Pinning synchronization of nonlinearly coupled complex networks with time-varying delays using M-matrix strategies

Neurocomputing 177 (2016) 89–97

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Pinning synchronization of nonlinearly coupled complex networks with time-varying delays using M-matrix strategies Jingyi Wang a, Jianwen Feng b,n, Chen Xu b,n, Yi Zhao b, Jiqiang Feng b a b

College of Information and Engineering, Shenzhen University, Shenzhen 518060, PR China College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 15 July 2015 Received in revised form 8 September 2015 Accepted 9 November 2015 Communicated by W. Lu Available online 26 November 2015

In this paper, we investigate the global synchronization problem for a class of nonlinearly-coupled complex networks (NCCNs) with mixed time-varying delays by using an M-matrix approach. The Mmatrix strategy allows many synchronization problems with much relaxed conditions to be considered and can be used to derive sufficient conditions for this new class of NCCNs by pinning control. We also propose an effective way of adapting the coupling strengths of complex networks and give numerical examples to illustrate the effectiveness of our theoretical results. & 2015 Elsevier B.V. All rights reserved.

Keywords: Nonlinearly-coupled complex networks Synchronization M-matrix Time-varying delays Pinning control

1. Introduction A complex network is a structure that is made up of a large set of nodes that are connected by a set of edges and they are ubiquitously found in nature and the modern world (for example, the Internet, mobile sensor networks, food networks and human communication networks). Complex networks have been extensively studied in the physical sciences and mathematics in the past few decades and much is already known about their dynamical behavior (see [1–7] and the references therein). Synchronization (first discovered by Pecora and Carroll in 1990) is a process in which two or more dynamical systems seek to adjust a certain property of their motion to a common behavior in the limit as time tends to infinity [8]. Synchronized complex networks have wide-ranging applications to many fields (such as population dynamics, power systems and automatic control [8–10]) and many synchronization patterns have been discovered in the past (for example, complete synchronization [11,12], cluster synchronization [13], phase synchronization [14] and so on). Techniques are also available that facilitate the attainment of synchronization such as by adaptive control [15,16], intermittent control [17], impulsive control [18] and pinning control [19–22]. As everyone knows, the real-world networks normally have a large number of nodes, and it is usually impractical to control a complex network by adding the controllers to all nodes [17]. n

Corresponding authors. E-mail addresses: [email protected] (J. Feng), [email protected] (C. Xu).

http://dx.doi.org/10.1016/j.neucom.2015.11.011 0925-2312/& 2015 Elsevier B.V. All rights reserved.

Pinning control is a particularly useful technique that applies controllers to only a small fraction of the nodes in a network (thus greatly reducing the number of controlled nodes in real-world complex networks). In fact, pinning control is so effective in some networks that only one controller is required for synchronization [19]. And the pinning scheme of the most highly connected nodes induced that a significant reduction in required controllers as compared to the traditional control scheme [9]. In general, the exact number of nodes that are required to be pinned for any network and coupling strength is also completely known [23,24]. Many networks in nature are nonlinearly-coupled (for example, neural and metabolic networks, in which the coupling configurations are continuously oscillating between two fixed states [25]). However, most synchronization studies that have been conducted so far have been confined to the smaller and less applicable class of complex networks that are linearly coupled [19,20,23] thus leaving the more important and naturally occurring nonlinearly-coupled ones virtually uninvestigated. This paper attempts to fill this gap in our knowledge by investigating the nonlinearly coupled problem via pinning control using the M-matrix approach. Pinning control is a very effective scheme for synchronization in complex networks by controlling only a small number of the system nodes [26]. The M-matrix is one of the most common ways of expressing the structure of a problem in the biological, physical and social sciences and it is also commonly used in mathematics to establish eigenvalue bounds. Song et al. recently proposed an Mmatrix method to investigate the first-order leader-follower synchronization problem in pinning-controlled multiagent systems

90

J. Wang et al. / Neurocomputing 177 (2016) 89–97

[27] and the leader–follower consensus problem for multi-agent systems with inherent nonlinear dynamics [28]. Consensus Tracking of multi-agent systems with Lipschitz-type node dynamics were studied by using tools M-matrix theory in [29]. The M-matrix provides a more general framework for the formulation and solution of nonlinearly-coupled systems and allows for the derivation of sufficient conditions for synchronization. The main aim of this paper is to extend the results of [27] to the more general problem of synchronizing nonlinearly coupled complex networks. In particular, we study the synchronization of nonlinearly-coupled complex networks with time-varying delays by using the pinning control method using the M-matrix approach. We also rigorously construct an adaptive feedback controller that can be used to completely synchronize any realworld network. Our results will be applicable to many synchronization problems in different fields of science and technology. This paper is organized as follows. In Section 2, we state some preliminary definitions and lemmas that will be needed for the rest of the paper. In Section 3, we introduce the general model for a nonlinearly coupled complex network with time-varying delayed dynamical nodes and establish some synchronization criteria for these complex dynamical networks. In Section 4, we discuss some numerical examples of the theoretical results. Section 5 is the conclusion of the paper.

