Cluster Synchronization in Directed Networks Via ... - Semantic Scholar

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 7, JULY 2011

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Cluster Synchronization in Directed Networks Via Intermittent Pinning Control Xiwei Liu and Tianping Chen

Abstract— In this paper, we investigate the cluster synchronization problem for linearly coupled networks, which can be recurrently connected neural networks, cellular neural networks, Hodgkin–Huxley models, Lorenz chaotic oscillators, etc., by adding some simple intermittent pinning controls. We assume the nodes in the network to be identical and the coupling matrix to be asymmetric. Some sufficient conditions to guarantee global cluster synchronization are presented. Furthermore, a centralized adaptive intermittent control is introduced and theoretical analysis is provided. Then, by applying the adaptive approach on the diagonal submatrices of the asymmetric coupling matrix, we also get the corresponding cluster synchronization result. Finally, numerical simulations are given to verify the theoretical results. Index Terms— Adaptive, cluster synchronization, consensus, dynamical networks, intermittent pinning control, neural networks.

I. I NTRODUCTION VER the past decades, increasing interest has been shown to the study of complex networks, especially the synchronization phenomenon. Many different synchronization protocols have been studied, such as complete synchronization, phase synchronization, cluster synchronization, lag synchronization, generalized synchronization, and so on. In the real world, synchronization of coupled oscillators can not only explain many natural phenomena [1], but also have many applications, such as image processing [2], secure communication [3], etc. Suppose the dynamical behavior of uncoupled system is described by

O

s˙ (t) = f (s(t)) (s 1 (t), . . . , s n (t))

(1)

∈ f (s(t)) = ( f 1 (s(t)), where s(t) = . . . , f n (s(t)))T : R n → R n is continuous. This general model (1) includes many systems as special cases: for example, recurrently connected neural networks, cellular neural networks [4], Rn ,

Manuscript received April 12, 2010; revised January 8, 2011; accepted March 23, 2011. Date of publication May 16, 2011; date of current version July 7, 2011. This work was supported in part by the National Science Foundation of China under Grant 60774074, Grant 60974015, Grant SGST 09DZ2272900, Grant 2010KJ002, and Grant 2010CB328101, the China Postdoctoral Science Foundation funded project under Grant 20080440079; and the Program for Young Excellent Talents in Tongji University. X. Liu is with the Key Laboratory of Embedded System and Service Computing, Ministry of Education, Department of Computer Science and Technology, Tongji University, Shanghai 200092, China (e-mail: [email protected]). T. Chen is with the School of Mathematical Sciences, Fudan University, Shanghai 200433, 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/TNN.2011.2139224

Hodgkin–Huxley models [5], Lorenz chaotic oscillators [6], Chua’s circuits [7], etc. In general, N linearly coupled such identical systems can be described as x˙i (t) = f (x i (t)) +

N 

ai j (x j (t) − x i (t))

j =1, j  =i

i = 1, . . . , N

(2)

where x i (t) = (x i1 (t), . . . , x in (t)) ∈ R n is the n-dimensional state variable of the i th node, the second term on the right side of (2) represents the linear and local coupling among pairs of connected oscillators, where the positive definite matrix  = diag(γ1 , . . . , γn ) denotes the inner coupling matrix, and the network topology is represented by the outer coupling matrix A = (ai j ) ∈ R N×N , e.g., if ai j = 0, then node i ’s dynamics is impacted by node j or, in other words, node i receives direct information from node j . If x i (t) − x j (t) → 0, i, j ∈ 1, . . . , N, as t → +∞, then we say the network (2) can realize complete synchronization. When f (·) = 0, then the synchronization problem becomes the consensus problem [8]–[11]. Hitherto, many approaches have been derived for complete synchronization. For example, in [12], the authors present a master-stability function based on the transverse Lyapunov exponents to study local synchronization, in [13]–[18] and references therein, the Lyapunov function approach is used to derive a sort of global synchronization criteria. For example, a distance from synchronization manifold to each state is defined in [13], and in [14]– [18], the left eigenvector corresponding to the zero eigenvalue of the diffusive coupling matrix is utilized to investigate the global synchronization. On the other hand, if the set of nodes can be divided into m clusters, i.e., {1, . . . , N} = C1 ∪ C2 ∪ · · · ∪ Cm where C1 = {1, . . . , r1 }, C2 = {r1 + 1, . . . , r2 } . . . , Cm = {rm−1 + 1, . . . , N}

(3)

such that coupled systems in the same cluster can be synchronized, but there is no synchronization among different clusters, then the network is said to realize cluster synchronization. In particular, when the number of cluster m = 1, cluster synchronization turns to complete synchronization. In the literature, many authors have investigated this phenomenon, [19]– [27]. Cluster synchronization is considered to be significant in biological sciences [28] and communication engineering [29]. In this paper, we will investigate the cluster synchronization

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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 7, JULY 2011

One Period

One Period

ω

ω

θ

work time

Fig. 1.

θ rest time

work time

rest time

. . .

Sketch map of the intermittent control.

of linearly coupled systems by adding some external controllers as x˙i (t) = f (x i (t)) +

N 

ai j (x j (t) − x i (t)) + u i

j =1, j  =i

i = 1, . . . , N

(4)

where u i is the external control. If controllers are added on only a small fraction of network nodes, then it is called the pinning control problem, [30]–[39]. For example, in [31], Chen et al. investigated the essence of the synchronization problem and pinning control problem, and found that adding the control only on just one node can synchronize the network, in [36]– [39], the authors explored a variety of pinning control methods for coupled nonlinear systems and proposed important criteria for node selection and drew important connections between controllability and network structure. In signal transmission, the signal will become weak due to diffusion, so an external control should be added until the strength of the signal reaches an upper level. Then, the external control can be removed considering the cost. Therefore, in comparison with continuous pinning control, discontinuous controllers, which include intermittent control and impulsive control, have attracted more interest due to its wide applications in engineering fields. In previous works, several authors have applied this approach successfully on cluster synchronization, [40]–[51] and references therein. For example, in [40] and [41], the authors proposed the “nodeto-node” intermittent control method, in [42] and [43], an intermittent linear error feedback coupling was introduced, and by using fast switching techniques [44], [45] the synchronization problem was investigated. Recently, another interesting intermittent control was introduced and studied [46]– [51], i.e., the control time is periodic, and in any period, the time is composed of “work time” and “rest time,” which can be found in the map (see Fig. 1). Obviously, when ω = θ , the intermittent control becomes the usual continuous control, while θ = 0, the intermittent control becomes the impulsive control. In the following, we will investigate synchronization problem under this type of intermittent control. The theoretical results for this type of intermittent control can apply to both impulsive control and continuous control. Until now, some topics for cluster synchronization have not been studied in detail. In this paper, we will focus on the following two problems:

