Graph-theoretic approach to exponential synchronization of discrete-time stochastic coupled systems with time-varying delay

Communicated by Dr. Yong Xu

Accepted Manuscript

Graph-theoretic approach to exponential synchronization of discrete-time stochastic coupled systems with time-varying delay Pengfei Wang, Zhangrui Chen, Wenxue Li PII: DOI: Reference:

S0925-2312(17)31517-5 10.1016/j.neucom.2017.08.069 NEUCOM 18880

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Neurocomputing

Received date: Revised date: Accepted date:

1 April 2017 5 July 2017 25 August 2017

Please cite this article as: Pengfei Wang, Zhangrui Chen, Wenxue Li, Graph-theoretic approach to exponential synchronization of discrete-time stochastic coupled systems with time-varying delay, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.08.069

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Graph-theoretic approach to exponential synchronization of discrete-time stochastic coupled systems with time-varying delay Pengfei Wang, Zhangrui Chen, Wenxue Li∗

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Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, 264209, PR China

Abstract

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In this paper, we investigate the exponential synchronization problem for discrete-time stochastic drive-response coupled systems with time-varying delay. By employing the Lyapunov method combined with Kirchhoff’s matrix tree theorem in graph theory as well as stochastic analysis technique, some novel sufficient criteria are established to guarantee the exponential synchronization of two identical delayed coupled systems with stochastic disturbances. These sufficient criteria have a close relationship with the topological property of the coupled network. Moreover, the theoretical results are applied to a coupled oscillators system to demonstrate the applicability of the proposed synchronization approaches. Finally, a numerical example is provided to illustrate the effectiveness of our theoretical results.

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Keywords: exponential synchronization, discrete-time coupled systems, stochastic disturbances, time-varying delay, graph theory

1. Introduction

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In the past few decades, there has been an increasing interest in coupled systems because of their extensive applications developed in physics [1, 2], neural networks [3, 4, 5, 6], biology [7, 8, 9], and engineering [10, 11]. Roughly speaking, the mathematical framework of coupled systems in the time domain can be classified into three categories (i.e. continuous time [12], discontinuous time [13] and discrete time [14]). Among them, discrete-time coupled systems have been widely investigated for their potential application prospects in biology, physics and communication networks [15, 16, 17, 18, 19]. These applications heavily depend on the dynamical behaviors such as synchronization, periodicity, bifurcation and so on. Particularly, synchronization has been regarded as one of the most effective ways to explore the collective phenomena of discrete-time coupled systems [20, 21, 22, 23, 24, 25]. In practice, time delay occurs frequently due to the finite speed of transmitting signals and traffic congestions. For example, in a multi-patch predator-prey system, a predator may need a period of time to grow up so that it is mature enough to prey. Moreover, due to the fact that time delay may lead to undesirable dynamic behaviors such as performance degradation, oscillation and even instability, the synchronization problem for coupled systems with time delay has attracted increasing research interests [26, 27]. Furthermore, time delay is usually time-varying and not identical. Hence, in order to model the real coupled systems better, we consider timevarying delay with the upper and lower bounds [28, 29]. Besides, stochastic disturbances in nature can not be ∗

Corresponding author. Tel.: +86 0631 5687035; fax: +86 0631 5687572. Email address: [email protected],[email protected] (Wenxue Li)

Preprint submitted to Elsevier

September 11, 2017

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neglected. For example, in reality, population systems are usually influenced by the environment noise, which is ubiquitous [7, 8, 30, 31]. Due to the environment noise, the birth rate, the mortality rate, competition coefficients and other parameters are associated with the system exhibit random fluctuation to some extent. Hence, it is of great significance to take both time-varying delay and stochastic interference into account when studying the synchronization problem for discrete-time coupled systems. The discussions above show the significance of the synchronization for discrete-time stochastic coupled systems with time-varying delay (DSCSVDs). Nowadays, using Lyapunov method has become a prevailing and effective approach to investigate the synchronization of the coupled systems. Nevertheless, the high dimensions of coupled systems make it difficult to construct a Lyapunov function. Hence, how to systematically construct a proper Lyapunov function for a high-dimensional system is an interesting topic. In [32], Liang et al. studied the synchronization problem of complex networks by using a Lyapunov-Krasovskii functional method combined with linear matrix inequalities, which then are extended to study many other complex systems such as complex dynamical networks with control packet loss and additive time-varying delays [33], neural networks with discontinuous activations [34]. Particularly, in paper [35], Li and Shuai provided a systematic method for constructing a global Lyapunov function for the coupled systems of differential equations on networks by using the results in graph theory. Then this method is extended to delayed coupled systems [11], stochastic coupled systems [38] and impulsive coupled systems [39]. In this paper, we extend this method to study the exponential synchronization of DSCSVDs. In detail, we model a DSCSVD on a weighted digraph (G, A). Here G is a digraph with a set of vertex L = {1, 2, . . . , l} and A = (ai j )l×l is the weighted matrix. Each vertex stands for an individual system called vertex system. For each vertex system, assume that there exists a Lyapunov function vi . P Then we construct global Lyapunov function V = li=1 ci vi , where ci has a close relationship with the topological structure of the DSCSVD. Then by combining Lyapunov method and Kirchhoff’s matrix tree theorem in graph theory, this paper proposes two sufficient criteria which are given in the form of vertex Lyapunov functions and the coefficients of the drive-response systems respectively. Compared with the results of previous works, the contributions of this paper are listed as follows.

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1. We investigate the global exponential synchronization for DSCSVDs by combining Lyapunov method and Kirchhoff’s matrix tree theorem in graph theory. 2. The theoretical results are applied to a stochastic coupled oscillations system with time-varying delay. At the same time, the related numerical simulation is also given to illustrate the effectiveness and feasibility of our theoretical results.

