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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 8, AUGUST 2015

Graph Theory-Based Approach for Stability Analysis of Stochastic Coupled Systems With Lévy Noise on Networks Chunmei Zhang, Wenxue Li, and Ke Wang

Abstract— In this paper, a novel class of stochastic coupled systems with Lévy noise on networks (SCSLNNs) is presented. Both white noise and Lévy noise are considered in the networks. By exploiting graph theory and Lyapunov stability theory, criteria ensuring pth moment exponential stability and stability in probability of these SCSLNNs are established, respectively. These principles are closely related to the topology of the network and the perturbation intensity of white noise and Lévy noise. Moreover, to verify the theoretical results, stochastic coupled oscillators with Lévy noise on a network and stochastic Volterra predator–prey system with Lévy noise are performed. Finally, a numerical example about oscillators’ network is provided to illustrate the feasibility of our analytical results. Index Terms— Lévy noise, networks, stability, stochastic coupled systems.

I. I NTRODUCTION

R

ECENTLY, coupled systems on networks (CSNs) have attracted a great deal of attention in physics, biology, neural networks, engineering fields, and so on [1]–[6]. It is well-known that stability analysis plays a key role in the applications of CSNs. An interesting question in some existing papers is how these stability criteria are related to the topological property of the network [7], [8]. Li and Shuai [8] developed a systematic method to study the global stability for CSNs by using results from graph theory. Their results

Manuscript received March 12, 2013; revised March 4, 2014; accepted August 19, 2014. Date of publication September 9, 2014; date of current version July 15, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 11301112, Grant 11171081, Grant 11171056, and Grant 11301207, in part by the Natural Science Foundation of Shandong Province under Grant ZR2013AQ003, in part by the China Post-Doctoral Science Foundation under Grant 2013M541352 and Grant 2014T70313, in part by the Fundamental Research Funds for the Central Universities under Grant HIT.IBRSEM.A.2014014, in part by the Key Project of Science and Technology of Weihai under Grant 2013DXGJ04, in part by the China Scholarship Council, in part by the Natural Science Foundation of Jiangsu Province under Grant BK20130411, and in part by the Natural Science Research Project of Ordinary Universities in Jiangsu Province under Grant 13KJB110002. C. Zhang is with the Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China, and also with the Department of Mathematics, University of Illinois at Urbana-Champaign, Champaign, IL 61801 USA (e-mail: [email protected]). W. Li is with the Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China (e-mail: [email protected]). K. Wang is with the Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China, and also with the School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2352217

have been applied to several classes of CSNs, such as coupled oscillators on a network, an n-patch predator–prey model, and a multigroup epidemic model with nonlinear incidence. Motivated by this pioneering work, then in [9]–[15], some stability problems for feedback control systems on networks, delayed CSNs, discrete-time CSNs, and impulsive CSNs were successfully investigated. In concrete applications, CSNs are inevitably subject to some kinds of environmental noise, such as white noise and Markovian jumps [16], [17]. Therefore, it is crucial to study the effects of noise perturbation on the stability of CSNs [18], [19]. By combining graph theory and stochastic stability theory, Li et al. [20], [22] and Ji et al. [21] have studied the stability of stochastic CSNs (SCSNs) and the stability of SCSNs with Markovian switching. However, there exists another different kind of noise in many coupled systems in science and engineering that is Lévy noise. It should be also noted that Lévy noise arises as a fluctuating driving process with heavy tails and jumps. Hence, it can better model the very rare yet extreme sudden event, such as earthquakes and epidemics, while deterministic systems or stochastic systems driven by white noise or Markovian jumps mentioned above cannot explain the sudden shocks. Such heavy-tailed fluctuations are abundant in neural networks, climate dynamics, financial time series, disease spreading, population dynamics problems, and so on [23], [24]. Hence, motivated to some extent by the work, such as [25] and [26], we can introduce stochastic systems driven by Lévy noise to better understand the sudden shocks phenomenon. Stochastic differential equations with Lévy noise have been widely applied in many fields, especially in economics and engineering [27], [28]. Stability analysis has been a focal subject for research, see for instance [29]–[34] and the references therein although the list is not the most inclusive. However, to the best of our knowledge, there exist few results so far about the stability for networks of stochastic coupled systems with Lévy noise based on the graph-theoretic approach [8], [20], [21]. Inspired by the above analysis, we can consider reasonably many large-scale dynamical systems in science and engineering as systems driven by Lévy noise on some directed graph, which can be called as stochastic coupled systems with Lévy noise on networks (SCSLNNs). Naturally, it is valuable to develop the stability theory for SCSLNNs. Hence, this paper is mainly concerned with the issues of exponential stability and stochastic stability of SCSLNNs by

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ZHANG et al.: GRAPH THEORY-BASED APPROACH FOR STABILITY ANALYSIS OF SCSLNNs

graph-theoretic method. The question, which we address in our present study, is what would be the action of the topological property of directed graph and the two kinds of noise, white noise and Lévy noise, on the stability of coupled networks. Compared with the existing results in the literature, especially in [20] and [22], contributions of this paper are threefold. 1) Lévy noise is introduced into stochastic coupled networks to describe vary rare yet extreme sudden environmental noise in the real applications. The implicit range for the noise intensity is also given, under which coupled systems can still be stable. 2) The issue about pth moment exponential stability of SCSLNNs is studied, which can show that the decay of pth moment of the solution is exponential under the perturbation of white noise and Lévy noise. 3) The results are applied to stochastic coupled oscillators with Lévy noise on a network and stochastic Volterra predator–prey system with Lévy noise. Numerical results are also provided to show the stability of coupled oscillators driven by Lévy noise. II. P RELIMINARIES AND M ODEL D ESCRIPTION In this section, we first introduce some basic notations and characteristics of Lévy process. Basic concepts on graph theory in this paper can be found in the Appendix. Then, we show how SCSLNNs could be constructed and finally give three kinds of stability definitions and an important lemma in graph theory. A. Lévy Process As usual, (, F , F, P) denotes a complete probability space with a filtration F = {Ft }t ≥0 satisfying the usual conditions (the filtration is right continuous and F0 contains all P-null sets), and W (·) is a scalar standard Brownian motion defined on (, F , F, P). Let E(·) be the mathematical expectation with respect to the given probability measure P and | · | be the Euclidean norm for vectors or the  trace norm for matrices. Throughout this paper, we set m = ni=1 m i for m i ∈ Z+ = m {1, 2, . . .} and Rm + = {x ∈ R : x i > 0, i = 1, 2 . . . , m}. Moreover, sgn(x) is the sign function of x and δr (a) = {x : |x − a| < r } for some r, a > 0. Let us now recall some fundamental characteristics of Lévy process associated with a given family of infinitely divisible distributions. For details, we refer [25] and [36]. Let X t , t ∈ T = [0, T ] (0 < T < ∞) be an R-valued stochastic process on the probability space (, F , P). It is called a Lévy process if the following three properties are satisfied. 1) X 0 = 0 a.s. Sample paths are right continuous with left-hand limits a.s. 2) Let X t − = lim↓0 X t − . Then, X t = X t − holds a.s. for any t. 3) It has independent increments, i.e., for any 0 ≤ t0 < t1 < · · · < tn ≤ T , the random variables X ti − X ti−1 , i = 1, . . . , n are independent.