Problem 1: In [26], the coupling matrix is assumed to be symmetric. What will happen when the coupling matrix is asymmetric? Can the corresponding cluster synchronization be realized for networks with asymmetric coupling? Problem 2: As for previous pinning control works, the corresponding adaptive synchronization control is also investigated. For example, [31], [52] gave the centralized adaptive method, while [53]– [57] presented various decentralized approaches, like node-based strategy, edge-based strategy, edge-snapping, etc. Although synchronization has been investigated under intermittent pinning control, [46]– [51], there is still no theoretical result for synchronization with the adaptive intermittent control. The rest of this paper is organized as follows. In Section II, some necessary definitions, lemmas, and notations are given. In Section III, we investigate the cluster synchronization problem with an asymmetric matrix and intermittent control for static network. By analyzing the obtained result, we find that one way to realize cluster synchronization is to make the diagonal matrices large enough. Therefore, in Section IV, how to apply the adaptive approach on the intermittent control is studied and rigorously proved. Numerical simulations to show the validity of obtained theoretical results are also presented in Section V. Finally, this paper is concluded in Section VI. II. P RELIMINARIES In this section, we present some definitions, lemmas, and notations, which will be useful throughout this paper. Definition 1: Matrix A = (ai j )i,N j =1 is said to belong to class A1, denoted as A ∈ A1, if [16], [18]:  1) ai j ≥ 0, i = j, aii = − Nj=1, j =i ai j , i = 1, . . . , N; 2) A is irreducible. If A ∈ A1 is symmetric, then we say that A belongs to class A2, denoted as A ∈ A2. N It is clear that if A ∈ A1, then j =1 ai j = 0 for i = 1, . . . , N. It is called the zero row-sum property. To generalize this property, we have Definition 2: Matrix A = (ai j ) ∈ R N1 ×N2 is said to belong to class  2A3, denoted as A ∈ A3, if its each row-sum is zero, ai j = 0, i = 1, . . . , N1 . i.e., Nj =1 Now, using the above types of matrices, we can define a new type of coupling matrix A for the following cluster synchronization analysis. Definition 3: Suppose A ∈ R N×N , the indexes {1, . . . , N} can be divided into m clusters as defined in (3), and the following form holds ⎞ ⎛ A11 A12 · · · A1m ⎜ A21 A22 · · · A2m ⎟ ⎟ A=⎜ (5) ⎝ ··· ··· ··· ··· ⎠ Am1 Am2 · · · Amm where Ai j ∈ R (ri −ri−1 )×(r j −r j −1 ) , Aii ∈ A1 and Ai j ∈ A3, i, j ∈ 1, . . . , m. Then matrix A is said to belong to class A4, denoted as A ∈ A4. Remark 1: In general, ai j > 0(or < 0), i = j is regarded as the cooperative (or competitive) relationship between node i and node j . Thus, A ∈ A4 means that nodes in the

LIU AND CHEN: CLUSTER SYNCHRONIZATION IN DIRECTED NETWORKS

same cluster only have cooperative relationships, while nodes belonging to different clusters can have both cooperative and competitive relationships. Definition 4: Network (4) with N nodes is said to realize cluster synchronization, if N nodes can be divided into m clusters as defined by (3), such that, for any node i ∈ Ck , k = 1, . . . , m lim x i (t) − sk (t) = 0

(6)

lim sk (t) − sl (t) = 0, l = k

(7)

t →+∞

and t →+∞

where  ·  is some norm, and sk (t) is a trajectory defined by s˙k (t) = f (sk (t)).

(8)

Notation 1: Throughout this paper, we denote the identity matrix by I with appropriate dimensions. If all eigenvalues of a matrix A ∈ R N×N are real, then we sort them as λ1 (A) ≤ λ2 (A) ≤ · · · ≤ λ N (A). We denote the symmetrical part of matrix A as As = (A + A T )/2. A symmetric real matrix A is positive definite (semidefinite) if x T Ax > 0(≥ 0) for all nonzero x, denoted as A > 0(A ≥ 0). The Kronecker product of an N by M matrix A = (ai j ) and a p by q matrix B is the N p by Mq matrix A ⊗ B, defined as ⎛ ⎞ a11 B · · · a1M B ⎜ ⎟ .. .. .. A⊗B =⎝ ⎠ . . . a N1 B

· · · aN M B

and the Kronecker product has the property (A ⊗ B)(C ⊗ D) = (AC) ⊗ (B D). The following lemma plays an important role in synchronization analysis with pinning controls. Lemma 1: Suppose A ∈ A1 and ε < 0. Then, there exists a positive definite diagonal matrix  = diag(φ1 , . . . , φ N ), such that Aε = A + diag(ε, 0, . . . , 0) is Lyapunov stable, i.e., [31] Aε + AεT  < 0.

(9)

III. C LUSTER S YNCHRONIZATION WITH S TATIC I NTERMITTENT C ONTROL A. Network Model In this paper, the cluster synchronization is realized by periodically intermittent pinning control (see Fig. 1). For simplicity, we assume that only the first node in every cluster is pinned with the same control gain. Therefore, the N linearly coupled systems with intermittent pinning control with clusters Ck , k = 1, . . . , m defined in (3), can be described as ⎧  ⎪ x˙rk−1 +1 (t) = f (xrk−1 +1 (t)) + Nj=1 ark−1 +1, j x j (t) ⎪ ⎪ ⎪ ⎪ + g(sk (t) − xrk−1 +1 (t)), ⎪ ⎪  ⎪ ⎪ x˙i (t) = f (x i (t)) + Nj=1 ai j x j (t), ⎨ for i = rk−1 + 2, . . . , rk , ⎪ ⎪ ⎪ if t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . ⎪ ⎪  ⎪ ⎪ x˙i (t) = f (x i (t)) + Nj=1 ai j x j (t), i = 1, . . . , N, ⎪ ⎪ ⎩ if t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, · · · (10)

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where sk (t) satisfies s˙k (t) = f (sk (t)), outer coupling matrix A = (ai j ) ∈ A4 has the form (5), and the inner coupling matrix  = diag(γ1 , . . . , γn ) is positive definite. Scalar g > 0 is the linear state feedback control gain, ω > 0 is the control period, and θ > 0 is called the control width (control duration). Denote ei (t) = x i (t) − sk (t), i = rk−1 + 1, . . . , rk , E k (t) = (erk−1 +1 (t)T , . . . , erk (t)T )T , F(E k (t)) = (( f (xrk−1 +1 (t)) − f (sk (t)))T , . . . , ( f (xrk (t)) − f (sk (t)))T )T , and E(t) = (E 1 (t)T , . . . , E m (t)T )T . Then, for 1 ≤ k ≤ m ⎧  E˙ k (t) = F(E k (t)) + mj=1 ( Aˆ kj ⊗ )E j (t) ⎪ ⎪ ⎨ if t ∈ [lω,  lω + θ ] (11) m ˙ ⎪ E (t) = F(E (t)) + k ⎪ j =1 (A kj ⊗ )E j (t) ⎩ k if t ∈ (lω + θ, (l + 1)ω) where submatrices Aˆ kj are defined as  Akj Aˆ kj = Akk − diag(g, 0, . . . , 0)

k = j k = j.