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The rest of this paper is outlined as follows. In Section 2, we give some basic preliminaries and model formulations of DSCSVDs. Then, the main results are presented in Section 3. In Section 4, we will apply our theoretical results to a coupled oscillators system with time-varying delay and a numerical example related to it is given in Section 5. Finally, we close this paper with a conclusion in Section 6. 2. Preliminaries and model formulations In this section, we first give some preliminaries and basic knowledge of graph theory in Subsection 2.1. Then in Subsection 2.2, we describe a general DSCSVD and propose some essential assumptions and a definition. 2.1. Preliminaries We define Rn as the n-dimensional Euclidean space and denote N = {0, 1, 2, . . .}, N+ = {1, 2, . . .} and L = {1, 2, . . . , l}. Let | · | denote the Euclidean norm for a vector or the spectral norm for a matrix. Let (Ω, F , F, P) 2

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stand for a complete probability space with filtration F = {Ft }t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). E{·} stands for the mathematical expectation with respect to the given probability measure P. Write L2F0 ([−τ, 0], Rn ) for the family of all F0 -measurable C([−τ, 0]; Rn )-value random variable $ = {$(s), −τ ≤ s ≤ 0} with the norm ||$|| = sup−τ≤s≤0 |$(s)|2 < ∞. Then some basic concepts on graph theory are provided. A digraph G = (L, E) with a set of vertices L and a set E of directed edge (i, j) from initial vertex i to terminal vertex j satisfying (i, i) < E for all i ∈ L. Digraph G can be weighted if each directed edge (i, j) is assigned a positive weight ai j . And the weighted matrix of G is A = (ai j )l×l satisfying ai j > 0 if and only if there exists an directed edge (i, j) in G. The weight W(G) of G is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices {i1 , i2 , . . . , i s } such that its set of arcs is {(ik , ik+1 ) : k = 1, 2, . . . , s − 1}. If i s = i1 , we call P a directed cycle. A connected subgraph T is a tree if it contains no cycles. A tree T is rooted at vertex i, called the root, if i is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph Q is unicyclic if there is a unique cycle. A digraph G is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. Denote the digraph with weighted matrix A as (G, A). A weighted digraph (G, A) is said to be balanced if W(C) = W(−C) for all directed cycles C. Here, −C denotes the reverse of C and is constructed by reversing the direction of all arcs in C. For a unicyclic graph Q with cycle CQ , let Q˜ ˜ be the unicyclic graph obtained by replacing CQ with −CQ . Suppose that (G, A) is balanced, then W(Q) = W(Q). Let L(G) = (ςi j )l×l stand for the Laplacian matrix of G where X   ais (i = j),    s,i ςi j =      − ai j (i , j). Other details on graph theory can be found in [35]. The following lemma will be needed in our main derivation.

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Lemma 1. [35] (Kirchhoff’s matrix tree theorem) Assume l ≥ 2. Let ci denote the cofactor of the ith diagonal element of L(G). Then the following identity holds ci ai j Fi j (xi , x j ) =

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l X

i, j=1

X

W(Q)

X

Frs (xr , x s ),

(s,r)∈E(CQ )

Q∈Q

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where Fi j , i, j ∈ L, are arbitrary functions, Q is the set of all spanning unicyclic graphs of (G, A), W(Q) is the weight of Q, CQ denotes the directed cycle of Q and E(CQ ) represents the set of all directed arcs in CQ . In particular, if (G, A) is strongly connected, then ci > 0 for i ∈ L.

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2.2. Model formulations To begin with, we describe a DSCSVD on a digraph G with l ≥ 2 vertices. More precisely, we assume that each vertex is described by the following difference equation with time-varying delay xi (k + 1) =xi (k) + fi (xi (k), xi (k − τ(k))) +

l X j=1

Hi j (xi (k), x j (k)), i ∈ L,

(1)

where xi (k) ∈ Rmi represents the state vector of the ith vertex system, fi : Rmi × Rmi → Rmi , Hi j : Rmi × Rm j → Rmi represents the influence of vertex j on vertex i and Hi j ≡ 0 if there exists no arc from j to i in G, τ(k) is the time-varying delay satisfying τm ≤ τ(k) ≤ τ M , where τm and τ M are positive integers. 3

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In this paper, we regard the system (1) as the drive system and the noise-perturbed response system is designed as yi (k + 1) =yi (k) + fi (yi (k), yi (k − τ(k))) +

l X j=1

Hi j (yi (k), y j (k)) + ui (k) + δi (yi (k) − xi (k))ω(k), i ∈ L,

(2)

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where δi : Rmi → Rmi is a continuous function representing the disturbance intensity of vertex i, ω(k) is a scalar Brownian motion on probability space (Ω, F , F, P) with n o E {ω(k)} = 0, E ω2 (k) = 1, E {ω(k1 )ω(k2 )} = 0 (k1 , k2 )

and ui is the state feedback controller which is introduced as

ui (k) = bi [ fi (yi (k), yi (k − τ(k))) − fi (xi (k), xi (k − τ(k)))],

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where bk , −1 is the gain constant to be scheduled. It should be noted that this kind of feedback control is very common and general, see [36, 37, 38]. For the brevity of the following analysis, we denote fi (xi (k), xi (k−τ(k))), fi (yi (k), yi (k−τ(k))), Hi j (xi (k), x j (k)), (k) Hi j (yi (k), y j (k)) by f x(k) , fy(k) , Hi(k) i ,τ i ,τ j,x , Hi j,y , respectively. Define ei (k) = yi (k) − xi (k) as the ith synchronization error state vector. Then we can get the following error system ei (k + 1) =ei (k) + (1 + bi )[ fy(k) − f x(k) ]+ i ,τ i ,τ

l h X j=1

i (k) Hi(k) − H j,y i j,x + δi (ei (k))ω(k), i ∈ L.