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It has been shown that any Lévy process consists of two components: a Brownian motion and a Poisson random measure. The law of a Lévy process X t is infinitely divisible and its characteristic function E[ei(α,X t ) ] is given by the Lévy–Khintchine formula  1 exp i (α, b(t)) − (α, A(t)α) 2  t + (ei(α,σ ) − 1)n(dsdσ ) 0

+

|σ |>1

 t 0

0 0, there exist positive constants c and δ, such that for some p > 0, it holds that Fig. 1. Architecture for a directed graph and a network of stochastic coupled systems formed by two vertex systems.

a SCSLNN on graph G as follows: ⎤ ⎡ ⎤ ⎡ n n dx i (t) = ⎣ f i + Hi j ⎦ dt + ⎣gi + Ni j ⎦ dW (t) ⎡

 +

Y

j =1

n ⎣γi + Ii j⎦ N˜ (dt, dσ ), i = 1, 2, . . . , n. (2) j =1

Write C 2,1 (Rm × R1+ ; R1+ ) for the family of all nonnegative functions V (x, t) on Rm × R1+ that are continuously twice differentiable in x and once in t. If Vi ∈ C 2,1 (Rm i × R1+ ; R1+ ), we define an operator LVi from Rm i × R1+ to R with respect to the i th equation of (2) by LVi (x i (t), t)

⎛ ⎞ n ∂ Vi (x i (t), t) ∂ Vi (x i (t), t) ⎝ + Hi j ⎠ = fi + ∂t ∂ xi j =1 ⎧⎡ ⎤T   ⎪ n 1 ⎨ ∂ 2 Vi (x i (t), t) + Tr ⎣gi + Ni j ⎦ (i) (i) 2 ⎪ ∂ x k ∂ xl ⎩ j =1 m i ×m i ⎫ ⎤ ⎡ ⎪ n ⎬ ⎦ ⎣ × gi + Ni j ⎪ ⎭ j =1 ⎛ ⎡ ⎛ ⎞ ⎞  n Ii j ⎠, t ⎠ + ⎣Vi ⎝ x i (t) + ⎝γi + Y

j =1

−Vi (x i (t), t)−

⎛

for any x 0 ∈ Rm satisfying |x 0 | < δ. Definition 3: The origin is said to be stochastically asymptotically stable if it is stochastically stable and, furthermore, for every ε ∈ (0, 1) and t0 ≥ 0, there exists a δ = δ(ε, t0 ) > 0 such that P( lim x(t; t0 , x 0 ) = 0) ≥ 1 − ε t →∞

Rm

for any x 0 ∈ satisfying |x 0 | < δ. Now, we give an important lemma in graph theory. Lemma 1 ([8, Th. 2.2]): Assume that n ≥ 2 and (G, A), A = (ai j )n×n , is a weighted digraph. Let Q be the set of all spanning unicyclic graphs Q of (G, A), CQ be the cycle of Q, W (Q) be the weight of Q, and ci be the cofactor of the i th diagonal element of the Laplacian matrix L for (G, A). Then for arbitrary functions Fi j (x i , x j ), 1 ≤ i, j ≤ n, it holds that n i, j =1

ci ai j Fi j (x i , x j ) =

Q∈Q

W (Q)



Frs (xr , x s ).

(s,r)∈E(C Q)

In addition, if (G, A) is strongly connected, then ci > 0 for i = 1, 2, . . . , n. III. S TABILITY A NALYSIS FOR SCSLNN S

⎞⎤

∂ Vi (x i (t), t) ⎝ γi + Ii j⎠⎦ λ(dσ ). ∂ xi n

for all t ≥ t0 and x 0 ∈ Rm . When p = 2, it is said to be exponentially stable in mean square. Definition 2: The origin is said to be stochastically stable if for every ε ∈ (0, 1), d > 0, and t0 ≥ 0, there exists a δ = δ(ε, d, t0 ) > 0 such that P(|x(t; t0 , x 0 )| < d, ∀t ≥ t0 ) ≥ 1 − ε

j =1

⎤

E|x 0 | p ≤ δ ⇒ E|x(t; t0 , x 0 )| p ≤ e−c(t −t0)

j =1

(3) For the aim of this paper, we assume that fi , gi , γi , Hi j , Ni j , and Ii j are such that initial-value problems associated with (1) and (2) have a unique solution. Assume also that f i (0, t) = gi (0, t) = γi (0, t, σ ) = 0 and Hi j (0, 0, t) = Ni j (0, 0, t) = Ii j (0, 0, t, σ ) = 0, which mean that x ≡ 0 is a trivial solution of (2). In system theory, this trivial

In this section, we establish two kinds of exponential stability principles for SCSLNNs with the help of graph theory and Lyapunov stability theory. One is given in the form of Lyapunov function, while the other is given with the coefficients of SCSLNNs, which is much more practicable. In addition, some new stochastic stability conditions for SCSLNNs are also provided. A. Lyapunov-Type Theorems Let us give two basic conditions first. A1. For any i ∈ {1, 2, . . . , n}, there exist positive constants αi , βi , and ςi , functions Vi (x i , t), Fi j (x i , x j , t),

