈𝑑 𝜌𝜈(𝜔)⟩ ≤ 0, (43)

and, consequently, 𝜌_𝜈(𝜔) ≤ 𝜌₁(𝜔) for 𝜈 ≥ 1. Hence,

𝐶𝑗,𝑘,󰜚_𝑇1,𝑇2(𝜈, 𝜔)is uniformly bounded in𝜈and

−󰜚𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗)𝜈 (𝑡) − 𝑥(𝑘)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝑑𝑘,𝑗_𝜈 (𝑡) ≤ 1 𝜈𝐶

𝑗,𝑘,󰜚

𝑇1,𝑇2(𝜈, 𝜔) (44)

uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]with

𝐶𝑗,𝑘,󰜚_𝑇1,𝑇2(𝜔) =sup

𝜈≥1𝐶

𝑗,𝑘,󰜚

𝑇1,𝑇2(𝜈, 𝜔) . (45)

Now let us estimate the difference between any two components of a solution of the coupled RODEs system (7) as𝜈 → ∞.

Lemma 6. Provided condition(2) is satisfied, then any two

components of a solution(𝑥_𝜈(1)(𝑡), 𝑥_𝜈(2)(𝑡), . . . , 𝑥(𝑁)_𝜈 (𝑡))Tof the

coupled RODEs system(7)uniformly vanish in any bounded

time interval when the coupling coefficient𝜈 → ∞; that is, for

any bounded interval[𝑇₁, 𝑇₂]and for all𝑡 ∈ [𝑇₁, 𝑇₂], it yields

lim

𝜈 → ∞󵄩󵄩󵄩󵄩󵄩𝑥

(𝑗)

𝜈 (𝑡) − 𝑥𝜈(𝑘)(𝑡)󵄩󵄩󵄩󵄩_{󵄩 = 0, ∀𝑗, 𝑘 ∈ {1, 2, . . . , 𝑁} .} (46)

Proof. The proof is quite similar to the proof of Lemma 4.2

in [15]. To prove the result, we can equivalently estimate the difference between any two adjacent components only because the first and the last components of the solution are considered to be adjacent. We will notice that only one new term appears in each step which continues the process, except the last step that ends the process.

For the difference of the first part of the solution

(𝑥(1)_𝜈 (𝑡), 𝑥(2)_𝜈 (𝑡), . . . , 𝑥(𝑁)_𝜈 (𝑡))T,

𝑑

𝑑𝑡₊󵄩󵄩󵄩󵄩󵄩𝑥(1)𝜈 (𝑡) − 𝑥(2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

= 2⟨𝑥(1)_𝜈 (𝑡) − 𝑥(2)_𝜈 (𝑡), 𝐹(1)(𝑥(1)_𝜈 , 𝑂(1)_𝑡 ) − 𝐹(2)(𝑥(2)_𝜈 , 𝑂_𝑡(2))⟩

− 6𝜈󵄩󵄩󵄩󵄩󵄩𝑥(1)𝜈 (𝑡) − 𝑥(2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 2𝜈 ⟨𝑥(1)_𝜈 (𝑡) − 𝑥(2)_𝜈 (𝑡), 𝑥(𝑁)_𝜈 (𝑡) − 𝑥(3)_𝜈 (𝑡)⟩ ≤ −5󵄩󵄩󵄩󵄩󵄩𝑥(1)

𝜈 (𝑡) − 𝑥𝜈(2)(𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑁)

𝜈 (𝑡) − 𝑥𝜈(3)(𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝑑1,2_𝜈 (𝑡)

≤ −𝛿𝜈󵄩󵄩󵄩󵄩󵄩𝑥(1)𝜈 (𝑡) − 𝑥(2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑁)𝜈 (𝑡) − 𝑥(3)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+1

𝜈𝐶1,2,5−𝛿𝑇1,𝑇2 (𝜔)

(47)

uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]by (44). Here, we can take

𝛿 = { { { { { { { { { { {

1 −cos 𝑁𝜋

𝑁 + 2, 𝑁is even, 1 −cos(𝑁 − 1)𝜋

𝑁 + 1 , 𝑁is odd.