(3)

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 T Let e(k) = eT1 (k), eT2 (k), . . . , eTl (k) ∈ Rm represent the global solution of error system (3), therefore, m = Pl + i=1 mi for mi ∈ N . The initial condition associated with error system (3) is given as e(s) = $(s), s ∈ N[−τ M , 0],

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where $(s) ∈ L2F0 ([−τ M , 0], Rm ) and N[−τ M , 0] = {−τ M , −τ M + 1, . . . , 0}. Assume that vector valued functions fi , Hi j (i, j ∈ L) satisfy Lipschitz condition, that is | fi (yi , y˜ i ) − fi (xi , x˜i )| ≤ γi |ei | + ξi |˜yi − x˜i |, |Hi j (yi , y j ) − Hi j (xi , x j )| ≤ Bi j (|ei | + |e j |)

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and functions δi (i ∈ L) satisfying

|δi (ei )|2 ≤ εi |ei |2 ,

(4) (5)

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where γi , ξi , Bi j and εi are known real scalar constants. It is noted that these assumptions are common and general. Then it is easy to see that e(k) = 0 is a trivial solution of system (3). Definition 1. [32] The drive system (1) and the response system (2) are said to be globally exponentially synchronized, if for a suitably designed feedback controller, the trivial solution of the error system (3) is globally exponentially stable in the mean square. That is, there exist constants ϑ > 0 and ν ∈ (0, 1) such that for sufficiently large integer M > 0, the inequality E|e(k)|2 ≤ ϑνk

sup s∈N[−τ M ,0]

holds for all k > M. 4

E|e(s)|2

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3. Main results In this section, we will propose two kinds of sufficient criteria to realize the synchronization between the drive system (1) and the noise-perturbed response system (2). Theorem 1. Assume that the following conditions hold.

|ei (k)|2 ≤ vi (ei (k)) ≤ |ei (k)|2 + ηi

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A1 For every i ∈ L, there are positive-definite functions vi , functions Fi j (ei , e j ), positive constants ϑi , κi and ηi and a matrix A = (ai j )l×l where ai j ≥ 0 such that k−1 X

s=k−τ M

and

l X

ai j Fi j (ei (k), e j (k)).

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E {∆vi (ei (k))} ≤ −ϑi E|ei (k)|2 − κi E|ei (k − τ(k))|2 +

|ei (s)|2

(6)

(7)

j=1

A2 For each directly cycle CQ of (G, A), the following inequality holds X Fi j (ei , e j ) ≤ 0.

(8)

(i, j)∈E(CQ )

and

V(e(k)) =

l X

ci vi (ei (k)) ≤

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i=1

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If digraph (G, A) is strongly connected, then the drive system (1) and the response system (2) are globally exponentially synchronized. P Proof. Let V(e) = li=1 ci vi (ei ), where ci is defined in Lemma 1. Since (G, A) is strongly connected, we have ci > 0 for all i ∈ L. By (6), we have V(e(k)) ≥ c0 |e(k)|2 (9) l X i=1

2

ci |ei (k)| +

l X

ci ηi

i=1

k−1 X

s=k−τ(k)

2

2

|ei (s)| ≤ c|e(k)| + η

k−1 X

s=k−τ M

|e(s)|2 ,

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where c0 = mini∈L {ci }, c = maxi∈L {ci } and η = maxi∈L {ci ηi }. Then by (7), calculating the difference of V and taking the mathematical expectation, it yields  l      X  E {∆V(e(k))} = E  c ∆v (e (k))  i i i     i=1  l  l      X X 2 2  ≤E  ci − ϑi |ei (k)| − κi |ei (k − τ(k))|  + ci ai j Fi j (ei (k), e j (k))     i=1

≤−

l X i=1

i, j=1

ϑi ci E|ei (k)|2 +

l X

ci ai j Fi j (ei (k), e j (k)).

i, j=1

5

(10)

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In view of Lemma 1 with weighted digraph (G, A), it is easy to show l X

ci ai j Fi j (ei , e j ) =

i, j=1

X

X

W(Q)

Fi j (ei , e j ).

( j,i)∈E(CQ )

Q∈Q

Noting that W(Q) ≥ 0 and making use of condition A2, one can arrive at E {∆V(e(k))} ≤ −ΘE|e(k)|2 ,

(11)

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where Θ = mini∈L {ci ϑi } > 0. Then by using (10) and (11), it yields that for any µ > 1, we have n o E µk+1 V(e(k + 1)) − µk V(e(k)) =µk+1 E {∆V(e(k))} + µk (µ − 1)E {V(e(k))} k−1 X k 2 k ≤h1 (µ)µ E|e(k)| + h2 (µ)µ E|e(s)|2 , s=k−τ M

(12)

N

k=0

 −1 s+τ N−1−τ s+τ  N−1 X N−1  X M XM X  X XM  k 2 µk E|e(s)|2 =  + +  µ E|e(s)| s=−τ M k=0

≤τ M

−1 X

µ

s=0

s+τ M

k=s+1 2

τM

N−1−τ XM

µ

s+τ M

2

E|e(s)| + τ M

s=0

2

τM

max E|e(s)| + τ M µ

s∈N[−τ M ,0]

N−1 X

N−1 X

µ s+τM E|e(s)|2

Then substituting (14) into (13), one can derive

µ s E|e(s)|2 .

N−1

 X k µ E|e(k)|2 . E µ V(e(N)) ≤ E {V(e(0))} + τ M µ h2 (µ) max E|e(s)| + h1 (µ) + τ M µτM h2 (µ) N

o

2

τM

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n

(14)

s=N−1−τ M

s=0

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≤τ M µ

s=N−τ M k=s+1

E|e(s)| + τ M

s=−τ M

(13)

k=0 s=k−τ M

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k=0 s=k−τ M

N−1 N−1 X k−1 X X o k 2 µk E|e(s)|2 , E µ V(e(N)) − E {V(e(0))} ≤ h1 (µ) µ E|e(k)| + h2 (µ)

n

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therein N−1 X k−1 X

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where h1 (µ) = −µΘ + (µ − 1)c and h2 (µ) = (µ − 1)η. Adding both side of (12) from 0 to N − 1 for k, we get

s∈N[−τ M ,0]

(15)

k=0

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By the definitions of h1 (µ) and h2 (µ), it is easy to see that there exists a µ0 > 1 such that h1 (µ0 )+τ M µτ0M h2 (µ0 ) = 0. Then let C = c + ητ M . According to (10), we get E {V(e(0))} ≤ C

max E|e(s)|2 .

s∈N[−τ M ,0]

This together with (9) and (15) implies  1 !N 1  τM C + τ M µ0 h2 (µ0 ) max E|e(s)|2 . E|e(N)| ≤ c0 µ0 s∈N[−τM ,0] 2