2.1. Notations Throughout this paper, Rn shall denote the n-dimensional Euclidean space and Rn
n the set of all n
n real matrices. For a real matrix A, let A > be its transpose and As ¼ ðA þ A > Þ=2 be its symmetric part. Let 1n ¼ ð1; 1; …; 1Þ > A Rn and In be the n-dimensional identity matrix. For symmetric matrices A, the notation A 4 0 (respectively, o 0) shall mean that A is a positive-definite (respectively, negative-definite) matrix and λmax ðAÞ and λmin ðAÞ shall denote the greatest and least eigenvalues of a symmetric matrix respectively. For a square matrix A A Cn
n , let λi ðAÞ be the ith eigenvalue, ρðAÞ ¼ max1 r i r n j λi ðAÞj be the spectral radius and J ðAÞ ¼ min1 r i r n Reðλi ðAÞÞ be the minimum of the real parts of all the eigenvalues. The symbol  denotes the Kronecker product. 2.2. Graph theory and matrix algebra Let G ¼ ðV; E; AÞ be a weighted directed graph composed of a set of nodes V ¼ fv1 ; …; vn g, a set of edges E ¼ V
V and a weighted adjacency matrix A ¼ ½aij  with non-negative adjacency elements aij. An edge of G is denoted by eij ¼ ðvi ; vj Þ A E, which means that node vj receives information from node vi (vi is called the parent of vj). The adjacency elements associated with the edges of the graph are positive, i.e., eij A E 3 aij 4 0. Moreover, we assume that aii ¼ 0 for all i A 1; …; n. The set of neighbors of node vi is denoted by N i ¼ fvj A V : ðvi ; vj Þ A Eg. The in-degree and out-degree of node vi are, respectively, defined as follows: deg in ðvi Þ ¼

aji ;

j¼1

Lemma 1 (Song et al. [27]). The Laplacian matrix L has a simple zero eigenvalue and all the other eigenvalues have positive real parts if and only if the diagraph associated with L has a directed spanning tree. Lemma 2 (Song et al. [27]). Let L be the Laplacian matrix of a diagraph G. The minimum number of directed trees in G is equal to the multiplicity of the zero eigenvalue of L. Definition 1 (Horn and Johnson [30]). The set Z n A Rn
n is defined by   Z n  A ¼ ½aij  A Rn
n : aij r 0; ia j; i; j ¼ 1; 2; …; n : Definition 2 (Horn and Johnson [30]). A matrix A A Rn
n is said to be positive stable if i þ ðAÞ ¼ n, where i þ ðAÞ is the number of eigenvalues of A, counting multiplicities, with positive real part. Definition 3 (Horn and Johnson [30]). A matrix A is called an Mmatrix if A A Z n and A is positive stable. Lemma 3 (Horn and Johnson [30]). If A A Z n , the following statements are equivalent: (1) (2) (3) (4)

2. Preliminaries

n X

Firstly, some results on M-matrix are provided to study the pinning control synchronization of NCCNS.

deg out ðvi Þ ¼

n X

aij :

j¼1

The degree matrix of digraph G is a diagonal matrix Δ ¼ ½Δij  where Δij ¼ 0 for all i a j, and Δii ¼ degout ðvi Þ. The graph Laplacian associated with the digraph G is defined as

A is positive stable, that is, A is an M-matrix. A ¼ αI  P, P Z 0, α 4 ρðPÞ. Every real eigenvalues of A is positive. There is a positive diagonal matrix D such that DA þ A > D is positive definite; that is, there is a positive diagonal dominant.