ZHANG et al.: GRAPH THEORY-BASED APPROACH FOR STABILITY ANALYSIS OF SCSLNNs

and a matrix A = (ai j )n×n such that for any t ≥ t0 , it follows: αi |x i | p ≤ Vi (x i , t) ≤ βi |x i | p

(4)

and

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and Lemma 1 LV (x, t) =

LVi (x i , t) ≤ −ςi |x i | p +

ci LVi (x i , t)

i=1

≤ n

n

n

⎛ ci ⎝−ςi |x i | p +

i=1

ai j Fi j (x i , x j , t). (5)

j =1

A2. Along each directed cycle C of the weighted digraph (G, A), it holds that Fi j (x i , x j , t) ≤ 0, t ≥ t0 . (6) ( j,i)∈E(C )

=− =− ≤−

n i=1 n i=1 n

ci ςi |x i | p +

n

⎞ ai j Fi j (x i , x j , t)⎠

j =1 n

ci ai j Fi j (x i , x j , t)

i, j =1

ci ςi |x i | p +





W (Q)

Q∈Q

Fi j (x i , x j , t)

( j,i)∈E(C Q )

ci ςi |x i | p

(8)

i=1

It should be pointed out that condition A2 holds for each directed cycle C and any x i and x j . Since the number of directed cycles may be very large and x i , x j are arbitrary, we must check A2 for infinite times. It seems to be so complicated. However, this problem can be successfully solved if we find some appropriate functions Fi j , i, j = 1, 2, . . . , n. For example, if for every i = 1, 2, . . . , n, there exists a function Pi , such that Fi j (x i (t), x j (t), t) ≤ Pi (x i (t))− P j (x j (t)),

j = 1, 2, . . . , n. (7)

Then, we have Fi j (x i (t), x j (t), t) ( j,i)∈E(C )

≤



(Pi (x i (t)) − P j (x j (t))) = 0, t ≥ t0

( j,i)∈E(C )

which shows that A2 is satisfied. Theorem 1: Let (G, A) be strongly connected. If conditions A1 and A2 hold, then there are constants C and δ ∈ (0, ςi /βi ] such that E|x(t; t0 , x 0 )| p ≤ Ce−δt for some p > 0 and all t ≥ t0 . That is, the origin is pth moment exponentially stable for SCSLNN (2). Proof: Let V (x, t) =

n

ci Vi (x i , t)

i=1

where ci is defined as Lemma 1. Then, we can see from (4) that there exist α ∗ , β ∗ > 0 such that α ∗ |x| p ≤ V (x, t) ≤ β ∗ |x| p for all t ≥ t0 . In what follows, we fix any x 0 ∈ Rm and write x(t; t0 , x 0 ) as x(t) for simplicity. For fixed t > 0, we can compute LV (x, t) using the fact W (Q) > 0, together with condition A2

≤ −ς |x| p

(9)

whereς = min1≤i≤n {ci ςi }. Then, it is obvious to see that V = ni=1 ci Vi is an exponential Lyapunov function for the origin of (2), and therefore exponential p-stability follows. For the details, we refer readers to the classical book [37]. This completes our proof. Corollary 1: Theorem 1 holds if (6) is replaced by (7). Condition A1 is applicable to the exponential stability of coupled systems in some applications, which can show that the decay of pth moment of the solution is exponential. Next, we give some weaker conditions than A1, which can be used efficiently to obtain the stochastic stability of coupled systems. A3. For any i ∈ {1, 2, . . . , n}, there exists a continuous nondecreasing function κi (x) on [0, +∞) such that κi (0) = 0, κi (x) > 0 if x > 0. Suppose also that there are functions Vi (x i , t), Fi j (x i , x j , t), and a matrix A = (ai j )n×n such that (G, A) is strongly connected and Vi (0, t) = 0, Vi (x i , t) ≥ κi (|x i |) n LVi (x i , t) ≤ ai j Fi j (x i , x j , t). j =1

A4. For any i ∈ {1, 2, . . . , n}, there exists continuous nondecreasing functions κi (x) and μi (x) on [0, +∞) such that κi (0) = μi (0) = 0, κi (x) > 0, and μi (x) > 0 if x > 0. In addition, there exist functions Vi (x i , t) and Fi j (x i , x j , t), a matrix A = (ai j )n×n , and a positive constant bi such that (G, A) is strongly connected and κi (|x i |) ≤ Vi (x i , t) ≤ μi (|x i |) n ai j Fi j (x i , x j , t). LVi (x i , t) ≤ −bi Vi (x i , t) + j =1

Hence, we can deduce two new criteria about stochastic stability and stochastically asymptotic stability. Remark 1: Under the conditions A2 and A3, the origin is stochastically stable for SCSLNN (2). In addition, if conditions A2 and A4 are satisfied, then the origin is stochastically asymptotically stable for SCSLNN (2). Both of them can be obtained easily by the method in [20].

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Fig. 2.

Balanced digraph G with three vertices.

Note that the fact that if (G, A) is balanced, then n i, j =1

=

ci ai j Fi j (x i , x j , t) 1 W (Q) 2 Q∈Q





 Fi j (x i , x j , t) + F j i (x j , x i , t) .

( j,i)∈E(C Q )

In this case, A2 can be replaced by the following. A2 . Along each directed cycle C of the weighted digraph (G, A), it holds that   Fi j (x i , x j , t) + F j i (x j , x i , t) ≤ 0. ( j,i)∈E(C)

For better understanding the new condition A2 and its actual advantage than A2, we first give a digraph Fig. 2. From Fig. 2, we know that there are two directed cycles C1 and C2 , where E(C1 ) = {(1, 2), (2, 3), (3, 1)} and E(C2 ) = {(2, 1), (1, 3), (3, 2)}. Hence, for any functions satisfying Fi j (x i , x j , t) ≤ Pi j (x i , x j , t) − P j i (x j , x i , t)

(10)



≤

≤

[(Pi j (x i , x j , t) − P j i (x j , x i , t))].

B2. There exist constants Ai j ≥ 0 and Bi j > 0, such that |Hi j (x, y, t)|∨|Ni j (x, y, t)|∨|Ii j (x, y, t)| ≤ Ai j |x|+ Bi j |y|.