(12)

We also make the following assumption for function f (·). Assumption 1: The function f (·) is said to satisfy the QUAD condition, denoted as f (·) ∈ QU AD(P, ), if there exist two positive definite diagonal matrices P = diag( p1 , . . . , pn ) and = diag(δ1 , . . . , δn ), such that for any x, y ∈ R n , the following condition holds: (x − y)T P( f (x) − f (y) − x + y) ≤ 0.

(13)

The QUAD condition (13) implies that the term x is a linear state feedback that globally stabilizes the system x(t) ˙ = f (x(t)), and this property is also known as quadratically stabilizable in the control literature. It can be shown that the QUAD assumption holds for several well-known chaotic oscillators, such as some cellular neural networks, the Lorenz system, and so on. Similar definitions have been widely adopted in the synchronization literature, [15], [16], [18]. B. Main Results In this subsection, we will find conditions for parameters ω, δ, and g such that linearly coupled networks (10) can realize cluster synchronization. Firstly, recalling the definition of Aˆ kk = Akk − diag(g, 0, . . . , 0), k = 1, . . . , m in (12), and combining with Lemma 1, we conclude that for any k = 1, . . . , m, there exist positive definite diagonal matrices k = diag(φrk−1 +1 , . . . , φrk )

(14)

T  < 0. Without loss of generality, we such that k Aˆ kk + Aˆ kk k N φi = 1. always assume that maxi=1 Denote a1 = maxni=1 δi /γi and ⎞ ⎛ T  1 A1m +Am1 m (1 Aˆ 11 )s + a1 1 · · · 2 ⎟ ⎜ .. .. .. ⎟ A=⎜ . . . ⎠ ⎝

⎛

T  m Am1 +A1m 1 2

(1 A11 )s + a1 1 ⎜ .. A=⎜ . ⎝ T  m Am1 +A1m 1 2

· · · (m Aˆ mm )s + a1 m ··· .. .

T  1 A1m +Am1 m 2

.. .

· · · (m Amm )s + a1 m

⎞ ⎟ ⎟. ⎠

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Theorem 1: Suppose f (·) ∈ QU AD(P, ). Then the coupled networks (10) with intermittent pinning control can realize cluster synchronization under the following condition: a2 θ + a3 (ω − θ ) < 0

where sk , k = 1, . . . , m are different constant vectors. By Theorem 1, it is easy to obtain the following result concerning the cluster consensus. Corollary 1: Denote ⎞ ⎛ T  1 A1m +Am1 m ··· (1 Aˆ 11 )s 2 ⎟ ⎜ .. .. .. ⎟ B=⎜ . . . ⎠ ⎝ T m Am1 +A1m 1 ··· (m Aˆ mm )s 2

⎛ ⎜ B=⎜ ⎝

(1 A11 )s .. .

T  m Am1 +A1m 1 2

··· .. . ···

x(t) ˙ = M(x(t)) − H (t)x(t)

(15)

where a2 = 2 minni=1 γi λ N (A), and a3 = 2 maxni=1 γi λ N (A). The proof will be given in Appendix 1. Remark 2: Combining condition (15) and definitions of A and A, it is clear that a2 must be negative, which indicates that the matrix A is negative definite. Combining with the fact that (k Aˆ kk )s , k = 1, . . . , m are negative definite (see Lemma 1), an effective way to achieve cluster synchronization is to make matrices (k Aˆ kk )s large enough. How to realize this idea is presented and rigorously proved in the next section by the adaptive approach. As a special case of cluster synchronization, we consider following cluster consensus model: ⎧  ⎪ x˙rk−1 +1 (t) = Nj=1 ark−1 +1, j x j (t) ⎪ ⎪ ⎪ ⎪ + g(sk − xrk−1 +1 (t)), ⎪ ⎪  ⎨ x˙i (t) = Nj=1 ai j x j (t), i = rk−1 + 2, . . . , rk , ⎪ if t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . ⎪ ⎪ N ⎪ ⎪ (t) = x ˙ ⎪ i j =1 ai j x j (t), i = 1, . . . , N, ⎪ ⎩ if t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, . . . (16)

and

we present and prove a useful lemma of adaptive intermittent control gain. Lemma 2: Suppose

T  1 A1m +Am1 m

2

.. .

⎞ ⎟ ⎟ ⎠

(m Amm )s

where Aˆ i j and i , i, j = 1, . . . , m are defined in Theorem 1. b2 = 2 minni=1 γi λ N (B), and b3 = 2 maxni=1 γi λ N (B). Then, under the condition b2 θ + b3 (ω − θ ) < 0

(17)

the network model (16) can realize cluster consensus, i.e., for any i = 1, . . . , N, if i ∈ Ck , we have lim x i (t) − sk  = 0.

t →+∞

IV. C LUSTER S YNCHRONIZATION WITH A DAPTIVE I NTERMITTENT C ONTROL In this section, we introduce the adaptive method [52] to intermittent control gain, and discuss the corresponding cluster synchronization. Before giving the theoretical result,

(18)

where x(t) ∈ R, M(·) : R → R is continuous and satisfies x M(x) ≤ L M x 2 , where L M > 0, and M(0) = 0. Function H (t) : {0 ∪ R+ } → R is the adaptive intermittent feedback control gain defined as ⎧ 0 t=0 ⎨ H (lω + θ ) t = (l + 1)ω, l = 0, 1, . . . H (t) = ⎩ 0 lω + θ < t < (l + 1)ω, l = 0, 1, . . . (19) and H˙ (t) = x 2 (t)

lω ≤ t ≤ lω + θ, l = 0, 1, . . .

(20)

lim x(t) = 0.

(21)

then t →∞

The proof will be given in Appendix 2. Remark 3: It can be seen that various adaptive rules (20) can be taken, for example H˙ (t) = h · |x(t)|q

lω ≤ t ≤ lω + θ, l = 0, 1, . . .