According to Definition 1, we have that the drive system (1) and the response system (2) are globally exponentially synchronized. This completes the proof. 6

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Remark 1. Note that if digraph (G, A) is balanced (i.e., W(C) = W(−C) for all directed cycles C, where −C is the reverse of C and is constructed by reversing the direction of arcs in C), then it holds l X

ci ai j Fi j (ei , e j ) =

i, j=1

X h i 1X W(Q) Fi j (ei , e j ) + F ji (e j , ei ) . 2 Q∈Q (i, j)∈E(C ) Q

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h i P Hence, it is obvious that (8) can be replaced by (i, j)∈E(CQ ) Fi j (ei , e j ) + F ji (e j , ei ) ≤ 0. Furthermore, if for each i, j ∈ L, there exist functions Ri (ei ) and R j (e j ) satisfying Fi j (ei , e j ) ≤ R j (e j ) − Ri (ei ), then (8) holds naturally. Remark 2. Theorem 1 offers a technique to systematically construct a Lyapunov function V for error system (3) by using Lyapunov functions vi of vertex systems and the topological structure of digraph (G, A). That is V(e) =

l X

ci vi (ei ).

i=1

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This technique avoids the difficulty to directly find the Lyapunov function of error system (3). Moreover, Theorem 1 uses a graph-theoretic approach to analyze the global exponential synchronization of DSCSVDs. This approach does not need us to solve any linear matrix inequality. It should be noted that condition (7) is not easy to verify. However, Theorem 1 is important since the next theorem is based on it. And the next theorem is easier to verify because its conditions are related to coefficients of the discussed system.

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Theorem 2. The noise-perturbed response system (2) is globally exponentially synchronized with the drive system (1) if for every i ∈ L, the following conditions hold. B1 There exist positive constants αi and βi such that

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eTi [ fi (yi , y˜ i ) − fi (xi , x˜i )] ≤ −αi |ei |2 + βi |˜yi − x˜i |2 .

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B2 There is a positive constant λi such that 4(1 + bi )2 ξi2 + 2(1 + bi )βi < λi and

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4(1 +

bi )2 γi2

l  X  + εi + 8lB2i j + 4Bi j + λi (τ M − τm + 1) < 2(1 + bi )αi . j=1

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(i) (i) Proof. Let vi (ei (k)) = v(i) 1 (ei (k)) + v2 (ei (k)) + v3 (ei (k)), where

v(i) 1 (ei (k))

2

= |ei (k)| ,

v(i) 2 (ei (k))

= λi

k−1 X

2

s=k−τ(k)

|ei (s)| ,

v(i) 3 (ei (k))

= λi

k−τ Xm

j=k−τ M +1 s= j

Calculating directly gives 2

2

|ei (k)| ≤ vi (ei (k)) ≤ |ei (k)| + λi (τ M − τm + 1)

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k−1 X

k−1 X

s=k−τ M

|ei (s)|2 ,

|ei (s)|2 .

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which means (6) holds. By calculating the difference of v(i) 1 (ei (k)) along the system (3) and taking the mathematical expectation, we have n o n o 2 2 E ∆v(i) 1 (ei (k)) = E |ei (k + 1)| − |ei (k)| X 2 l  (k) (k) 2 (k) (k) 2 ≤E 2(1 + bi ) | fyi ,τ − f xi ,τ | + 2 [Hi j,y − Hi j,x ] + εi |ei (k)|2 + 2(1 +

bi )eTi (k)[ fy(k) i ,τ

f x(k) ] i ,τ

−

+

2eTi (k)

l X

[Hi(k) j,y

j=1



Hi(k) j,x ]

−

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j=1

l  X 2 2 2 2 2 2 B2i j (|ei (k)| + |e j (k)|)2 + εi |ei (k)|2 ≤E 4(1 + bi ) γi |ei (k)| + 4(1 + b1 ) ξi |ei (k − τ(k))| + 2l j=1

2

2

j=1

 Bi j (|ei (k)| + |e j (k)|)

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− 2(1 + bi )αi |ei (k)| + 2(1 + bi )βi |ei (k − τ(k))| + 2|ei (k)|

l X

(16)

l   X  ≤ − 2(1 + bi )αi + 4(1 + bi )2 γi2 + εi + 8lB2i j + 4Bi j E|ei (k)|2 j=1

l X n o   + 4(1 + bi )2 ξi2 + 2(1 + bi )βi E|ei (k − τ(k))|2 + (4lB2i j + Bi j )E |e j (k)|2 − |ei (k)|2 . j=1

k X

|ei (s)|2 −

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   E ∆v(i) (e (k)) =λ E i i 2

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(i) Similarly, calculating the difference of v(i) 2 (ei (k)) and v3 (ei (k)) respectively and taking the mathematical expectation yield

s=k+1−τ(k+1)

k−1 X

s=k−τ(k)

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 =λi E |ei (k)|2 − |ei (k − τ(k))|2 −

 n (i) o E ∆v3 (ei (k)) =λi E

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and

CE

 ≤λi E |ei (k)|2 − |ei (k − τ(k))|2 + k+1−τ Xm

k X

j=k+2−τ M s= j

|ei (s)| − 2

=λi (τ M − τm )E|ei (k)| + λi     =λi E  (τ − τm )|ei (k)|2 −   M

k−1 X

k−τ Xm

k+1−τ Xm

k−1 X

k−1 X

j=k+2−τ M s= j

s=k−τ M +1

8

|ei (s)|2 +

(|ei (s)|)2

s=k+1−τ M

j=k−τ M +1 s= j

k−τ Xm



s=k+1−τ(k)

k−τ Xm

2

|ei (s)|2

2

|ei (s)|





2

E|ei (s)| − λi

    |ei (s)|2  .  

k−1 X

s=k+1−τ(k+1)

k−τ Xm

|ei (s)|2

k−1 X

j=k−τ M +1 s= j



E|ei (s)|2

(17)

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Combining (16), (17) and (18) gives l    X  E {∆vi (ei (k))} ≤ − 2(1 + bi )αi + 4(1 + bi )2 γi2 + εi + 8lB2i j + 4Bi j + λi (τ M − τm + 1) E|ei (k)|2 j=1

l X   ai j Fi j (ei (k), e j (k)), + 4(1 + bi )2 ξi2 + 2(1 + bi )βi − λi E|ei (k − τ(k))|2 + j=1

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where ai j = 4lB2i j + Bi j and Fi j (ei (k), e j (k)) = E|e j (k)|2 − E|ei (k)|2 . It shows that (7) holds. Making use of Remark 1, we can draw a conclusion that among each directed cycle CQ of the weighted digraph (G, A), it holds X  X  Fi j (ei (k), e j (k)) = E|e j (k)|2 − E|ei (k)|2 ≤ 0. (i, j)∈E(CQ )

(i, j)∈E(CQ )

4. An application to a coupled oscillators system

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By now, all the conditions in Theorem 1 have been verified. According to Theorem 1, we can get the desired results.