⎛

2(αi − ξi ) > λ¯ ⎝μi +

+(P j i (x j , x i , t) − Pi j (x i , x j , t))] k = 1, 2



x T f i (x, t) ≤ −αi |x|2 , |gi (x, t)|2 ≤ ξi |x|2 .

B4. It holds that

[(Pi j (x i , x j , t) − P j i (x j , x i , t))

n j =1

⎞ νi j ⎠ +

n

(K i j + L i j )

j =1

where

which means that A2 is satisfied. However, for k = 1, 2 Fi j (x i , x j , t) ( j,i)∈E(C k )

We first make some assumptions on coefficients of (2). The following conditions hold for every i = 1, 2, . . . , n, t ≥ t0 , and x ∈ Rm i , y ∈ Rm j , σ ∈ Y. B1. There are positive constants αi and ξi such that

|γi (x, t, σ )| ≤ ηi |x|.

( j,i)∈E(C k )

= 0,

B. Coefficients-Type Theorem

B3. There exists a positive number ηi such that

where i, j ∈ {1, 2, . . . , n}, t ≥ t0 , it holds that [Fi j (x i , x j , t) + F j i (x j , x i , t)] ( j,i)∈E(C k )

1) Theorem 1 still holds if A2 is replaced by A2 . 2) The origin is stochastically stable for SCSLNN (2) if A3 and A2 hold. 3) The origin is stochastically asymptotically stable for SCSLNN (2) if A4 and A2 hold. Obviously, (10) is a general case of (7). However, (7) is applicable for the coupled systems on strongly connected graph, while (10) can be only used for the coupled systems on strongly connected and balanced graph. Remark 3: Constructing an appropriate Lyapunov function for SCSLNNs plays a key role in the study of stability for SCSLNNs. Lyapunov function for (2) possesses the character that it can be easily constructed by exploiting individual Vi and some network properties. In particular, when γi = 0 and Ii j = 0 (i, j = 1, 2, . . . , n), global stability results for SCSNs have been given in [20]. In addition, when γi = gi = 0 and Ni j = Ii j = 0 (i, j = 1, 2, . . . , n), global stability problem for coupled systems of differential equations on networks have been fully discussed in [8]. These can fully show that the results obtained here generalize those given in the previous investigations to some extent. We remark that [8] is the pioneering work about the graph theory-based approach, and contains extensive applications of graph theory to coupled systems of differential equations on networks.

K i j = 2 Ai j + Bi j + 4n A2i j , L i j = Bi j + 4n Bi2j n n μi = ηi2 + 2 (ηi + n Ai j )Ai j + ηi Bi j (11)

( j,i)∈E(C k )

Clearly, summation (11) may be greater than zero, which means that A2 does not always stand up. All of these can clearly show the advantage of A2 . Remark 2: If (G, A) is strongly connected and balanced, it is straightforward to get a conclusion as follows.

j =1

j =1

and νi j = (ηi + 2n Bi j )Bi j . In what follows, we shall seek an easy-to-check criterion for exponential stability of (2) in mean square, which is given with the coefficients of SCSLNN (2). Theorem 2: Let conditions B1–B4 hold. Then, the origin is exponentially stable in mean square for (2).

ZHANG et al.: GRAPH THEORY-BASED APPROACH FOR STABILITY ANALYSIS OF SCSLNNs

Proof: Define Vi : Rm i → R1+ by Vi (x i ) = |x i |2 . Using the operator (3), we can compute LVi (x i ): LVi (x i ) 2 ⎡ ⎤    n n   T ⎣  = 2x i Hi j ⎦ +  g i + Ni j  fi +   j =1 j =1 ⎡ 2     n   ⎢ + ⎣ − |x i |2 + x i + γi + Ii j  Y   j =1 ⎛ ⎞⎤ n −2x iT ⎝γi + Ii j ⎠⎦ λ(dσ ) j =1

+ 2n

(Ai j |x i | + Bi j |x j |) + 2ξi x i2

2 A2i j x i2 + 2Bi2j x 2j



j =1

⎤  ⎛ ⎞2   n n  ⎥ ⎢ Ii j ⎠ + 2 γi Ii j ⎦ λ(dσ ) + ⎣γi2 + ⎝ Y  j =1  j =1 ⎡



≤

−2αi x i2

+2

n

j =1 n

+ 2ξi x i2 + 4n +

Y

n

⎣ηi2 x i2 + n

Bi j x i2

+

j =1

A2i j x i2 + 4n

j =1

⎡



+

Ai j x i2

n

Bi j x 2j

Bi2j x 2j

n  2 2  2 Ai j x i + 2Bi2j x 2j

Y

⎣ηi2 x i2 + 2n

n

A2i j x i2 + 2n

j =1

+ 2ηi

n j =1

Ai j x i2 +ηi

n

j =1

Bi j x i2 +ηi

L i j + λ¯ νi j



⎞⎤ νi j ⎠⎦ x i2

j =1

 2

x 2j − x i

j =1

 −ςi x i2 +

n

ai j Fi j (x i , x j , t)

j =1

n j =1

  1 log E|x(t)|2 ≤ −δ, for some δ > 0 t

lim sup t →∞

Bi2j x 2j

j =1 n

(K i j + L i j ) + λ¯ ⎝μi +

n

can imply that

j =1

+

+

t →∞

 Bi j + 4n Bi2j x 2j

⎡

j =1

j =1 n 

lim sup

j =1



⎛

 ¯ i j x 2j L i j + λν

Then

j =1

j =1

n 

= ⎣2(ξi − αi ) +

n

n 

 p  p E|x(t)| p = E (|x(t)|2 ) 2 < E|x(t)|2 2 .