(22)

where h > 0 and q > 0, q = 1, 1/2, etc. Moreover, H˙ (t) can also be a combination of these cases: for example, when lω ≤ t ≤ lω + θ, l = 0, 1, . . .  if |x(t)| < 1 |x(t)|1/2 ˙ (23) H (t) = |x(t)|2 if |x(t)| ≥ 1. As for the control gain H (t), it can also have the following form:  0 t = 0, H (t) = (24) H (lω + θ ) lω + θ ≤ t ≤ (l + 1)ω and when δ = ω, the adaptive intermittent method (24) becomes the adaptive method discussed in [31] and [52]. In [58], the authors investigate the parameter drift problem, which can be stated as follows. While external disturbance is ignored, the system error e(t) can be regarded as the approximate error of adaptive system. In case the error of the system is smaller than the dead-zone threshold (ed z ), the adaptive updating (or learning) is stopped, that is  Enabled if e(t) ≥ ed z Learning Disabled if e(t) < ed z . Similar to the analysis in Appendix 2. Corollary 2: Suppose all assumptions in Lemma 2 are satisfied. Function H (t) : {0 ∪ R+ } → R is the adaptive intermittent feedback control gain defined as follows: ⎧ 0 t=0 ⎨ H (lω + θ ) t = (l + 1)ω, l = 0, 1, . . . H (t) = ⎩ 0 lω + θ < t < (l + 1)ω, l = 0, 1, . . . (25) and when lω ≤ t ≤ lω + θ, l = 0, 1, . . .  2 x (t) if |x(t)| ≥ ebound ˙ H (t) = 0 if |x(t)| < ebound

(26)

LIU AND CHEN: CLUSTER SYNCHRONIZATION IN DIRECTED NETWORKS

where ebound ≥ 0 is the threshold. Then, the solution x(t) of the system (18) is finally bounded limt →∞ |x(t)| ≤ ebound .

(27)

In case ebound = 0, (26) becomes the adaptive rule (20). Now, let us consider N linearly coupled systems (10) with an adaptive intermittent pinning control, i.e., for any node in clusters Ck , k = 1, . . . , m, where Ck are defined by (3), its dynamical behavior is described as follows: ⎧  x˙rk−1 +1 (t) = f (xrk−1 +1 (t)) + j ∈Ck ark−1 +1, j x j (t) ⎪ ⎪  ⎪ ⎪ ⎪ + H (t) j ∈Ck ark−1 +1, j x j (t) ⎪ ⎪ ⎪ ⎪ + g H (t)(s k (t) − x rk−1 +1 (t)) ⎪ ⎪ ⎪ ⎪ x ˙ (t) = f (x (t)) + i i ⎨ j  ∈Ck ai j x j (t)  + H (t) j ∈Ck ai j x j (t) ⎪ ⎪ ⎪ for i = rk−1 + 2, . . . , rk , ⎪ ⎪ ⎪ ⎪ if t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . ⎪  ⎪ ⎪ ⎪ x˙i (t) = f (x i (t)) + Nj=1 ai j x j (t), i = 1, . . . , N ⎪ ⎪ ⎩ if t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, . . . (28) With the same notations as those in Theorem 1, for 1 ≤ k ≤ m, we have  ⎧ E˙ k (t) = F(E k (t)) + mj=1, j =k ( Aˆ kj ⊗ )E j (t) ⎪ ⎪ ⎪ ⎪ ⎨ +H (t)( Aˆ kk ⊗ )E k (t), if t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . ⎪ ⎪ ⎪ E˙ k (t) = F(E k (t)) + mj=1 (Akj ⊗ )E j (t), ⎪ ⎩ if t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, . . . (29) where the adaptive control gain ⎧ 0 t =0 ⎨ H (t) = H (lω + θ ) t = (l + 1)ω, l = 0, 1, . . . ⎩ 0 lω + θ < t < (l + 1)ω, l = 0, 1, . . . (30) with the adaptive rule H˙ (t) = h ·

m 

rk 

decentralized) adaptive approaches for complete synchronization, [53], [55], [57]. The main reasons why we adopt the centralized approach instead of the decentralized one are the following. 1) Generally, in previous works in the literature, the Lyapunov theorem is used. However, here we do not use the Lyapunov theorem. Because for the intermittent control with centralized/decentralized adaptive coupling strength, we cannot find a suitable Lyapunov function. 2) If we adapt every ai j (t) by (for example) a˙ i j (t) = x i (t) − x j (t), where node i and node j are neighbors, the weights of some edges might not converge. Similar simulation results have been reported in [55]. This is because we do not know whether increasing (or decreasing) individual ai j (t), when i and node j may belong to different clusters, will benefit cluster synchronization. 3) Besides, the computational cost is very high to adapt at least 2r weights ai j (t). Here, we propose the following decentralized adaptive strategy, which might be more effective for cluster synchronization. Suppose Hi (t) ∈ R, i = 1, . . . , m. Denote the strength matrix H (t) = diag(H1 (t)Ir1 , H2 (t)Ir2 , . . . , Hm (t)Irm ). With the same notations as those in Theorem 1 and 2, for 1 ≤ k ≤ m, we have  ⎧ E˙ k (t) = F(E k (t)) + mj=1, j =k ( Aˆ kj ⊗ )E j (t) ⎪ ⎪ ⎪ ⎪ ⎨ + (Hk (t) Aˆ kk ⊗ )E k (t), if t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . ⎪ m ⎪ ˙ ⎪ (t) = F(E (t)) + E k ⎪ j =1 (A kj ⊗ )E j (t), ⎩ k if t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, . . . (32) where the adaptive control gains ⎧ 0 t =0 ⎨ Hk (t) = Hk (lω + θ ) t = (l + 1)ω, l = 0, 1, . . . ⎩ 0 lω + θ < t < (l + 1)ω, l = 0, 1, . . . (33) with the adaptive rule

x i (t) − sk (t)2

k=1 i=rk−1 +1

lω ≤ t ≤ lω + θ, l = 0, 1, . . .

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(31)

where Aˆ kj , k, j = 1, . . . , m are defined by (12), and h > 0. Applying Lemma 2 to (28), we can state the following theorem concerning the cluster synchronization with adaptive intermittent control. Theorem 2: If function f (·) satisfies the QUAD condition (13), then coupled network (29) with adaptive rules (30)–(31) can realize cluster synchronization. The proof can be found in Appendix 3. Remark 4: Following two cases are of special importance: 1) m = 1. In this case, the coupled network (29) with adaptive rules (30)–(31) leads to complete synchronization with adaptive intermittent control; 2) f = 0. In this case, the coupled network (29) with adaptive rules (30)–(31) can realize cluster consensus. Remark 5: It should be pointed out that the adaptive rule is centralized. In fact, there also exist many distributed (or

H˙ k (t) = h k ·

rk 

x i (t) − sk (t)2

i=rk−1 +1

lω ≤ t ≤ lω + θ, l = 0, 1, . . .