In this section, we consider a coupled oscillators system on a weighted digraph (G, A) with l ≥ 2 vertices and A = (ai j )l×l where ai j ≥ 0. Each independent vertex is assigned by the following second order oscillator

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x¨i (t) + ζi x˙i (t) + xi (t) + ηi xi (t − τ(t)) = Pi (t), t ≥ 0, i ∈ L,

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where ζi ≥ 0 is damping coefficient, ηi > 0, and τ(t) stands for time-varying delay satisfying τm ≤ τ(t) ≤ τ M , where τm , τ M are known real scalar constants. Suppose the influence from vertex j on vertex i is ai j ( x˙i (t) − x˙ j (t)) + ai j (θi xi (t) − θ j x j (t)), where θi , θ j are known constants. Thus, we obtain the following systems

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x¨i (t) + ζi x˙i (t) + xi (t) + ηi xi (t − τ(t)) +

l X j=1

ai j ( x˙i (t) − x˙ j (t)) +

l X j=1

ai j (θi xi (t) − θ j x j (t)) = Pi (t), i ∈ L.

(19)

Making a transformation x˜i (t) = x˙i (t) + θi xi (t), then system (19) can be rewritten as

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x˙i (t) = x˜i (t) − θi xi (t),

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x˙˜i (t) = (ζi θi − θi2 − 1)xi (t) + (θi − ζi ) x˜i (t) − ηi xi (t − τ(t)) + Pi (t) −

l X j=1

ai j ( x˜i (t) − x˜ j (t)),

(20)

where i ∈ L. Then we can get the following discrete-time coupled oscillators by using Euler method to system (20). xi (k + 1) =xi (k) + h( x˜i (k) − θi xi (k)), x˜i (k + 1) = x˜i (k) + h(ζi θi − θi2 − 1)xi (k) + h(θi − ζi ) x˜i (k) − hηi xi (k − τ(k)) l X −h ai j ( x˜i (k) − x˜ j (k)) + hPi (k), j=1

9

(21)

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in which i ∈ L, h > 0 is the time step size, xi (k) is the numerical approximations to xi (kh). We regard system (21) as drive system and the response system can be described as yi (k + 1) =yi (k) + h(˜yi (k) − θi yi (k)) + hu(1) i (k) + di (yi (k) − xi (k))ω(k),

y˜ i (k + 1) =˜yi (k) + h(ζi θi − θi2 − 1)yi (k) + h(θi − ζi )˜yi (k) − hηi yi (k − τ(k)) + hu(2) i (k) + hPi (k) l X ai j (˜yi (k) − y˜ j (k)) + pi (˜yi (k) − x˜i (k))ω(k), −h

(22)

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j=1

(2) where i ∈ L, di , pi > 0 and u(1) i , ui are the controllers. Let Xi (k) = (xi (k), x˜i (k))T , Hi j (Xi (k), X j (k)) = (0, −hai j ( x˜i (k) − x˜ j (k)))T , fi (Xi (k), Xi (k − τ(k))) = ( fi1 (k), fi2 (k))T , where fi1 (k) = h( x˜i (k) − θi xi (k)) and fi2 (k) = h(ζi θi − θi2 − 1)xi (k) + h(θi − ζi ) x˜i (k) − hηi xi (k − τ(k) + hPi (k)) and !   u(1) (k) i ui (k) = = bi fi (Yi (k), Yi (k − τ(k))) − fi (Xi (k), Xi (k − τ(k))) , (2) ui (k)

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where bi , −1 is the gain constant to be designed. Then the drive system (21) can be rewritten as Xi (k + 1) = Xi (k) + fi (Xi (k), Xi (k − τ(k))) +

l X

Hi j (Xi (k), X j (k)).

(23)

Hi j (Yi (k), Y j (k)) + ui (k) + δi (Yi (k) − Xi (k))ω(k),

(24)

j=1

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Similarly, the response system (22) is recast as

l X

Yi (k + 1) =Yi (k) + fi (Yi (k), Yi (k − τ(k))) +

! di (yi (k) − xi (k)) . pi (˜yi (k) − x˜i (k))

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where

j=1

δi (Yi (k) − Xi (k)) =

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Theorem 3. Suppose that digraph (G, A) is strongly connected and for each i ∈ L, it holds that 0 ≤ θi (ζi −θi ) ≤ 1, 4(1 + bi )2 h2 η2i + 2(1 + bi )βi < λi and

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bi )2 h2 ζ˜i2

(1 +

l X + εi + (8lh2 a2i j + 4hai j ) + λi (τ M − τm + 1) < 2(1 + bi )αi , j=1

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q n o hη2 in which ζ˜i = Re(ζi + ζi2 − 4), αi = min hθ2 i , 14 h(ζi − θi ) , βi = ζi −θi i and εi = max{di2 , p2i }. Then the drive system (23) and the response system (24) are globally exponentially synchronized. Proof. Denote Ei (k) , (ei (k), e˜ i (k))T , where ei (k) = yi (k) − xi (k) and e˜ i (k) = y˜ i (k) − x˜i (k). Obviously, Ei (k) = Yi (k) − Xi (k). Thus, we get the following error system   Ei (k + 1) =Ei (k) + (1 + bi ) fi (Yi (k), Yi (k − τ(k))) − fi (Xi (k), Xi (k − τ(k))) + δi (Ei (k))ω(k) l X (25) + [Hi j (Yi (k), Y j (k)) − Hi j (Xi (k), X j (k))]. j=1