(48)

In fact, from [15, Lemma 4.1], we can take any 𝛿 ∈

(−2cos(𝑁𝜋/(𝑁 + 2)), 2) when 𝑁 is even and any 𝛿 ∈

(−2cos((𝑁 − 1)𝜋/(𝑁 + 1)), 2)when𝑁is odd.

We have seen that the estimations in (47) generate𝑥(3)_𝜈 (𝑡)−

𝑥(𝑁)

𝜈 (𝑡). Now, we have

𝑑

𝑑𝑡₊󵄩󵄩󵄩󵄩󵄩𝑥(3)𝜈 (𝑡) − 𝑥(𝑁)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

= 2⟨𝑥(3)_𝜈 (𝑡) − 𝑥(𝑁)_𝜈 (𝑡), 𝐹(3)(𝑥(3)_𝜈 , 𝑂_𝑡(3)) − 𝐹(𝑁)(𝑥(𝑁)_𝜈 , 𝑂(𝑁)_𝑡 )⟩

− 4𝜈󵄩󵄩󵄩󵄩󵄩𝑥(3)𝜈 (𝑡) − 𝑥(𝑁)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 2𝜈⟨𝑥(3)

𝜈 (𝑡) − 𝑥(𝑁)𝜈 (𝑡), 𝑥(2)𝜈 (𝑡) − 𝑥(1)𝜈 (𝑡)⟩

+ 2𝜈⟨𝑥(3)_𝜈 (𝑡) − 𝑥(𝑁)_𝜈 (𝑡), 𝑥(4)_𝜈 (𝑡) − 𝑥(𝑁−1)_𝜈 (𝑡)⟩ ≤ −𝛿𝜈󵄩󵄩󵄩󵄩󵄩𝑥(3)𝜈 (𝑡) − 𝑥(𝑁)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩

2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(1)𝜈 (𝑡) − 𝑥(2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(4)𝜈 (𝑡) − 𝑥(𝑁−1)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+1_𝜈𝐶3,𝑁,2−𝛿_𝑇1,𝑇2 (𝜔)

(49)

Note that 𝑥(1)_𝜈 (𝑡) − 𝑥(2)_𝜈 (𝑡) has been fixed and 𝑥(4)_𝜈 (𝑡) −

𝑥(𝑁−1)

𝜈 (𝑡)is generated. Similarly, it yields

𝑑

𝑑𝑡₊󵄩󵄩󵄩󵄩󵄩𝑥(4)𝜈 (𝑡) − 𝑥(𝑁−1)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

≤ −𝛿𝜈󵄩󵄩󵄩󵄩󵄩𝑥(4)𝜈 (𝑡) − 𝑥(𝑁−1)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(3)𝜈 (𝑡) − 𝑥(𝑁)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(5)𝜈 (𝑡) − 𝑥(𝑁−2)𝜈 (t)󵄩󵄩󵄩󵄩_󵄩 2

+ 1

𝜈𝐶4,𝑁−1,2−𝛿𝑇1,𝑇2 (𝜔)

(50) uniformly for𝑡 ∈ [𝑇₁, 𝑇₂].

Continuing such estimations, for𝑗 = 2, 3, . . ., we get

𝑑

𝑑𝑡₊󵄩󵄩󵄩󵄩󵄩𝑥(𝑗+3)𝜈 (𝑡) − 𝑥(𝑁−𝑗)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩

2

≤ −𝛿𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗+3)𝜈 (𝑡) − 𝑥(𝑁−𝑗)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗+2)𝜈 (𝑡) − 𝑥(𝑁−𝑗+1)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+ 𝜈󵄩󵄩󵄩󵄩󵄩𝑥(𝑗+4)𝜈 (𝑡) − 𝑥(𝑁−𝑗−1)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

+1

𝜈𝐶𝑗+3,𝑁−𝑗,2−𝛿𝑇1,𝑇2 (𝜔)

(51) uniformly for𝑡 ∈ [𝑇₁, 𝑇₂].