Before proceeding further, the following essential lemmas are presented, which are used in proving the main result of this paper. Lemma 4 (Wang et al. [25]). Let 1n ¼ ð1; 1; …; 1Þ > , I n ¼ diagf1; 1; …; 1g A Rn and Q ¼ ðqij Þ ¼ I n  ð1=NÞ1n  1n> . For all zero-row-sum matrices M A Rm
n and θ 40, we have   1 1 > x MM > x þ θy > Q y : x > My ¼ x > MQ y r 2 θ Lemma 5 (Li et al. [31]). If Q A Rn
n is such that qij ¼ qji and P qii ¼  nj¼ 1;i a j qij , i; j ¼ 1; 2; …; n, then u > Q  Pv ¼

n X n X

ui qij Pvj ¼ 

i¼1j¼1

X

qij ðui  uj ÞPðvi vj Þ;

j4i

for all matrices P A Rm
m and vectors u ¼ ðu1> ; u2> ; …; un> Þ > and v ¼ ðv1> ; v2> ; …; vn> Þ > , where ui ¼ ðui1 ; ui2 ; …; uim Þ > and vi ¼ ðvi1 ; vi2 ; …; vim Þ > . Lemma 6 (Horn and Johnson [30]). Let A A Rn
n and B A Rm
m . If λðAÞ is an eigenvalue of A and λðBÞ is an eigenvalue of B, then λðAÞλðBÞ is an eigenvalue of A  B. Lemma 7. For matrices A; B; C and D with appropriate dimensions, one has (1) ðA  BÞ > ¼ A >  B > ; (2) ðA þ BÞ  C ¼ A  C þ B  C; (3) ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ. 2.3. Model description

LðGÞ ¼ L ¼ Δ  A:

Consider a complex network consisting of N identical nodes with nondelayed and time-varying delayed nonlinear coupling

A reversal of a graph is obtained by reversing the orientation of all the edges. If a graph has adjacency matrix A, then its reversal has adjacency matrix A > .

x_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ þ

N X j ¼ 1;i a j

c1 aij Γ ½gðxj ðtÞÞ  gðxi ðtÞÞ

J. Wang et al. / Neurocomputing 177 (2016) 89–97

þ

N X

c2 bij Γ ½hðxj ðt  τ2 ðtÞÞÞ  hðxi ðt  τ2 ðtÞÞÞ;

constants η 4 0; θ 4 0 such that   ðx yÞ > P ðf ðt; x; zÞ  f ðt; y; wÞÞ  ΔΓ ðx  yÞ

i ¼ 1; 2; …; N;

j ¼ 1;i a j

ð1Þ >

 τ r s r 0;

i ¼ 1; 2; …; N;

s_ ðtÞ ¼ f ðt; sðtÞ; sðt  τ1 ðtÞÞÞ

ð2Þ

by adding pinning controllers to some of the nodes. Without loss of generality, let the first l nodes be controlled. Then the network is described by x_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞþ

c1 aij Γ ðgðxj ðtÞÞ gðxi ðtÞÞÞ

N X

c2 bij Γ ðhðxj ðt  τ2 ðtÞÞÞ  hðxi ðt  τ2 ðtÞÞÞÞ þui ðtÞ;

j ¼ 1;i a j

i ¼ 1; 2; …; N;

ð3Þ

where ui(t) ði ¼ 1; 2; …; NÞ are the linear state feedback controllers that are defined by (  c1 εi Γ ðxi ðtÞ  sðtÞÞ; i ¼ 1; 2; …; l; ui ðtÞ ¼ ð4Þ 0; i ¼ l þ 1; l þ 2; …; N; for some control gains εi 40 ði ¼ 1; 2; …; lÞ, denoted by Ξ ¼ diagfε1 ; ε2 ; …; εl ; 0; …; 0g A Rn
n . Define ei ðtÞ ¼ xi ðtÞ  sðtÞ ði ¼ 1; 2; … ; NÞ as the synchronization error. Then, according to the controller (4), the error system is e_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ  f ðt; si ðtÞ; si ðt  τ1 ðtÞÞÞ N X þ aij Γ ðhðxj ðtÞÞ  hðsðtÞÞÞ N X

bij Γ ðgðxj ðt  τ2 ðtÞÞÞ  gðsðt  τ2 ðtÞÞÞÞ þ ui ðtÞ;

j¼1

i ¼ 1; 2; …; N:

Remark 1. A continuous function f is said to belong to the function class QUAD, denoted by f A QUADðP; Δ; η; ζ Þ, if f ðÞ satisfies Assumption 1. One can easily verify that the function class QUAD includes almost all the chaotic systems such as the Lorenz system, the Rössler system, the Chen system, the delayed Chua's circuit, the logistic delayed differential system, the delayed Hopfield neural network and the delayed CNNs. Assumption 2 (Wang et al. [25]). For each nonlinear coupling function hi ðÞ and g i ðÞ in network (1), there exist two nonnegative h g scalers βih, εih, βig and εig such that hi ðxÞ  β i x and g i ðxÞ  βi x satisfies the Lipschitz condition j hi ðx1 Þ  hi ðx2 Þ  β i ðx1 x2 Þj r εhi j x1  x2 j and j g i ðx1 Þ  g i ðx2 Þ  βi ðx1  x2 Þj r εgi j x1  x2 j ; g

respectively, for all x1 ; x2 A R, i ¼ 1; 2; …; n.