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Next, we will verify that the error system (25) satisfies the conditions in Theorem 2. At first, it is straightforward to show that the error system (25) satisfies (5). Moreover, according to |˜ei (k)| ≤ |Ei (k)|, it is easy to see H (Y (k), Y (k)) − H (X (k), X (k)) ≤ ha (|˜e (k)| + |˜e (k)|) ≤ ha (|E (k)| + |E (k)|). ij

i

j

ij

i

j

ij

i

j

ij

i

j

Then calculating yields

EiT (k)[ fi (Yi (k), Yi (k − τ(k))) − fi (Xi (k), Xi (k − τ(k)))]

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| fi (Yi (k), Yi (k − τ(k))) − fi (Xi (k), Xi (k − τ(k)))| = |Ai Ei (k) + Bi Ei (k − τ(k))| ≤ kAi k|Ei (k)| + |Bi ||Ei (k − τ(k))|, ! ! hηi −hθi −h and Bi = . Then we have kAi k = h2 ζ˜i and |Bi | = hηi . This implies where Ai = 0 hζi θi − hθi2 − h hθi − hζi that (4) holds. As for condition B1, we have ≤h(ζi θi − θi2 )ei (k)˜ei (k) + hηi |ei (k − τ(k))|˜ei (k)| − hθi |ei (k)|2 − h(ζi − θi )|˜ei (k)|2 .

By using inequalities

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θi |ei (k)|2 |˜ei (k)|2 + , 2 2θi (ζi − θi )|˜ei (k)|2 ηi |ei (k − τ(k))|2 ei (k − τ(k))˜ei (k) ≤ + 4ηi ζi − θi and 0 ≤ θi (ζi − θi ) ≤ 1, we have ei (k)˜ei (k) ≤

hη2i h(ζi − γi ) hθi h(ζi − θi ) (ζi θi − θi2 )|ei (k)|2 + |˜ei (k)|2 + |˜ei (k)|2 + |ei (k − τ(k))|2 2 2 4 ζi − θi − hθi |ei (k)|2 − h(ζi − θi )|˜ei (k)|2

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≤

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EiT (k)[ fi (Yi (k), Yi (k − τ(k))) − fi (Xi (k), Xi (k − τ(k)))]

hη2i 1 hθi |ei (k)|2 − h(ζi − γi )|˜ei (k)|2 + |ei (k − τ(k))|2 2 4 ζi − θi ≤ − αi |Ei (k)|2 + βi Ei2 (k − τ(k)).

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≤−

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Then it is easy to see that the condition B1 in Theorem 2 has been satisfied. And condition B2 follows naturally. According to the Theorem 2, the drive system (23) and the response system (24) are globally exponentially synchronized. This completes the proof.

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5. Numerical example

In this section, we will give a numerical example to show the feasibility and effectiveness of our results above. At first, we consider a weighted digraph with 6 vertices (see Figure 1) and the weighted matrix is shown as follows.   0 0.07 0 0 0   0   0.04 0 0 0 0.04 0    0 0 0 0 0 0.07   A = (ai j )6×6 =   0 0   0 0.08 0.02 0   0 0 0 0.05 0 0.01    0 0 0 0.05 0 0 11

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a25 5

a54

a42 a21

4 a64

6

1

a13 a43

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a56

2

3

a36

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Figure 1: A coupled oscillators system with 6 oscillators

Obviously, the digraph (G, A) is strongly connected. k √ Then, take Pi (k) = 32×cos and h = 0.01. Let τ(k) = d4| sin k|e + 1, one can get τ M = 5 and τm = 1. Choose 4 k values for coefficients θi , ζi , ηi , εi , ϕi and λi shown in Table 1. Table 1: The value of parameters θi , ζi , ηi , di , pi bi and λi , i = 1, 2, . . . , 6.

θi

1 2 3 4 5 6

0.8 2 0.9 2 0.7 2.1 1 2 0.9 2 1.1 1.9

ηi

di

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pi

bi

0.1 0.012 0.015 -0.2 0.05 0.014 0.01 -0.2 0.1 0.02 0.015 -0.1 0.11 0.01 0.01 -0.2 0.12 0.017 0.013 -0.3 0.1 0.01 0.01 -0.3

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ζi

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i

λi

0.0002 0.0001 0.0002 0.0002 0.0002 0.0002

and

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P For the sake of simplicity, introduce Λi = (1 + bi )2 h2 ζ˜i2 + εi + lj=1 (8lh2 a2i j + 4hai j ) + λi (τ M − τm + 1) and Ξi = 4(1 + bi )2 h2 η2i + 2(1 + bi )βi . By direct calculation, it yields the values of θi (ζi − θi ), Ξi , Λi and 2(1 + bi )αi in Table 2. It is easy to see that the following inequalities hold for i = 1, 2, . . . , 6, 0 ≤ θi (ζi − θi ) ≤ 1,

(1 + bi )2 h2 ζ˜i2 + εi +

l X j=1

4(1 + bi )2 h2 η2i + 2(1 + bi )βi ≤ λi

(8lh2 a2i j + 4hai j ) + λi (τ M − τm + 1) < 2(1 + bi )αi .

This implies that all conditions in Theorem 3 hold. Therefore we can conclude that the drive system (23) and the response system (24) are globally exponentially synchronized. 12

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Table 2: The values of θi (ζi − θi ), 3ξi + 2βi − λi and Λi for i = 1, 2, . . . , 6.