We can divide the situation into two cases:𝑁is even and

𝑁is odd, which is just as the same as [15] did. When𝑁is even, we can rewrite the inequalities in the matrix form

̇

u(𝑡) ≤H_𝜈u(𝑡) +1

𝜈C (52)

uniformly for𝑡 ∈ [𝑇₁, 𝑇₂], where for𝑡 ∈R,

u(𝑡) = (󵄩󵄩󵄩󵄩_󵄩𝑥𝜈(1)(𝑡) − 𝑥(2)_𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩2_{, 󵄩󵄩󵄩󵄩󵄩𝑥}(3)_𝜈 (𝑡) − 𝑥(𝑁)_𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩2, . . . , 󵄩󵄩󵄩󵄩

󵄩𝑥(𝑁/2+1)𝜈 (𝑡) − 𝑥𝜈(𝑁/2+2)(𝑡)󵄩󵄩󵄩󵄩_󵄩

2

)T,

C= (𝐶1,2,5−𝛿_𝑇1,𝑇2 (𝜔), 𝐶3,𝑁,2−𝛿_𝑇1,𝑇2 (𝜔), . . . ,

𝐶𝑁/2,𝑁/2+3,2−𝛿_𝑇1,𝑇2 (𝜔), 𝐶𝑁/2+1,𝑁/2+2,5−𝛿_𝑇1,𝑇2 (𝜔))T

(53)

are(𝑁/2)-dimensional vectors, and

H_𝜈= ((

(

−𝛿𝜈 𝜈 0 ⋅ ⋅ ⋅ 0

𝜈 −𝛿𝜈 𝜈 . .. ...

0 𝜈 . .. ... 0

..

. . .. ... −𝛿𝜈 𝜈

0 ⋅ ⋅ ⋅ 0 𝜈 −𝛿𝜈

) )

)(𝑁/2)×(𝑁/2)

. (54)

ByLemma 2, it follows from (52) that

u(𝑡) ≤ 𝑒(𝑡−𝑡0)H𝜈_u(𝑡

0) +1_𝜈∫

𝑡

𝑡0

𝑒(𝑡−𝑠)H𝜈_{C𝑑𝑠. (55)}

By [15, Lemma 4.1] again,(1/𝜈)H_𝜈is negative definite, then we have

󵄩󵄩󵄩󵄩

󵄩𝑒(𝑡−𝑡0)H𝜈u(𝑡0)󵄩󵄩󵄩󵄩󵄩 ≤ 𝑒(𝑡−𝑡0)𝜇max󵄩󵄩󵄩󵄩u(𝑡0)󵄩󵄩󵄩󵄩 , (56)

where𝜇max = −𝛿 − 2cos(𝑁𝜋/(𝑁 + 2)) < 0is the maximal

eigenvalue of(1/𝜈)H_𝜈. Thus, (55) implies that

u(𝑡) 󳨀→0 as 𝜈 󳨀→ ∞, 󵄩󵄩󵄩󵄩

󵄩𝑥(1)𝜈 (𝑡) − 𝑥(2)𝜈 (𝑡)󵄩󵄩󵄩󵄩󵄩 2

󳨀→ 0, 󵄩󵄩󵄩󵄩

󵄩𝑥(𝑁/2+1)𝜈 (𝑡) − 𝑥(𝑁/2+2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩

2

󳨀→ 0

(57)

uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as𝜈 → ∞.

Similarly, when𝑁is odd, we can rewrite the inequalities in the matrix form

̇

v(𝑡) ≤ ̃H_𝜈v(𝑡) +1

𝜈C̃ (58)

uniformly for𝑡 ∈ [𝑇₁, 𝑇₂], where for𝑡 ∈R,

v(𝑡) = (󵄩󵄩󵄩󵄩_󵄩𝑥(1)_𝜈 (𝑡) − 𝑥(2)_𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩2_{, 󵄩󵄩󵄩󵄩󵄩𝑥𝜈}(3)(𝑡) − 𝑥(𝑁)_𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩2, . . . , 󵄩󵄩󵄩󵄩

󵄩𝑥((𝑁+1)/2)𝜈 (𝑡) − 𝑥((𝑁+1)/2+2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩

2

)T, ̃

C= (𝐶1,2,5−𝛿_𝑇1,𝑇2 (𝜔), 𝐶3,𝑁,2−𝛿_𝑇1,𝑇2 (𝜔), . . . ,

𝐶(𝑁−1)/2,(𝑁+1)/2+3,2−𝛿_𝑇1,𝑇2 (𝜔), 𝐶(𝑁+1)/2,(𝑁+1)/2+2,5−𝛿_𝑇1,𝑇2 (𝜔))T

(59) are((𝑁 − 1)/2)-dimensional vectors, and

̃

H_𝜈= ((

(

−𝛿𝜈 𝜈 0 ⋅ ⋅ ⋅ 0

𝜈 −𝛿𝜈 𝜈 . .. ...