(34)

where Aˆ kj , k, j = 1, . . . , m are defined by (12), and h k > 0. Though simulations indicate that the proposed decentralized adaptive methods (32)–(34) can realize cluster synchronization, it is still a tough open problem to prove it rigorously. V. N UMERICAL E XAMPLES In this section, we give some numerical examples to demonstrate the effectiveness of the theoretical results for cluster synchronization. A. Cluster Synchronization in a Network with Five Nodes Consider a linearly coupled network with five nodes, and suppose the network can be divided into two clusters: C1 = {1, 2} and C2 = {3, 4, 5}. If the intermittent controllers with

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control gain g are added only to nodes 1 and 3, then the network can be described as follows: ⎧  x˙1 (t) = f (x 1 (t)) + 5j =1 a1 j x j (t) + g(s1 (t) − x 1 (t)) ⎪ ⎪ ⎪ 5 ⎪ ⎪ ⎪ j =1 a3 j x j (t) + g(s2 (t) − x 3 (t)) ⎪ x˙3 (t) = f (x 3 (t)) +  ⎨ i = 2, 4, 5, x˙i (t) = f (x i (t)) + 5j =1 ai j x j (t), ⎪ t ∈ [2l, 2l + 1], l = 0, 1, 2, . . . ⎪ ⎪ N ⎪ ⎪ (t) = f (x (t)) + a x (t), i = 1, . . . , 5, x ˙ ⎪ i j j =1 i j ⎪ ⎩ i t ∈ (2l + 1, 2(l + 1)), l = 0, 1, 2, . . . where si (t), i = 1, 2, satisfying s˙i (t) = f (si (t)). Denoting E 1 (t) = ((x 1 (t) − s1 (t))T , (x 2 (t) − s1 (t))T )T , E 2 (t) = ((x 3 (t) − s2 (t))T , (x 4 (t) − s2 (t))T , (x 5 (t) − s2 (t))T )T , F(E 1 (t)) = (( f (x 1 (t)) − f (s1 (t)))T , ( f (x 2 (t)) − f (s1 (t)))T )T , F(E 2 (t)) = (( f (x 3 (t))− f (s2 (t)))T , ( f (x 4 (t))− f (s2 (t)))T , ( f (x 5 (t)) − f (s2 (t)))T )T , the above coupled network can be written as ⎧ E˙ 1 (t) = F(E 1 (t)) + ( Aˆ 11 ⊗ )E 1 (t) + ( Aˆ 12 ⊗ ))E 2 (t) ⎪ ⎪ ⎪ ⎪ ⎪ E˙ (t) = F(E 2 (t)) + ( Aˆ 21 ⊗ )E 1 (t) + ( Aˆ 22 ⊗ ))E 2 (t) ⎪ ⎨ 2 in case t ∈ [2l, 2l + 1], l = 0, 1, . . . ˙ ⎪ (t) = F(E E 1 1 (t)) + (A 11 ⊗ I3 )E 1 (t) + (A 12 ⊗ I3 ))E 2 (t) ⎪ ⎪ ⎪ ˙ ⎪ E (t) = F(E 2 (t)) + (A 21 ⊗ I3 )E 1 (t) + (A 22 ⊗ I3 ))E 2 (t) ⎪ ⎩ 2 in case t ∈ (2l + 1, 2(l + 1)), l = 0, 1, . . . (35) In the following, we first assume g = 1 and the coupling matrix:     −1 1 1 0 ˆ − A11 = A11 − diag(g, 0) = 2 −2 0 0   −2 1 = 2 −2  2 3 −5 ˆ A12 = A12 = ⎛ 0 4 −4⎞ −2 2 Aˆ 21 = A21 = ⎝ −2 2 ⎠ 3 −3 A22 − diag(g, 0, 0)⎞ ⎛ Aˆ 22 = ⎛ ⎞ 1 0 0 −1 1 0 = ⎝ 0 −1 1 ⎠ − ⎝ 0 0 0 ⎠ 0 0 0 1 −2 ⎞ ⎛ 1 −2 1 0 = ⎝ 0 −1 1 ⎠. 1 1 −2 The fourth-order Runge–Kutta scheme is used to solve all the ordinary differential equations in our numerical simulations with step-size 0.001. Simulation 1: Neural networks. In this case, we consider a 3-D neural network [59] s˙ (t) = f (s(t)) = −s(t) + T L(s(t))

(36)

where s(t) = (s 1 (t), s 2 (t), s 3 (t))T , L(s(t)) = (l(s 1 (t)), l(s 2 (t)), l(s 3 (t)))T , l(v) = (|v + 1| − |v − 1|)/2, and ⎛ ⎞ 1.25 −3.2 −3.2 T = ⎝ −3.2 1.1 −4.4 ⎠. −3.2 4.4 1 This neural network has a double-scrolling chaotic attractor, see Fig. 2.

1.5 1 0.5 0 −0.5 −1 −1.5 −2

1.5

−1 0.5

0 1 2

−1.5

−1

−0.5

1

0

Fig. 2. Chaotic attractor of uncoupled 3-D neural network (36) with initial values (0.1, 0.1, 0.1)T .

In this example, the initial values were taken as: s1 (0) = (0.0355, 2.2272, −0.0692)T , s2 (0) = (−0.5073, 0.2358, 0.2458)T , x 1 (0) = (0.0700, −0.6086, −1.2226)T , x 2 (0) = (0.3165, −1.3429, −1.0322)T , x 3 (0) = (1.3312, −0.4189, −0.1403)T , x 4 (0) = (0.8998, −0.3001, 1.0294)T , and x 5 (0) = (−0.3451, 1.0128, 0.6293)T . According to the theoretical results reported in [60], function f (·) satisfies the QUAD condition. Moreover, we assume the inner coupling matrix  = I . Notations E1(t) and E2(t) are used to denote the synchronization errors in clusters C1 and C2 as follows:   2  x i (t) − s1 (t)2 (37) E1(t) =  i=1

  5  E2(t) =  x i (t) − s2 (t)2 .

(38)

i=3

Fig. 3 indicates that for error systems (35) with constant intermittent control gain g = 1, cluster synchronization cannot be realized. Instead, by adapting the intermittent control gain ⎧ E˙ 1 (t) = F(E 1 (t)) + ( Aˆ 11 ⊗ )E 1 (t) + ( Aˆ 12 ⊗ ))E 2 (t) ⎪ ⎪ ⎪ ˙ ⎪ ⎪ E (t) = F(E 2 (t)) + ( Aˆ 21 ⊗ )E 1 (t) + ( Aˆ 22 ⊗ ))E 2 (t) ⎪ ⎪ 2 ⎪ ⎪ in case t ∈ [2l, 2l + 1], l = 0, 1, . . . , ⎪ ⎨ ˙ E 1 (t) = F(E 1 (t)) + (A11 ⊗ I3 )E 1 (t) ⎪ + H (t)(A12 ⊗ I3 ))E 2 (t) ⎪ ⎪ ⎪ ˙ ⎪ (t) = F(E 2 (t)) + (A21 ⊗ I3 )E 1 (t) E ⎪ ⎪ 2 ⎪ ⎪ + H (t)(A22 ⊗ I3 ))E 2 (t) ⎪ ⎩ in case t ∈ (2l + 1, 2(l + 1)), l = 0, 1, . . . (39)

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40

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15 E1(t)

E1(t)

E2(t)

E2(t)

35

10

30

5

25 0

0

5

10

15

20

15

10

10

5

0

H(t)

15

5

0

5

10

0

15

0

5

10

15

Time (t)

Time (t) Fig. 3. Trajectories of cluster errors E 1 (t) and E 2 (t) for coupled neural networks’ error systems (35) and (36) with constant intermittent control gain, which means that cluster synchronization cannot be realized.