1 2 3 4 5 6

θi (ζi − θi ) 0.96 0.99 0.98 1 0.99 0.88

Ξi

Λi

2(1 + bi )αi

1.35 × 10−4 0.37 × 10−4 1.32 × 10−4 1.97 × 10−4 1.86 × 10−4 1.77 × 10−4

0.0028 0.0027 0.0027 0.0025 0.0026 0.0026

0.0048 0.0044 0.0063 0.004 0.0039 0.0039

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i

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Then taking initial values in Table 3, we can get the simulation results for drive system, response system and synchronization error system shown in Figures 2-4, respectively. In Figure 2, the mean square trajectories of the drive system are given, and plot values of E(|xi (t)|2 ) and E(| x˜i (t)|2 ), i = 1, 2, . . . , 6, on the vertical Yaxis against time on the horizontal X-axis. In Figure 3, the mean square trajectories of the response system are given. The horizontal X-axis represents time, and the vertical Y-axis stands for values of E(|yi (t)|2 ) and E(|˜yi (t)|2 ), i = 1, 2, . . . , 6. In Figure 4, the mean square trajectories of the error system are given. Also, we plot on the X-axis time, and plot on the Y-axis values of E(|ei (t)|2 ) and E(|˜ei (t)|2 ), i = 1, 2, . . . , 6. We can conclude from these figures that the error state goes to zero in a short time. This means that the drive system and the response system are exponentially synchronized in mean square, which illustrate the effective of our theoretical results. Table 3: The initial values (k = −5, −4, . . . , 0).

−11.03 cos k −3.27 sin −6.66 cos k −12.01 sin k −16.01 cos k 6.68 cos k

x˜i (k)

y˜ i (k)

16.31 sin2 k 7.32 cos2 k 7.31 sin2 14.28 cos2 4.01 sin2 k −4.31 sin k

−3.2 cos2 k −8.88 sin2 k −3.17 cos2 k −16.31 sin2 k −4.27 cos2 k −8.86 sin2 k

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1 7.7 sin k 2 4.49 cos k 3 12.31 sin k 4 16.11 cos k 5 11.04 sin k 6 6.07 cos k

yi (k)

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xi (k)

i

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1500

E|x1 (t)|2 E|˜ x1 (t)|2 E|x2 (t)|2 E|˜ x2 (t)|2 E|x3 (t)|2 E|˜ x3 (t)|2 E|x4 (t)|2 E|˜ x4 (t)|2 E|x5 (t)|2 E|˜ x5 (t)|2 E|x6 (t)|2 E|˜ x6 (t)|2

1000

500

0

0

50

100

Figure 2: The dynamical behavior of the drive system.

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1500 E|y1 (t)|2 E|˜ y1 (t)|2 E|y2 (t)|2 E|˜ y2 (t)|2 E|y3 (t)|2 E|˜ y3 (t)|2 E|y4 (t)|2 E|˜ y4 (t)|2 E|y5 (t)|2 E|˜ y5 (t)|2 E|y6 (t)|2 E|˜ y6 (t)|2

1000

0

0

50

100

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150

Figure 3: The dynamical behavior of the response system.

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100

80

60

40

0

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20

0

50

100

E|e1 (t)|2 E|˜ e1 (t)|2 E|e2 (t)|2 E|˜ e2 (t)|2 E|e3 (t)|2 E|˜ e3 (t)|2 E|e4 (t)|2 E|˜ e4 (t)|2 E|e5 (t)|2 E|˜ e5 (t)|2 E|e6 (t)|2 E|˜ e6 (t)|2

150

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Figure 4: The dynamical behavior of the error system.

6. Conclusion

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In this paper, we investigate the problem of exponential synchronization of discrete-time stochastic coupled systems with time-varying delay. Nowadays, many researchers have devoted themselves to this kind of problem by the aid of linear matrix inequality, while we employing the Lyapunov method combined with Kirchhoff’s matrix tree theorem in graph theory as well as stochastic analysis technique. Two different sufficient criteria are given to guarantee two identical delayed coupled systems with stochastic disturbances to be exponentially synchronized in mean square, which are easily verified and do not need to solve any linear matrix inequalities. Finally, we apply our theoretical results to a coupled oscillators system and a numerical example related to the application is given. Our future work will concentrate on taking both probabilistic time-varying delay and Markovian switching into account. Acknowlegments

The authors sincerely thanks the associated editor and reviewers for their valuable comments and suggestions. This work was supported by the NNSF of China (Nos. 11301112 and 11401136), the NSF of Shandong Province (Nos.ZR2013AQ003 and ZR2014AQ010), China Postdoctoral Science Foundation funded project (No.2014T70313) and HIT. IBRSEM. A. 2014014. 14

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References [1] Y. Tang, Z. Wang, W. Wong, J. Kurths, J. Fang, Multiobjective synchronization of coupled systems, Chaos 21 (2011) 025114. [2] G. Barlev, M. Girvan, E. Ott, Map model for synchronization of systems of many coupled oscillators, Chaos 20 (2010) 023109.

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[3] P. Shi, Y. Zhang, M. Chadli, R. Agarwal, Mixed H-infinity and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays, IEEE Trans. Neural Netw. Learn. Syst. 27(4) (2016) 903-909. [4] K. Balasundarama, R. Rajab, Q. Zhu, S. Chandrasekarane, H. Zhou, New global asymptotic stability of discrete-time recurrent neural networks with multiple time-varying delays in the leakage term and impulsive effects, Neurocomputing, 214 (2016) 420-429.

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[5] X. Yang, J. Lu, Finite-time synchronization of coupled networks with markovian topology and impulsive effects, IEEE Trans. Autom. Control 61 (2016) 2256-2261. [6] X. Liu, J. Cao, W. Yu, Nonsmooth finite-time synchronization of switched coupled neural networks, IEEE T. Cybern. 46 (2016) 2360-2371. [7] M. Liu, C. Bai, A remark on stochastic logistic model with diffusion, Appl. Math. Comput. 228 (2014) 141-146.

M

[8] M. Liu, M. Deng, B. Du, Analysis of a stochastic logistic model with diffusion, Appl. Math. Comput. 266 (2015) 169-182.

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[9] Y. Guo, S. Liu, X. Ding, The existence of periodic solutions for coupled rayleigh system, Neurocomputing 191 (2016) 398-408.

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[10] J. Qiu, K. Lu, P Shi, S. Mahmoud, Robust exponential stability for discrete-time interval BAM neural networks with delays and Markovian jump parameters, Int. J. Adapt. Control Signal Process. 24 (2010) 760-785.