0 𝜈 . .. ... 0

..

. . .. ... −𝛿𝜈 𝜈

0 ⋅ ⋅ ⋅ 0 𝜈 −𝛿𝜈

) )

)_{((𝑁−1)/2)×((𝑁−1)/2)}

.

(60) ByLemma 2, it follows from (58) that

v(𝑡) ≤ 𝑒(𝑡−𝑡0)̃H𝜈v(𝑡

0) +1_𝜈∫

𝑡

𝑡0𝑒

(𝑡−𝑠)̃H𝜈̃C𝑑𝑠. ₍₆₁₎

Just like the even case, for uniform𝑡 ∈ [𝑇₁, 𝑇₂], we have

󵄩󵄩󵄩󵄩

󵄩𝑥(1)𝜈 (𝑡) − 𝑥(2)𝜈 (𝑡)󵄩󵄩󵄩󵄩_󵄩 2

󳨀→ 0, as𝜈 󳨀→ ∞. (62)

For other adjacent components, the process above can be repeated. Hence, we can draw a conclusion that the difference between any adjacent components of a solution of the coupled RODEs system (7) tends to zero uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as the coupling coefficient goes to infinity which completes the proof.

We know that all components of a solution of system (7) have the same limit uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as𝜈 → ∞. Now, we are in the position to find what they converge to.

Lemma 7. If assumptions(2)and(6)hold, the c`adl`ag random

dynamical system 𝜑(𝑡, 𝜔) generated by the solution of the

averaged RODEs system

𝑑𝑍 𝑑𝑡₊ =

1 𝑁

𝑁

∑

𝑗=1

𝑒𝑂(𝑗)𝑡 𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑍) + 1

𝑁

𝑁

∑

𝑗=1

𝑂(𝑗)_𝑡 𝑍 (63)

has a singleton sets random attractor denoted by {𝑍(𝜔)}.

Furthermore,

𝑍(𝜃_𝑡𝜔)exp(1

𝑁

𝑁

∑

𝑗=1

𝑂(𝑗)_𝑡 (𝜔)) (64)

is the stationary stochastic solution of the equivalently averaged SODE system

𝑑𝑧 =_𝑁1∑𝑁

𝑗=1

𝑒−𝜁𝑡(𝑗)𝑓(𝑗)(𝑒𝜁(𝑗)𝑡 𝑧)𝑑𝑡 + 1

𝑁

𝑚

∑

𝑖=1 𝑁

∑

𝑗=1

𝑐_𝑖(𝑗)𝑧 ⬦ 𝑑𝐿(𝑖)_𝑡 , (65)

where𝜁_𝑡(𝑗)= ∑𝑁_𝑘=1(𝑂(𝑗)_𝑡 − 𝑂(𝑘)_𝑡 ), 𝑗 = 1, . . . , 𝑁.

Proof. Assume that𝑍₁(𝑡)and𝑍₂(𝑡)are two solutions of (63),

we have

𝑑

𝑑𝑡₊󵄩󵄩󵄩󵄩𝑍1(𝑡) − 𝑍2(𝑡)󵄩󵄩󵄩󵄩2

≤ (−2𝐿 +_𝑁2∑𝑁

𝑗=1

𝑂(𝑗)_{𝑡 ) 󵄩󵄩󵄩󵄩𝑍1}(𝑡) − 𝑍_{2(𝑡)󵄩󵄩󵄩󵄩}2.

(66)

It follows from Gronwall’s lemma (see [27, Lemma 2.8]) that

󵄩󵄩󵄩󵄩𝑍1(𝑡) − 𝑍2(𝑡)󵄩󵄩󵄩󵄩2

≤ 𝑒−2𝑡(𝐿−(1/𝑁) ∑𝑁𝑗=1(1/𝑡) ∫0𝑡𝑂(𝑗)𝑠 𝑑𝑠)󵄩󵄩󵄩󵄩𝑍

1(0) − 𝑍2(0)󵄩󵄩󵄩󵄩2,

(67)

which implies that lim

𝑡 → ∞󵄩󵄩󵄩󵄩𝑍1(𝑡) − 𝑍2(𝑡)󵄩󵄩󵄩󵄩

2_{= 0.}

(68) Then, all solutions of (63) converge pathwise to each other.