 ⎤  n    +2ηi x i (Ai j |x i | + Bi j |x j |)⎦ λ(dσ )  j =1  ⎤ ⎡ n   2 Ai j + Bi j + 4n A2i j ⎦ x i2 ≤ ⎣2(ξi − αi ) + +

¯ i ⎦ x i2 + K i j + λμ

j =1

⎡

j =1 n

⎤

Obviously, the conclusion of this theorem follows immediately from Theorem 1. Remark 4: Conditions B2 and B3 are the conditions for the coupling strength and jump strength, respectively. B4 seems to be complex because our model is a network of coupled systems, where there are interactions among n vertex dynamical systems and both white noise and Lévy noise are considered. From B4, we can clearly see that if both the strength of coupling and noise are small, the stability of SCSLNNs can be achieved. In addition, the new stability criterion can be helpful and significant for us to design some stable coupled networks by adjusting the coupling matrix. Remark 5: When 0 < p < 2, by Jensen’s inequality, the following inequality holds:

j =1 n 

= ⎣2(ξi − αi ) +

n

( j,i)∈E(C Q )

j =1

≤ −2αi x i2 + 2|x i |

⎡

where ai j = L i j + λ¯ νi j > 0, Fi j (x i , x j , t) = x 2j − x i2 , and ςi > 0 by B4, i, j = 1, 2, . . . n. Hence, along each directed cycle C of weighted digraph (G, A) (A = (ai j )n×n ), it follows that: Fi j (x i , x j , t) = 0.

2    n   Hi j + 2|gi |2 + 2  Ni j  ≤ 2x iT f i + 2x iT   j =1 j =1 ⎡ 2    n  ⎢ + ⎣x i + γi + Ii j  − |x i |2 Y   j =1 ⎛ ⎞⎤ n ⎥ − 2x iT ⎝γi + Ii j ⎠ ⎦ λ(dσ ) n

n

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⎤ Bi j x 2j⎦ λ(dσ )

1 pδ log(E|x(t)| p ) ≤ − . t 2

In other words, under the same conditions with Theorem 2, the trivial solution of (2) is also pth moment exponentially stable when 0 < p < 2. Remark 6: Actually, Theorem 1 is shown in terms of Lyapunov functions and topological structure. Hence, it is necessary for us to find some appropriate functions to check the availability of Theorem 1, which is the task of Theorem 2. Theorem 2 clearly shows the detail how to find such functions Fi j to verify conditions (5) and (6) in Theorem 1. This further confirms that conditions (5) and (6) are available and practicable.

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IV. N ETWORK OF S TOCHASTIC C OUPLED O SCILLATORS W ITH L ÉVY N OISE A simple oscillator reads x(t) ¨ + α x˙ (t) + x(t) = 0 which can be rewritten as  dx(t) = (y(t) − ωx(t))dt dy(t) = [(−ω2 + αω − 1)x(t) + (ω − α)y(t)]dt

(12)

where α ≥ 0 is the damping coefficient and y(t) = x(t) ˙ + ωx(t), ω > 0. Oscillators models have been studied extensively [20], [38], [39]. We stochastically perturb (12) by white noise and Lévy noise into the stochastic oscillator ⎧ ⎨ dx(t) = (y(t) − ωx(t))dt + (ω − α)y(t)]dt (13) dy(t) = [(−ω2 + αω − 1)x(t)  ⎩ −βy(t)dW (t) + Y γ (x(t), σ ) N˜ (dt, dσ ). Here, W˙ (t) is scalar white noise, i.e., W (t) is a 1-D Brownian motion, which is independent of Poisson counting measure N. In this section, we concentrate on the exponential stability of stochastic coupled oscillators with Lévy noise on a network. Given a weighted digraph (G, A) with n (n ≥ 2) vertices and A = (ai j )n×n , a network of coupled oscillators with Lévy noise on G can be built like this: each vertex denotes a stochastic oscillator with Lévy noise described by (13) and the coupling from vertex j to vertex i is given by ai j (yi − y j ) [8], [20], [40]. Here, weight constants ai j ≥ 0, and ai j = 0 if and only if no arc exists from j to i in G. Hence, a network of stochastic coupled oscillators with Lévy noise is arrived as ⎧ dx i (t) = (yi (t) − ωi x i (t))dt, i = 1, 2, . . . , n, ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ − ωi2 + αi ωi − 1 x i (t) + (ωi − αi )yi (t) ⎪ dyi (t) = ⎨ ! n (14) ⎪ ⎪ − a (y (t) − y (t)) dt i j i j ⎪ ⎪ ⎪ ⎪ j =1 ⎪  ⎩ −βi yi (t)dW (t) + Y γi (x i (t), σ ) N˜ (dt, dσ ). Let X i = (x i , yi )T , G i (X i ) = (0, βi yi )T , i (X i , σ ) = (0, γi (x i , σ ))T , and Pi (X i ) = (yi −ωi x i , (−ωi2 +αi ωi −1)x i + (ωi − αi )yi − nj =1 ai j (yi − y j ))T . Then, coupled system (14) is given by d X i (t) = Pi (X  i (t))dt + G i (X i (t))dW (t) + i (X i (t), σ ) N˜ (dt, dσ ), i = 1, 2, . . . , n. (15) Y

We first make some assumptions on the jump strength of (15). The following conditions hold for every i = 1, 2, . . . , n, x i ∈ Rm i and σ ∈ Y. C1. γi (0, σ ) = 0. C2. There exists a positive constant ηi such that |γi (x i , σ )| ≤ ηi |x i |. C3. It holds that ¯ i2 < ωi βi2 + λη

(16)

where ωi satisfying (αi − ωi )ωi ≤ 1 and 2ωi ≤ αi .

= It can be checked that, under C1, X ∗ (x 1 , y1 , . . . , x n , yn )T = (0, . . . , 0)T is a trivial solution of coupled system (15). It should be pointed out that C1 means when x i = 0, γi (x i , σ ) = 0. It is totally different with γi = 0, which means γi (x i , σ ) = 0 for all x i . Theorem 3: Suppose that (G, A) is strongly connected and C1–C3 hold. Then, X ∗ is exponentially stable in mean square. Proof: Let Vi (X i ) = |X i |2 /2. Then, we can apply the operator (3) to it to have LVi (X i ) as follows: LVi (X i ) ! ∂ Vi (X i ) ∂ Vi (X i ) Pi (X i (t)) = , ∂ xi ∂yi " ! # 1 ∂ 2 Vi (X i ) + Tr G iT (X i (t)) G i (X i (t)) 2 ∂ x i ∂yi 2×2  " + Vi (X i + i (X i , σ )) − Vi (X i ) Y # ! ∂ Vi (X i ) ∂ Vi (X i ) , − i (X i , σ ) λ(dσ ) ∂ xi ∂yi = −ωi x i2 + (−ωi2 + αi ωi )x i yi + (ωi − αi )yi2  n γi2 (x i , σ ) 1 λ(dσ ) − ai j yi (yi − y j ) + βi2 yi2 + 2 2 Y j =1 ! ωi (αi − ωi ) 1 2 ωi x i2 + ≤ −ωi x i2 + yi + (ωi − αi )yi2 2 ωi n 1 1¯ 2 2 − ai j yi (yi − y j ) + βi2 yi2 + λη x 2 2 i i j =1 ! 1 ωi (αi − ωi ) = −ωi 1 − x i2 − (αi − ωi )yi2 2 2 n 1 2 2 1 2 2 − ai j yi (yi − y j ) + βi yi + λ¯ ηi x i 2 2 j =1