Fig. 4. (Above) The trajectories of cluster errors E 1 (t) and E 2 (t), which means that cluster synchronization is realized. (Below) The trajectory of adaptive intermittent control gain H (t) under adaptive rule (41).

20 E1(t)

where  H (t) =

15

0 H (lω + θ )

t =0 t = (l + 1)ω, l = 0, 1, . . .

(40)

H˙ (t) = 0.1

i=1

10 5

with the following adaptive rules:  2 

E2(t)

x i (t) − s1 (t) + 2

t ∈ [2l, 2l + 1] l = 0, 1, . . .

5 



0

x i (t) − s2 (t)

2

0

5

10

15

20

25

i=3

(41)

20

Simulations show that cluster synchronization can be realized, and H (t) → 13.1465, t → +∞, see Fig. 4. Moreover, as for networks (35), we can also apply the decentralized adaptive method (33) and (34) to the intermittent control gain ⎧ E˙ 1 (t) = F(E 1 (t)) + (A11 ⊗ )E 1 (t) + (A12 ⊗ ))E 2 (t) ⎪ ⎪ ⎪ ⎪ ⎪ E˙ 2 (t) = F(E 2 (t)) + (A21 ⊗ )E 1 (t) + (A22 ⊗ ))E 2 (t) ⎪ ⎪ ⎪ ⎪ in case t ∈ [2l, 2l + 1], l = 0, 1, . . . ⎪ ⎪ ⎨ E˙ (t) = F(E (t)) + H (t)( Aˆ ⊗ I )E (t) 1 1 1 11 3 1 ˆ 12 ⊗ I3 ))E 2 (t) + ( A ⎪ ⎪ ⎪ ⎪ ⎪ E˙ 2 (t) = F(E 2 (t)) + H2 (t)( Aˆ 21 ⊗ I3 )E 1 (t) ⎪ ⎪ ⎪ ⎪ + ( Aˆ 22 ⊗ I3 ))E 2 (t) ⎪ ⎪ ⎩ in case t ∈ (2l + 1, 2(l + 1)), l = 0, 1, . . . (42)

15

H1(t) H2(t)

10 5 0

0

5

10

15

20

25

Fig. 5. (Above) Trajectories of cluster errors E 1 (t) and E 2 (t), which means that cluster synchronization is realized. (Below) Trajectory of adaptive intermittent control gain H1 (t) and H2 (t) under adaptive rule (44).

evolve with the following adaptive rules: H˙ 1(t) = 0.05

2 

x i (t) − s1 (t)2

i=1

where the adaptive control gains  0 t =0 Hi (t) = Hi (lω + θ ) t = (l + 1)ω, l = 0, 1, . . .

H˙ 2(t) = 0.1 (43)

5 

x i (t) − s2 (t)2

i=3

t ∈ [2l, 2l + 1], l = 0, 1, . . .

(44)

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−60

−60

40 E1(t)

30

−60.5 V [mV]

V [mV]

−60.5 s 1

−61

E2(t)

20 −61 10

−61.5 0.436

0.44

0.444

0.448

−61.5

0

(a) 50

100 (b)

200

0

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

600

50

H(t)

s 2

−50

−100

0

0.2

0.4 (c)

0.6

V [mV]

V [mV]

0

0

400

−50

200

−100

0

100 (d)

200

Fig. 6. Figure showing that both the stable fixed point and the stable limit circle can coexist for Hodgkin–Huxley (45) with I0 = 7. More concretely, subplots (a) and (b) describe the stable fixed point case on the h−V phase and t−V phase, the corresponding trajectory is denoted as s1 (t) with the initial value (−60.6, 0.08, 0.44, 0.38)T , while subplots (c) and (d) describe the stable limit circle case on the h−V phase and t−V phase, the corresponding trajectory is denoted as s2 (t) with the initial value (−25, 0.3804, 0.5678, 0.0759)T .

180 E1(t) E2(t)

160 140 120 100

0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 Time (t)

Fig. 8. (Above) Trajectories of cluster errors E 1 (t) and E 2 (t), which means that cluster synchronization is realized. (Below) Trajectory of adaptive intermittent control gain H (t) under adaptive rule (46).

Simulation 2: Hodgkin–Huxley model. In this case, we consider the linearly and partially coupled neural networks error systems (35), where the single neural equation is the Hodgkin–Huxley equation [5], which models action potentials in squid giant axons as follows: ⎧ dV C dt = I0 − g Na m 3 h(V − VNa ) − g K n 4 (V − VK ) ⎪ ⎪ ⎪ ⎪ −g L (V − VL ) ⎨ dm = α (45) m (V )(1 − m) − βm (V )m dt ⎪ dh ⎪ ⎪ = α (V )(1 − h) − β (V )h h h ⎪ dt ⎩ dn dt = αn (V )(1 − n) − βn (V )n where the parameters g Na = 120 mS/cm2 , g K = 36 mS/cm2 , g L = 0.3 mS/cm2 , VNa = 50 mV, VK = −77 mV, VL = −54.4 mV, C = 1 μF/cm2 , and

80

0.1(V + 40) 1 − exp[−(V + 40)/10] βm (V ) = 4exp[−(V + 65)/18] αh (V ) = 0.07exp[−(V + 65)/20]

60

αm (V ) =

40 20 0

0

0

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 Time (t)

Fig. 7. Trajectories of cluster errors E 1 (t) and E 2 (t) for coupled H–H networks’ error systems (35) and (36) with constant intermittent control gain, which means that cluster synchronization cannot be realized.

Simulations show that cluster synchronization can also be realized, and as t → +∞, H1 (t) → 10.8574, and H2 (t) → 6.7603, see Fig. 5.