Now, we have to give what they converge to based on the theory of c`adl`ag random dynamical systems. Let𝑍(𝑡)be a solution of (63), we get

𝑑

𝑑𝑡₊‖𝑍(𝑡)‖2≤ (−2𝐿 + 2 𝑁

𝑁

∑

𝑗=1

𝑂(𝑗)_𝑡 ) ‖𝑍(𝑡)‖2 +_𝐿𝑁1 ∑𝑁

𝑗=1𝑒

−2𝑂(𝑗)𝑡 󵄩󵄩󵄩󵄩

󵄩𝑓(𝑗)(0)󵄩󵄩󵄩󵄩󵄩2.

(69)

From Gronwall’s lemma in [27], again, it yields for−𝑡₀, 𝑡 > 𝑇_𝜔,

‖𝑍(𝑡)‖2

≤ 𝑒−𝐿(𝑡−𝑡0)+(2/𝑁) ∑𝑁𝑗=1∫_𝑡0𝑡𝑂(𝑗)𝑠 𝑑𝑠󵄩󵄩󵄩󵄩𝑍(𝑡

0)󵄩󵄩󵄩󵄩2+_𝐿𝑁1

×∑𝑁

𝑗=1󵄩󵄩󵄩󵄩󵄩𝑓

(𝑗)₍₀₎󵄩󵄩󵄩󵄩

󵄩2∫

𝑡

𝑡0

𝑒−2𝑂𝜏(𝑗)−𝐿(𝑡−𝜏)+(2/𝑁) ∑𝑁𝑘=1∫𝜏𝑡𝑂(𝑘)𝑠 𝑑𝑠𝑑𝜏.

(70) By pathwise pullback convergence with 𝑡₀ → −∞, the random closed ball centered as the origin with random radius

̃𝑅(𝜔)is a pullback absorbing set of𝜑(𝑡, 𝜔), where

̃𝑅2_{(𝜔) = 1 +} 1

𝐿𝑁

𝑁

∑

𝑗=1󵄩󵄩󵄩󵄩󵄩𝑓

(𝑗)₍₀₎󵄩󵄩󵄩󵄩

󵄩2∫

0

−∞𝑒

𝐿𝜏−2𝑂(𝑗)

𝜏 +(2/𝑁) ∑𝑁𝑘=1∫𝜏0𝑂(𝑘)𝑠 𝑑𝑠𝑑𝜏.

(71) Obviously, byLemma 1, the integral defined in the right hand side is well defined.

ByProposition 3, there exists a random attractor{𝑍(𝜔)} for 𝜑(𝑡, 𝜔). Since all solutions of (63) converge pathwise to each other, the random attractor{𝑍(𝜔)}is composed of singleton sets.

Note that the averaged RODE (63) is transformed from the averaged SODE (65) by the following transformation:

𝑍(𝑡, 𝜔) = 𝑧exp(−1

𝑁

𝑁

∑

𝑗=1

𝑂_𝑡(𝑗)(𝜔)) , (72) so the pathwise singleton sets attractor

𝑍(𝜃_𝑡𝜔)exp((1/𝑁) ∑𝑁_𝑗=1𝑂(𝑗)_𝑡 (𝜔)) is a stationary solution of the averaged SODE (65) since the Ornstein-Uhlenbeck process is stationary.

Now, we will present another main solution of this work.

Theorem 8. Let

(𝑥(1)_𝜈𝑛(𝑡, 𝜔), 𝑥(2)_𝜈𝑛(𝑡, 𝜔), . . . , 𝑥(𝑁)_𝜈𝑛 (𝑡, 𝜔))T

= (𝑥(1)_𝜈𝑛(𝜃𝑡𝜔), 𝑥(2)𝜈𝑛(𝜃𝑡𝜔) , . . . , 𝑥(𝑁)𝜈𝑛 (𝜃𝑡𝜔))T

(73)

be the singleton sets random attractor of the c`adl`ag random

dynamical system𝜑(𝑡, 𝜔)generated by the solution of RODEs

system(7), then

((𝑥(1)_𝜈𝑛(𝑡, 𝜔), 𝑥(2)_𝜈𝑛(𝑡, 𝜔), . . . , 𝑥(𝑁)_𝜈𝑛 (𝑡, 𝜔))T) 󳨀→ (𝑍(𝑡, 𝜔), 𝑍(𝑡, 𝜔) , . . . , 𝑍(𝑡, 𝜔))T