λ¯ 2 2 1 2 2 η x + βi yi 2 i i 2 !  1 1 − (yi − y j )2 + y 2j − yi2 2 2 j =1 ! ! n λ¯ 2 1 2 ωi 1 2 1 2 ηi + βi − y j − yi ai j X i2 + 2 2 2 2 2

ωi 2 X + 2 i n + ai j

≤−

≤

j =1

 −ςi X i2 +

n

ai j Fi j (yi , y j ).

j =1

Above, we use the inequality 2x i yi = 2(εx i )(yi /ε) ≤ ε2 x i2 + ¯ 2 < ωi , we have yi2 /ε2 and let ε2 = ωi . By condition βi2 + λη i ςi > 0. Along every directed cycle C of the weighted digraph (G, A) ! 1 2 1 2 y j − yi = 0. Fi j (yi , y j ) = 2 2 ( j,i)∈E(C Q )

( j,i)∈E(C Q )

Making use of Theorem 1 yields that X ∗ is exponentially stable in mean square. Remark 7: Obviously, X ∗ is stochastically asymptotically stable by Remark 1. In addition, Theorem 3 shows that, in a network of stochastic coupled oscillators with Lévy noise,

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exponential stability of a trivial solution to (15) cannot be reversed by white noise and Lévy noise, if the perturbation intensity is such that condition βi2 + λ¯ ηi2 < ωi holds. Remark 8: Clearly, assume that γi = 0 for i = 1, 2, . . . , n, that is, there is no Lévy noise considered in the model. Then the stochastic coupled oscillators with Lévy noise on network (15) degenerates to d X i (t) = P(X i (t))dt + G(X i (t))dW (t), i = 1, 2, . . . , n which is fully studied in [20]. Furthermore, suppose that γi = 0 and βi = 0 for i = 1, 2, . . . , n, that is, there is no noise considered. Then, network (15) turns into d X i (t) = P(X i (t))dt, i = 1, 2, . . . , n which can be investigated by the method in [8]. V. S TOCHASTIC VOLTERRA P REDATOR –P REY S YSTEM W ITH L ÉVY N OISE Volterra predator–prey equation [8] and the corresponding stochastic model driven by white noise [20] can be, respectively, described as ⎛ ⎞ n x˙ i = x i ⎝ei + pi j x j ⎠, i = 1, 2, . . . , n (17) j =1

and

⎡⎛

⎞ n n dx i (t) = x i (t) ⎣⎝ei (t)+ pi j x j (t)⎠ dt + αi j Ui j (x i (t) j =1

⎤

j =1

−x i∗ , x j (t)−x ∗j )dW (t)⎦, i = 1, . . . , n

(18)

where x i represents population density of the i th species, ei , pi j ∈ R. αi j is white noise intensity and W˙ (t) is 1-D white noise. There exist many studies on (17) and (18), readers can see [41] and [42] and the references cited therein. However, as alluded to in Section I, systems driven by white noise or Markovian switching cannot interpret sudden environmental shocks well. From a modeling point of view, it is more realistic to model this phenomenon by stochastic system that incorporates both white noise and Lévy noise. Hence, take a further step by considering Lévy noise into (18), we can get dx i (t)

⎡⎛

= x i (t) ⎣⎝ei +

n

⎞ pi j x j (t)⎠ dt

n

αi j Ui j (x i (t) − x i∗ , x j (t) − x ∗j )dW (t)

j =1

+

 n Y j =1

βi j γi j (x i (t)−x i∗ , x j (t)−x ∗j , σ ) N˜ (dt, dσ )⎦

in which βi j is the intensity of Lévy noise.

for those x i , x j ∈ R with % $    max x i − x i  , x j − x j  ≤ k and  & ' max (1 + γi (x i , σ ))0.5 − 1 − 0.5γi (x i , σ ) λ(dσ ), Y   (γi (x i , σ ) − ln(1 + γi (x i , σ ))) λ(dσ ) ≤ ρi x i2 +νi . Y

n Here, γi (x i , σ ) = j =1 βi j γi j (x i , x j , σ ) > −1 for brevity. D2. For any i, j = 1, 2, . . . , n, αii > 0 and αi j ≥ 0 if i = j . D3. Ui j (x, y) = |x|α |y|β , x, y ∈ R, where α, β ≥ 1. D4. For i, j = 1, 2, . . . , n, there exist positive constants r , L, and Mi j such that |Ui j (x, y)|2 ≤ L|x y| and  ⎛ ⎞    n   n  λ(dσ )  ⎝ ⎠ β γ (x, y, σ )−ln 1+ β γ (x, y, σ ) ij ij ij ij   Y  j =1  j =1 ≤

n

Mi j |x y|, (x, y) ∈ δr (0) × δr (0), σ ∈ Y.

j =1

Remark 9: Condition D4 seems to be complicated and not easy to be checked. Actually, it can be verified by√some simple functions. For example, we take Ui j = γi j = |x y|, ∀i, j . Obviously, we can get L = 1. There exists a positive constant r such n that√for (x, y) ∈ δr (0) × δr (0), it follows that 0 ≤ j =1 βi j |x y| ≤ 1 and