βh (V ) = {1 + exp[−(V + 35)/10]}−1 0.01(V + 55) αn (V ) = 1 − exp[−(V + 55)/10] βn (V ) = 0.125exp[−(V + 65)/80]. The dynamics of the Hodgkin–Huxley model exhibits a complex and rich behavior which sensitively depends on the model parameters, for example, as described in [61]. When the external stimulus I0 is constant and I0 ∈ (6.26, 9.763) μA/cm2 , then both the stable fixed point (which means the inhibition state) and the stable limit circle (which means the active state) can coexist.

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300

300 E1 E2 E3

250 200

250

150

150

100

100

50

50

0

0

Ei(t), i = 1, 2, 3

200

1000 2000 3000 4000 5000 6000 7000 8000 9000

0 25

50

Hi(t), i = 1, 2, 3

H (t)

40

20

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5

0

0

1

2

3

4

5

6

7

0

8

0

1

2

3

4

5

6

7

8

Fig. 9. Trajectories of cluster errors E 1 (t), E 2 (t), E 3 (t) and the centralized adaptive coupling strength H (t) for coupled Lorenz networks with adaptive intermittent control method (29)–(31), where h = 0.001.

Fig. 10. Trajectories of cluster errors E 1 (t), E 2 (t), E 3 (t) and the centralized adaptive coupling strength H1 (t), H2 (t), H3 (t) for coupled Lorenz networks with adaptive intermittent control method (32)–(34), where h 1 = 0.002, h 2 = 0.001, h 3 = 0.001.

Thus, in the following, we will always choose I0 = 7 μA/cm2 . In Fig. 6(a) and (b), describe a trajectory, denoted as s1 (t), which converge to a stable fixed point under the initial value as (−60.6, 0.08, 0.44, 0.38)T , while (c) and (d) describe another trajectory, denoted as s2 (t), which converge to a stable limit circle under another initial value as (−25, 0.3804, 0.5678, 0.0759)T . Choose the initial values for x i (0), i = 1, . . . , 5 as x 1 (0) = (−65, 0.3063, 0.5085, 0.5108)T , x 2 (0) = (−57, 0.8176, 0.7948, 0.6443)T , x 3 (0) = (−25, 0.3786, 0.8116, 0.5328)T , x 4 (0) = (−28, 0.3507, 0.9390, 0.8759)T , and x 5 (0) = (−20, 0.5502, 0.6225, 0.5870)T . Here, according to the application in real practice, we assume the inner coupling matrix  = diag(1, 0, 0, 0). Then, for error systems (35) with constant intermittent control gain, simulation shows that cluster synchronization cannot be realized, see Fig. 7. Therefore, we should apply the adaptive method to the control gain. For error systems (39) under intermittent control gain (30) with the adaptive rule   2 5   2 2 x i (t) − s1 (t) + x i (t) − s2 (t) H˙ (t) = 0.5

in [62]. As for the adaptive method, we adopt the centralized strategy (29)–(31), and the decentralized strategy (32)–(34), respectively, the corresponding simulation results are depicted in Figs. 9 and 10.

i=1

VI. C ONCLUSION In this paper, cluster synchronization with intermittent control was investigated. It was assumed that the dynamics on each node is identical, and all nodes are linearly coupled. First, we gave a criterion for cluster synchronization with an asymmetric coupling matrix, which has broader applications than previous works for symmetric coupling matrix. Secondly, we pointed out that to realize cluster synchronization, enlarging the couplings of nodes in the same cluster is the key point. Then, we proposed a centralized adaptive intermittent control approach to realize cluster synchronization. The derivation used in this paper is different from the routine method by using the Lyapunov function. Simulations also confirmed the effectiveness of the proposed scheme. We also discussed a decentralized adaptive algorithm for intermittent control. However, its rigorous proof is still an open problem.

i=3

t ∈ [2l, 2l + 1], l = 0, 1, . . .

(46)

simulations show that cluster synchronization can be realized, see Fig. 8, where E1(t), E2(t) are defined in (37) and (38). B. Cluster Synchronization in a Small-World Network In this subsection, we consider cluster synchronization in a small-world network with 600 nodes, and suppose its nodes can be divided into three clusters: C1 = {1, . . . , 200}, C2 = {201, . . . , 400}, C3 = {401, . . . , 600}. Assume that the original dynamics of each node is governed by Lorenz system, whose QUAD property is proved

ACKNOWLEDGMENT The authors would like to thank the reviewers and editor for their valuable comments and suggestions which improved the presentation and theoretical results of this paper. A PPENDIX 1 P ROOF OF T HEOREM 1 Proof: At first, from the definition of positive diagonal matrices and , one can get, for any i = 1, . . . , n, δi = γi · δi /γi , therefore, ≤ a1 , where a1 = maxni=1 δi /γi .

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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 7, JULY 2011

Then, for lω ≤ t ≤ lω + θ with l > l ∗ , we have

Now, define the Lyapunov function as 1 E k (t)T (k ⊗ P)E k (t) 2 m

V (t) =

k=1

where P is defined in (13). Then its derivative with respect to time t along with solutions of (11) can be calculated as follows. When t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . V˙ (t) =

m  k=1

= =

m  k=1 m 

W˙ (t) ≤ 2(L M − L M ω/θ − ε)W (t)

(51)

W (t) ≤ W (lω)e2(L M −L M ω/θ−ε)(t −lω).

(52)

(47) and

Instead, when t ∈ (lω + θ, (l + 1)ω), we have W˙ (t) ≤ 2L M W (t)

E k (t)T (k ⊗ P) E˙ k (t) 

 m  E k (t) (k ⊗ P) F(E k (t)) + ( Aˆ kj ⊗ )E j (t) T

j =1 rk 

W (t) ≤ W (lω + θ )e2L M (t −lω−θ) ≤ W (lω)e2(L M −L M ω/θ−ε)θ e2L M (t −lω−θ) ≤ W (lω)e−2εθ .

k=1 j =1

≤

m 

W (t) ≤ W (l ∗ ω)e−2(l−l

T

E k (t) k=1 m  m 

+

[k ⊗ (P )]E k (t)

lim x(t) = 0.

≤ E(t) [ A ⊗ (P)]E(t) ≤ 2 min γi λ N (A)V (t) = a2 V (t) i

which implies that, when t ∈ [lω, lω + θ ], l = 0, 1, 2, . . .