CE

[11] W. Li, S. Wang, H. Su, K. Wang, Global exponential stability for stochastic networks of coupled oscillators with variable delay, Commun. Nonlinear Sci. Numer. Simul. 22 (2015) 877-888.

AC

[12] T. Chen, R. Wang, B. Wu, Synchronization of multi-group coupled systems on networks with reaction diffusion terms based on the graph-theoretic approach, Neurocomputing 227 (2017) 54-63. [13] M. Forti, M-matrices and global convergence of discontinuous neural networks, Int. J. Circuit Theory Appl. 35(2) (2007) 105-130. [14] H. Su, W. Li, K. Wang, Global stability analysis of discrete-time coupled systems on networks and its applications, Chaos 22 (2012) 033135. [15] H. Su, P. Wang, X. Ding, Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application, Discrete Contin. Dyn. Syst.-Ser. B 21 (2016) 253-269. 15

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[16] X. Su, P. Shi, L. Wu, Y. Song, A novel approach to filter design for TCS fuzzy discrete-time systems with time-varying delay, IEEE Trans. Fuzzy Syst. 20(6) (2012) 1114-1129. [17] H. Jiang, L. Zhang, Z. Teng, Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. 16(6) (2005) 1340-1351.

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[18] H. Li, H. Jiang, C. Hu, Existence and global exponential stability of periodic solution of memristor-based BAM neural networks with time-varying delays, Neural Netw. 75 (2016) 97-109. [19] J. Wang, H. Jiang, C. Hu, Existence and stability of periodic solutions of discrete-time Cohen-Grossberg neural networks with delays and impulses, Neurocomputing 142 (2014) 542-550. [20] Z. Wang, Y. Wang, Y. Liu, Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays, IEEE Trans. Neural Netw. Learn. Syst. 21(1) (2010) 11-25.

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[21] W. He, F. Qian, J. Lam, G. Chen, Q. Han, J. Kurths, Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design, Automatica, 62 (2015), 249-262. [22] H. Liu, Z. Wang, B. Shen F. Alsaadi, H-infinity state estimation for discrete-time memristive recurrent neural networks with stochastic time-delays, Int. J. Gen. Syst. 45(5) (2016) 633-647.

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[23] W. Chen, S. Luo, W. Zheng, Impulsive synchronization of reaction-diffusion neural networks with mixed delays and its application to image encryption, IEEE Trans. Neural Netw. Learn. Syst. 27(12) (2016) 2696-2710.

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[24] X. Yang, Z. Feng, J. Feng, J. Cao, Synchronization of discrete-time neural networks with delays and Markov jump topologies based on tracker information, Neural Netw. 85 (2017) 157-164.

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[25] H. Chen, J. Liang, T. Huang, J. Cao, Synchronization of arbitrarily switched boolean networks, IEEE Trans. Neural Netw. Learn. Syst. 28 (2017) 612-619.

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[26] Q. Zhu, J. Cao, R. Rakkiyappan, Exponential input-to-state stability of stochastic Cohen-Grossberg neural networks with mixed delays, Nonlinear Dyn. 79(2) (2015) 1085-1098.

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[27] Q. Zhu, R. Rakkiyappan, A. Chandrasekar, Stochastic stability of Markovian jump BAM neural networks with leakage delays and impulse control, Neurocomputing 136 (2014) 136-151. [28] S. Lakshmanan, K. Mathiyalagan, J.H. Park, R. Sakthivel, F.A. Rihan, Delay-dependent H∞ state estimation of neural networks with mixed time-varying delays, Neurocomputing, 129 (2014) 392-400. [29] L. Shi, H. Zhu, S. Zhong, Y. Zeng, J. Chen, Synchronization for time-varying complex networks based on control, J. Comput. Appl. Math. 301 (2016) 178-187. [30] M. Liu, P. Mandal, Dynamical behavior of a one-prey two-predator model with random perturbations, Commun. Nonlinear Sci. Numer. Simul. 28 (2015) 123-137. 16

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[31] W. Li, H. Su, K. Wang, Global stability analysis for stochastic coupled systems on networks, Automatica 47 (2011) 215-220. [32] J. Liang, Z. Wang, X. Liu, Exponential synchronization of stochastic delayed discrete-time complex networks, Nonlinear Dyn. 53 (2008) 153-165.

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[33] R. Rakkiyappan, N. Sakthivel, J. Cao, Stochastic sampled-data control for synchronization of complex dynamical networks with control packet loss and additive time-varying delays, Neural Netw. 66 (2015) 46-63. [34] X. Liu, J.H. Park, N. Jiang, J. Cao, Nonsmooth finite-time stabilization of neural networks with discontinuous activations, Neural Netw. 52 (2014) 25-32. [35] M.Y. Li , Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ. 248 (2010) 1-20.

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[36] Q. Zhu, J. Cao, pth moment exponential synchronization for stochastic delayed Cohen-Grossberg neural networks with Markovian switching, Nonlinear Dyn. 67 (2012) 829-845. [37] Z. Liu, S. L¨u, S. Zhong, M. Ye, pth moment exponential synchronization analysis for a class of stochastic neural networks with mixed delays, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 1899-1909. [38] C. Zhang, W. Li, K. Wang, Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching, Nonlinear Anal.-Hybrid Syst. 15 (2015) 37-51.

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[39] J. Suo, J. Sun, Y. Zhang, Stability analysis for impulsive coupled systems on networks, Neurocomputing 99 (2013) 172-177.

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Pengfei Wang was born in 1992. He received his B.S. degree from Harbin Institute of Technology at Weihai, China, in 2016. He is currently a M.S. Student in Harbin Institute of Technology, China. His current research interests include stability theory for stochastic differential equations and difference equations.

Zhangrui Chen was born in 1996. He is currently an undergraduate student in the Department of Mathematics, HarbinInstitute of Technology, China. His current research interests include dynamic behaviors of discrete-time coupled systems.

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Wenxue Li was born in 1981. He received his Ph.D. degree from Harbin Institute of Technology, China, in 2009. He is currently an associate professor in Harbin Institute of Technology at Weihai. He current research interests include stability theory for stochastic differential and integral equations.

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