(74)

pathwise uniformly for𝑡belongs to any bounded time interval

[𝑇₁, 𝑇₂] for any sequence 𝜈_𝑛 → ∞, where 𝑍(𝑡, 𝜔) =

𝑍(𝜃_𝑡𝜔)is the solution of the averaged RODE(63)and𝑍(𝜔)

is the singleton sets random attractor of the c`adl`ag random

dynamical system𝜑(𝑡, 𝜔)which is generated by the solution of

Proof. Define

𝑍_𝜈(𝜔) = _𝑁1∑𝑁

𝑗=1

𝑥(𝑗)

𝜈 (𝜔), (75)

where{𝑥(1)_𝜈 (𝜔), 𝑥(2)_𝜈 (𝜔), . . . , 𝑥(𝑁)_𝜈 (𝜔)}is the singleton sets ran-dom attractor of the c`adl`ag RDS generated by RODEs system (7). Thus,𝑍_𝜈(𝑡, 𝜔) = 𝑍_𝜈(𝜃_𝑡𝜔)satisfies

𝑑𝑍𝜈(𝑡, 𝜔)

𝑑𝑡₊ = 1

𝑁

𝑁

∑

𝑗=1

(𝑒−𝑂𝑡(𝑗)𝑓(𝑗)(𝑒𝑂𝑡(𝑗)𝑥(𝑗)

𝜈 (𝑡, 𝜔)) + 𝑂(𝑗)𝑡 𝑥(𝑗)𝜈 (𝑡, 𝜔)) .

(76) Then, we get

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩

𝑑𝑍_𝜈(𝑡, 𝜔) 𝑑𝑡₊ 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩

2

≤ _𝑁2∑𝑁

𝑗=1

(𝑒−2𝑂(𝑗)𝑡 󵄩󵄩󵄩󵄩

󵄩󵄩󵄩𝑓(𝑗)(𝑒𝑂(𝑗)𝑡 𝑥(𝑗) 𝜈 (𝑡, 𝜔))󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

+󵄨󵄨󵄨󵄨_󵄨󵄨𝑂(𝑗)_𝑡 󵄨󵄨󵄨󵄨_󵄨󵄨2󵄩󵄩󵄩󵄩_󵄩𝑥(𝑗)_𝜈 (𝑡, 𝜔)󵄩󵄩󵄩󵄩_󵄩2) ,

(77)

by the continuous property of the solutions and the fact that these solutions belong to the compact ballB₁(𝜔), it follows that

sup

𝑡∈[𝑇1,𝑇2]

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩

𝑑𝑍_𝜈(𝑡, 𝜔) 𝑑𝑡₊ 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩_󵄩≤ (

2 𝑁

𝑁

∑

𝑗=1

󰜚

4C𝑗,∙,󰜚𝑇1,𝑇2(𝜔))

1/2

< ∞. (78) By the Ascoli-Arzel`a theorem in a Skorohod space of bounded time intveral 𝐷([𝑇₁, 𝑇₂],R𝑑) in [23], there exists a subsequence 𝜈_𝑛𝑘 → ∞ such that 𝑍_𝜈

𝑛𝑘(𝑡, 𝜔)converges to

𝑍(𝑡, 𝜔)as𝑛_𝑘 → ∞.