Case 2: For all t > 0, H (t) ≤ L M ω/θ . In this case, we have ∞  lω+θ  x 2 (s)ds < ∞. (56) l=1

(48)

Therefore  lω+θ

Similarly, when t ∈ (lω + θ, (l + 1)ω), one has V˙ (t) ≤ E(t)T [ A ⊗ (P)]E(t) ≤ a3 V (t), which implies: when t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, . . . (49)

Combining with inequalities (48) and (49), we have V ((l + 1)ω) ≤ V (lω + θ )ea3 (ω−θ) ≤ V (lω)ea2 θ+a3 (ω−θ) ≤ · · · ≤ V (0)e(a2θ+a3 (ω−θ))(l+1) which means ei (t) → 0, when t → +∞, i = 1, . . . , N, i.e., cluster synchronization is realized. The proof is completed. A PPENDIX 2 P ROOF OF L EMMA 2 Proof: It is clear that the time series {H (lω), l = 0, 1, 2, . . .} is positive and monotonically increasing. Denoting W (t) = x 2 (t) W˙ (t) = 2[x(t)M(x(t)) − H (t)x 2(t)].

(55)

t →∞

k=1 j =1 T

V (t) ≤ V (lω + θ )ea3 (t −lω−θ) .

∗ )εθ

which means

E k (t)T [k Aˆ kj ⊗ (P)]E j (t)

V (t) ≤ V (lω)ea2 (t −lω) .

(54)

Combining (52) and (54), we conclude that for all t ∈ [lω, (l + 1)ω], there holds

E k (t)T [k Aˆ kj ⊗ (P)]E j (t)

+

and

φi (x i (t) − sk (t))T P( f (x i (t)) − f (sk (t)))

k=1 i=rk−1 +1 m m  

(53)

(50)

Case 1: There exists some l ∗ > 0, such that H (l ∗ω) > L M ω/θ . In this case, pick a positive constant 0 < ε < H (l ∗ ω) − L M ω/θ , such that for all l ≥ l ∗ LMω + ε, lω ≤ t ≤ lω + θ. H (t) > θ

lω

x 2 (s)ds → 0, as l → +∞.

(57)

lω

We claim limt →∞ x(t) = 0. Otherwise, for any constant  > 0, there exist a sequence T1 < · · · < Tn , . . ., such that Tk < [(lk − 1)ω, (lk + 1)ω] < Tk+1 , and a sequence tk ∈ [lk ω, (lk + 1)ω], such that x 2 (tk ) ≥ ε. From (53), we have W˙ (t) ≤ 2L M W (t), which implies that for any t ≤ tk x 2 (tk ) ≤ x 2 (t)e2L M (tk −t ) x 2 (t) ≥ x 2 (tk )e−2L M (tk −t ) ≥ εe−2L M (tk −t ) .

(58) (59)

Therefore, in the interval t ∈ [(lk − 1)ω, lk ω], we have x 2 (t) ≥ εe−4L M ω , which implies 

(lk −1)ω+θ (lk −1)ω

x 2 (s)ds ≥ εθ e−4L M ω

(60)

which is a contradiction to (57). Therefore, in this case, we still have lim x(t) = 0.

t →∞

The proof is completed.

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A PPENDIX 3 P ROOF OF T HEOREM 3 Proof: At first, (29) can be rewritten as

⎧ ˙ E(t) = F(E(t)) + ( A˜ ⊗ )E(t) + H (t)( Aˆ ⊗ )E(t) ⎪ ⎪ ⎨ if t ∈ [lω, lω + θ ], l = 0, 1, 2, . . . ˙ ⎪ E(t) = F(E(t)) + (A ⊗ )E(t) ⎪ ⎩ if t ∈ (lω + θ, (l + 1)ω), l = 0, 1, 2, . . . (61) where A˜ = A − Aˆ and Aˆ = diag( Aˆ 11 , . . . , Aˆ mm ). Define a function F(E(t))  F(E(t)) + (A ⊗ )E(t), = F(E(t)) + ( A˜ ⊗ )E(t),

t ∈ (lω + θ, (l + 1)ω) t ∈ [lω, lω + θ ]

˜ 2 }. and define L = max1≤i≤n δi +max1≤i≤n γi ·max{A2 ,  A Then E(t)T (I ⊗ P)F(E(t)) ≤ L E(t)T (I ⊗ P)E(t).

(62)

Now, (61) can be stated as ˙ E(t) = F(E(t)) + H (t)( Aˆ ⊗ )E(t).

(63)

E(t)T (

Define W (t) = ⊗ P)E(t), where  diag(1 , . . . , m ) is defined in (14), then

=

˙ (t) W

= 2E(t)T ( ⊗ P)(F(E(t)) + H (t)( Aˆ ⊗ )E(t)) = 2E(t)T ( ⊗ P)F(E(t)) + 2H (t)E(t)T ( ⊗ P)( Aˆ ⊗ )E(t) ≤ 2E(t)T (I ⊗ P)F(E(t)) − 2|λ | min γi H (t)W (t) 1≤i≤n

λ

}s

where = maxk=1 , . . . , m λrk {k Akk < 0. Following the proof of Lemma 2, we can obtain lim W (t) = 0

(64)

lim E(t) = 0

(65)

t →∞

which means t →∞

i.e., cluster synchronization is finally realized. The proof is completed. R EFERENCES [1] R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. Appl. Math., vol. 50, no. 6, pp. 1645– 1662, 1990. [2] G. W. Wei and Y. Q. Jia, “Synchronization-based image edge detection,” Europhys. Lett., vol. 59, no. 6, pp. 814–819, 2002. [3] Q. Xie, G. Chen, and E. M. Bollt, “Hybrid chaos synchronization and its application in information processing,” Math. Comput. Model., vol. 35, nos. 1–2, pp. 145–163, Jan. 2002. [4] J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Nat. Acad. Sci., vol. 79, no. 8, pp. 2554–2558, Apr. 1982. [5] A. Hodgkin and A. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” J. Physiol., vol. 117, pp. 500–544, Aug. 1952. [6] E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci., vol. 20, no. 2, pp. 130–141, 1963. [7] T. Matsumoto, L. Chuo, and M. Komuro, “The double scroll,” IEEE Trans. Circuits Syst., vol. 32, no. 8, pp. 797–818, Aug. 1985.

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Xiwei Liu received the Ph.D. degree in applied mathematics from Fudan University, Shanghai, China, in 2008. He was a Post-Doctoral Researcher at the Department of Physics, Fudan University, Shanghai, from 2008 to 2010. He is currently a Lecturer in the Department of Computer Science and Technology, Tongji University, Shanghai. His current research interests include nonlinear dynamical systems, complex networks, and neural networks.

Tianping Chen is a Professor in the Department of Mathematics, Fudan University, Shanghai, China. His current research interests include harmonic analysis, approximation theory, neural networks, signal processing, dynamical systems, and complex networks. He is a recipient of several awards, including the Second Prize of National Natural Sciences Award of China in 2002, the Outstanding Paper Award of IEEE T RANSACTIONS ON N EURAL N ETWORKS in 1997, and the Best Paper Award of the Japanese Neural Network Society in 1997.