Y

⎤

i = 1, 2, . . . , n

Suppose that pii ≤ 0, and pi j p j i < 0 if pi j = 0 for i = j , which is the assumption that restricts (19) to interactions of the predator–prey type. Let x ∗ = (x 1∗ , . . . , x n∗ ), x i∗ > 0 be the positive equilibrium of (17). Suppose also that Ui j (0, 0) = 0 and γi j (0, 0, σ ) = 0 (i, j = 1, 2, . . . , n), which imply that two kinds of noise retain the equilibrium point x ∗ . Here, γi j (0, 0, σ ) = 0 means γi j (x i − x i∗ , x j − x ∗j , σ ) = 0, while x i = x i∗ , x j = x ∗j . We next make some assumptions on the coefficients of (19). D1. For i, j = 1, 2, . . . , n, there exist positive numbers ρi , νi , and L (k) i j such that for σ ∈ Y, x i , x j ∈ R, it follows:      max γi j (x i , x j , σ ) − γi j (x i , x j , σ )λ(dσ ), Y       γi j (x i , x j , σ ) − γi j (x i , x j , σ )λ(dσ ) Y   % $     (k) ≤ L i j max x i − x i , x j − x j 

  n n    βi j γi j (x, y, σ ) − ln 1 + βi j γi j (x, y, σ ) λ(dσ ) 

j =1

+

1705

(19)

j =1

j =1

 n

2 √ j =1 βi j |x y|

n n λ¯ 2 βi j |x y|  Mi j |x y|. 2 2 j =1 j =1 In the remainder of this section, we focus on a meaningful topic, that is, stochastic stability of equilibrium point x ∗ . We can also assume that the system has a unique global positive solution. This assumption is reasonable because it

≤

λ¯ ≤

n

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could be guaranteed by some conditions, which are shown as follows. Proposition 1: Suppose that conditions D1–D3 hold. Then, for any given initial value x 0 ∈ Rn+ , there is a unique solution x(t) to (19) on t ≥ 0 and the solution x(t) ∈ Rn+ for all t ≥ 0 with probability 1. This proposition can be proved in a similar procedure as [43, Th. 2.1] and [44, Th. 2.1]. Therefore, we omit its proof. It should be mentioned that D1–D3 are just one kind of conditions ensuring the unique global positive solution of (19), while D4 is a condition for stability of the equilibrium. Let G be a digraph with n vertices. Then, (19) can be regarded as a stochastic coupled system with Lévy noise on G. In other words, each vertex i represents i th equation of (19). Denote Hi j = pi j x i x j , Ni j = αi j x i Ui j (x i − x i∗ , x j − x ∗j ), and Ii j = βi j x i γi j (x i − x i∗ , x j − x ∗j ), which stand for the influence of vertex j on vertex i . For i, j = 1, 2, . . . , n, set  !   ∗ nL 2  α − Mi j  si j =  pi j + x i 2 ij  !   nL 2 ri j = − pi j + x i∗ αi j − Mi j  2 and S = (si j )n×n , R = (ri j )n×n . Thus, the following result can be derived. Proposition 2: Let D4 hold. If (G, S) and (G, R) are strongly connected and balanced and  if pi j = 0 αi j = Mi j = 0,  (20) ∗ 2   x i n Lαi j − 2Mi j < 2| pi j |, if pi j < 0 x∗

of (19) is stochastically then, the positive equilibrium stable. Proof: Let Vi (x i ) = x i −x i∗ −x i∗ ln x i /x i∗ for i = 1, 2, . . . , n. We next check that Vi (x i ) satisfies conditions A2 and A3 listed in Section III. From D4 and (20), we have LVi (x i )

⎛ ⎞ ⎡ ⎤2 ! n n x i∗ x i∗ ⎝ei x i + = 1− pi j x i x j ⎠ + 2 ⎣ αi j x i Ui j ⎦ xi 2x i j =1 j =1 !  xi ∗ ∗ − x i − x i − x i ln ∗ λ(dσ ) xi Y ⎡   n x i + x i nj =1 βi j γi j ∗ ⎣ βi j γi j − x i ln + xi + xi x i∗ Y j =1 ⎤ n ∗! x −x i∗ − x i βi j γi j 1 − i ⎦ λ(dσ ) xi j =1 ⎛ ⎞ n n nx ∗ 2 2 pi j x j ⎠ + i αi j Ui j ≤ (x i − x i∗ ) ⎝ei + 2 j =1 j =1 ⎛ ⎛ ⎞⎞  n n ∗⎝ + xi βi j γi j − ln ⎝1 + βi j γi j ⎠⎠ λ(dσ ) Y

j =1

j =1

=

n

pi j (x i − x i∗ )(x j − x ∗j )

j =1

+ x i∗

n

Mi j |x i − x i∗ ||x j − x ∗j |

j =1

+

n n Lx i∗ 2 αi j |x i − x i∗ ||x j − x ∗j | 2 j =1

=

n

sgn[(x i − x i∗ )(x j − x ∗j )] pi j |x i − x i∗ ||x j − x ∗j |

j =1

! n nL 2 αi j − Mi j |x i − x i∗ ||x j − x ∗j | 2 j =1 !# n " nL 2 αi j − Mi j = sgn[(x i − x i∗ )(x j − x ∗j )] pi j + x i∗ 2 j =1     ×x i − x i∗ x j − x ∗j  n n = μi j |x i − x i∗ ||x j − x ∗j | = ai j Fi j (x i , x j ) + x i∗

j =1

j =1

where   a i j = μ i j  =



si j , ri j ,

sgn[(x i − x i∗ )(x j − x ∗j )] > 0 sgn[(x i − x i∗ )(x j − x ∗j )] < 0

and Fi j (x i , x j ) = sgn[μi j ]|x i − x i∗ ||x j − x ∗j |. This together with (20) can imply immediately that Fi j (x i , x j ) = −F j i (x j , x i ) for i = j . Then, we can say that the positive equilibrium x ∗ of (19) is stochastically stable by Remark 2. That ends the proof. Remark 10: Condition (20) gives the bound for perturbation intensity of white noise and Lévy noise, under which (19) still remain to be stable. Obviously, if βi j = 0 for i, j = 1, 2, . . . , n, then (19) becomes (18). In this Brownian motion case, the results in Proposition 2 coincide with the results in [20]. If αi j = βi j = 0 for i, j = 1, 2, . . . , n, then (19) is changed as (17). For the deterministic case, the result in Proposition 2 is in accord with the conclusion in [8]. These facts can show that our results are the generalization of earlier work. VI. S IMULATION R ESULTS A numerical example is shown in this section to demonstrate the practical applicability of the theoretical results. Example: Let n = 6. Take γi = ηi x i and then consider the oscillators network (14) with the parameters in Table I. The weighted matrix A = (ai j )6×6 is as follows: ⎛ ⎞ 0 3 1 2 2 1 ⎜ 2 0 2 2 3 1.3 ⎟ ⎜ ⎟ ⎜ 2 2 0 1 1 2 ⎟ ⎜ ⎟. A=⎜ 3 0 2 1 ⎟ ⎜ 1 2 ⎟ ⎝ 1.5 2 1 1 0 2 ⎠ 2 1 1.5 1.2 1.1 0

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TABLE I PARAMETERS FOR N ETWORK (14)

Fig. 5.