Since difference between any two components of a solu-tion of the coupled RODEs system (7) tends to zero uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as𝜈 → ∞, from (75), we have

𝑥(𝑗)_𝜈

𝑛𝑘(𝑡, 𝜔) = 𝑍𝜈𝑛𝑘(𝑡, 𝜔)

+ 1 𝑁_𝑗∑󸀠 _̸=𝑗

∑

𝑗󸀠󸀠 _̸=𝑗󸀠

(𝑥(𝑗_𝜈󸀠󸀠)

𝑛𝑘(𝑡, 𝜔) − 𝑥

(𝑗󸀠₎

𝜈_𝑛𝑘(𝑡, 𝜔))

󳨀→ 𝑍 (𝑡, 𝜔)

(79)

uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as𝜈_𝑛𝑘 → ∞for𝑗 = 1, . . . , 𝑁. Furthermore, it follows from (76) that, for𝑡 ≥ 𝑇₁,

𝑍𝜈(𝑡, 𝜔) = 𝑍𝜈(𝑇1, 𝜔) +_𝑁1 𝑁

∑

𝑗=1

∫𝑡

𝑇1(𝑒

−𝑂(𝑗)

𝑠 𝑓(𝑗)(𝑒𝑂(𝑗)𝑠 𝑥(𝑗)

𝜈 (𝑠, 𝜔))

+ 𝑂_𝑠(𝑗)𝑥(𝑗)_𝜈 (𝑠, 𝜔))𝑑𝑠.

(80)

Thus,

𝑍(𝑡, 𝜔) = 𝑍(𝑇1, 𝜔) + _𝑁1 𝑁

∑

𝑗=1

∫𝑡

𝑇1

(𝑒−𝑂(𝑗)𝑠 𝑓(𝑗)(𝑒𝑂𝑠(𝑗)𝑍(𝑠, 𝜔))

+ 𝑂(𝑗)_𝑠 𝑍(𝑠, 𝜔) )𝑑𝑠,

(81) uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as𝜈_𝑛𝑘 → ∞, which implies that

𝑍_𝜈(𝑠, 𝜔)solves RODE (63). Then, we note that all possible sequences of 𝑍_𝜈𝑛𝑘(𝑡, 𝜔) converge to the same limit 𝑍(𝑡, 𝜔) uniformly for𝑡 ∈ [𝑇₁, 𝑇₂] as𝜈_𝑛 → ∞. Since the RDS𝜑 generated by the solutions of RODE (63) has a singleton sets random attractor {𝑍(𝜔)}, the stationary stochastic process

𝑍(𝜃_𝑡𝜔)must be equal to𝑍(𝑡, 𝜔), that is, 𝑍(𝑡, 𝜔) = 𝑍(𝜃_𝑡𝜔), which completes the proof.

As an obvious result ofTheorem 8, we get the following.

Corollary 9.

((𝑥(1)

𝜈 (𝑡, 𝜔), 𝑥(2)𝜈 (𝑡, 𝜔), . . . , 𝑥(𝑁)𝜈 (𝑡, 𝜔))

T ) 󳨀→ (𝑍(𝑡, 𝜔), 𝑍(𝑡, 𝜔) , . . . , 𝑍(𝑡, 𝜔))T

(82)

in Skorohod metric pathwise uniformly for𝑡 ∈ [𝑇₁, 𝑇₂]as𝜈 →

∞.

Remark 10. The results in this paper hold just in almost

everywhere because𝜔 ∈ ΩinLemma 1, and we still useΩ instead ofΩ.

Remark 11. Although the same results hold when the systems

perturbed by non-Gaussian noises (see, e.g., [17] and this paper for𝛼-stable L´evy noises and [16,18] for additive L´evy noises), there exists some difference between dealing with such stochastic systems when driven by Bronian motions and L´evy motions. Firstly, to some extent, the cases of L´evy noises have more general sense than the Bronian motions. For example, when𝛼 = 2, the L´evy noise is the standard Brownian motion and the Marcus integral is reduced to the Stratonovich stochastic integral, that is, the case of multiplicative white noise (see [11, 15]). Here we only are restricted to1 < 𝛼 < 2. Secondly, We need to consider the c`adl`ag functions in the Skorohod metric, which are different from the continuous cases in the metric under the compact-open topology. Last but not least, we have to consider the solutions in the sense of Carth´eodory and the right hand derivatives.

Acknowledgments

The author would like to thank the anonymous referees for their helpful comments and suggestions which largely improved the quality of the paper. This work is partially sup-ported by NSF of China under Grant no. 11071165, Guangxi Provincial Department of Research Project (201010LX166),

and Program to Sponsor Teams for Innovation in the Con-struction of Talent Highlands in Guangxi Institutions of Higher Learning under Grant no.[2011]47.

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