Some sample paths of (14) with the parameters in Table I.

Poisson jumps, respectively. Therein, the initial value is (0.03, −0.01, 0.03, 0.02, −0.05, −0.03, 0.04, 0.02, 0.01, 0.03, −0.02, −0.01)T. VII. C ONCLUSION Fig. 3.

Poisson process N (t).

Fig. 4.

Second moment of solution to (14) with the parameters in Table I.

First, we show the simulation of Poisson process in Fig. 3.  Then, take a Lyapunov function V = 6i=1 ci (x i + yi )2 /2, we can easily check that all conditions in Theorem 3 are satisfied with the parameters in Table I. Therefore, we can deduce that the trivial solution X ∗ = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T is exponentially stable in mean square and stochastically asymptotically stable by applying Theorem 3 and Remark 7, respectively. We can see the simulation results clearly in Figs. 4 and 5. They present second moment of solution and some sample paths of (14) driven by

This paper investigates in detail the stability of SCSLNNs. It has applications in many branches of science and industry. Based on graph theory and Lyapunov stability theory, some criteria about the pth moment exponential stability and stability in probability of SCSLNNs are derived, respectively. These criteria can be successfully applied to any other stochastic coupled networks with Lévy noise. In addition, we have given a network of exponentially stable stochastic coupled oscillators with Lévy noise, and obtained an allowable bound of perturbation intensity of white noise and Lévy noise for stability of such oscillators. Moreover, stochastic Volterra predator–prey system with Lévy noise has been studied. We have provided the existence and uniqueness result for global positive solution and given the stochastic stability of positive equilibrium. Above two well-known examples fully demonstrate the applicability and effectiveness of our results in this paper. In addition, simulation results in this paper can be helpful in understanding the role of Lévy noise in network stability. This paper is not a simple modification of [20] and other relevant papers. On the one hand, Lévy noise has been firstly incorporated into the coupled networks in this paper, which is much more realistic in the concrete applications. It should be mentioned that the stability analysis for the model including Lévy noise in our paper is more complicated than one without Lévy noise, since we must deal with the jump part. On the other hand, [20] mainly studied the stochastic stability, while our paper aims to investigate both the moment exponential stability and stochastic stability. In a word, our results improve and extend the earlier work. In our future work, we will further look into a more complicated Lyapunov function for coupled systems, not just at the weighted sum of the given ones. In addition, we will

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ACKNOWLEDGMENT The authors really appreciate the reviewers’ valuable comments. We also would like to express our deep gratitude to Prof. R. Song at University of Illinois for his helpful suggestions to improve this paper. R EFERENCES

Fig. 6.

Strongly connected but nonbalanced digraph G with three vertices.

theoretically address some sensitivity analysis and robustness issues for stochastic coupled systems of differential equations on networks under Lyapunov conditions in view of the work for stochastic difference equations under Lyapunov conditions in [34]. A PPENDIX To analyze the stability for SCSLNNs over directed network topology, we now state some basic concepts of graph theory. For detail, we refer the reader to [8], [20], and [35]. A digraph G = (U, E) contains a set U = {1, 2, . . . , n} of vertices and a set E of arcs (i, j ) leading from initial vertex i to terminal vertex j . A subgraph H of G is said to be spanning if H and G have the same vertex set. A digraph G is weighted if each arc ( j, i ) is assigned a positive weight ai j . Here, ai j > 0 if and only if there exists an arc from vertex j to vertex i in G, and we call A = (ai j )n×n as the weighted matrix. 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 {i 1 , i 2 , . . . , i s } such that its set of arcs is {(i k , i k+1 ) : k = 1, 2, . . . , s − 1}. If i s = i 1 , we call P a directed cycle. A connected subgraph T of G 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 it is a disjoint union of rooted trees whose roots form a directed cycle. A digraph G is strongly connected if, for each ordered pair u, v of vertices, there is a path from u to v. Denote a digraph G with weighted matrix A as (G, A). A weighted digraph (G, A) is said to be balanced if −C exists and 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 ˜ The Laplacian that (G, A) is balanced, then W (Q) = W (Q). ) , matrix of (G, A) is defined as L = ( p i j n×n where pi j = −ai j  for i = j and pi j = k=i aik for i = j . From above definitions, we know that Fig. 6 is strongly connected, but is not balanced. However, Fig. 2 in Section III is really a strongly connected and balanced graph.

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ZHANG et al.: GRAPH THEORY-BASED APPROACH FOR STABILITY ANALYSIS OF SCSLNNs

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Chunmei Zhang was born in 1987. She received the B.S. and M.S. degrees from the Harbin Institute of Technology, Weihai, China, in 2005 and 2009, respectively. She is currently pursuing the Ph.D. degree with the University of Illinois at UrbanaChampaign, Champaign, IL, USA. Her current research interests include stochastic differential equations, coupled systems on networks, Lévy process, and dynamical systems on time scales.

Wenxue Li was born in 1981. He received the Ph.D. degree from the Harbin Institute of Technology, Weihai, China, in 2009. He is currently an Associate Professor with the Department of Mathematics, Harbin Institute of Technology. He has authored or co-authored many papers. His current research interests include stochastic differential and integral equations, coupled systems on networks, time-delay systems, and reaction-diffusion systems.

Ke Wang received the M.S. degree from Northeast Normal University, Changchun, China, in 1984, and the Ph.D. degree from Jilin University, Changchun, in 1995. He is currently a Professor with the Harbin Institute of Technology, Weihai, China. His current research interests include ordinary and functional differential equations, stochastic differential equations, and random dynamical systems.