3. Main results

Theorem 1. Under Assumptions 1 and 2, if τ1 ðtÞ and τ2 ðtÞ are bounded and continuously differentiable functions such that τ_ 1 ðtÞ o τ 1 o1 and τ_ 2 ðtÞ o τ 2 o 1 and β

J ð c1 ðA  Υ g P Γ  Ξ  P Γ ÞÞ 4 ρ;

ð7Þ 1

where ρ ¼ λmax ð  ηΨ  I n þ Ψ  P ΔΓ þ c1 Ψ AA > Ψ  P ΓΘg þ 2   ε 1 1 ð1=NÞ I N  ðΥ g Þ2 P ΓΘg þ c2 Ψ BB > Ψ  P ΓΘh þ Ψ  Q þ Ψ  g g ε β ε g g RÞ, Υ g ¼ diagfε1 ; …; εn g, Υ g ¼ diagfβ1 ; …; βn g, Υ h ¼ diagfεh1 ; …; εhn g, h h β Υ h ¼ diagfβ1 ; …; βn g, Q ¼ ðζ =ð1  τ 1 ÞÞI n þ ν1 In , R ¼ 2c2 λmax ðΨ ÞððN  ε β 1Þ= Nð1  τ 2 ÞÞðΥ h þ Υ h Þ2 P ΓΘh þ ν2 I n , Θh ; Θg are positive definite matrices, Ψ is a positive definite diagonal matrix and ν1 and ν2 are positive scalars, then the pinning-controlled complex network (3) globally asymptotically synchronizes to the isolated node (2). β

j¼1

þ

Γ and all x; y; z; w A R .

In the section, we derive a sufficient condition for the attainment of synchronization by pinning control with the M-matrix approach.

j ¼ 1;i a j

þ

for some given matrix

ð6Þ n

h

where τ ¼ maxfτ1 ðtÞ; τ2 ðtÞg and φi A Cð½  τ ; 0; Rn Þ with the norm J φi J 2 ¼ sup  τ r s r 0 φi ðsÞ > φi ðsÞ and our objective is to control system (1) so that it stays in the trajectory sðtÞ A Rn of the system

N X

r  ηðx  yÞ > ðx  yÞ þ ζ ðz  wÞ > ðz  wÞ

n

where xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ; …; xin ðtÞÞ A R is the state vector of the ith node of the network, f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ ¼ ½f 1 ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ; f 2 ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ; …; f n ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ > is a continuous vector-valued function, Γ ¼ diagðγ 1 ; γ 2 ; …; γ n Þ is an inner coupling of the network that satisfies γ j 40, j ¼ 1; 2; …; n, A ¼ ½aij  A Rn
n and B ¼ ½bij  A Rn
n are the outer coupling matrices of the network at time t and t  τ2 ðtÞ respectively such that aij Z0 for P PN i a j, aii ¼  N j ¼ 1;j a i aij , bij Z0 for i aj and bii ¼  j ¼ 1;j a i bij , τ 1 ðtÞ is the inner time-varying delay, τ2 ðtÞ is the coupling timevarying delay and the nonlinear coupling functions hðÞ : Rn -Rn and gðÞ : Rn -Rn are continuous and of the form gðxi ðtÞÞ ¼ ðg 1 ðxi1 ðtÞÞ; g 2 ðxi2 ðtÞÞ; …; g n ðxin ðtÞÞÞ > and hðxi ðtÞÞ ¼ ðh1 ðxi1 ðt  τ2 ðtÞÞÞ; h2 ðxi2 ðt  τ2 ðtÞÞÞ; …; hn ðxin ðt  τ2 ðtÞÞÞÞ > . The initial conditions associated with (1) are xi ðsÞ ¼ φi ðsÞ;

91

ð5Þ

Proof. Let λi be an eigenvalue of c1 A  Υ g P Γ  Ξ  P Γ . Then the existence of a directed spanning tree in G ensures that Rðλi Þ 40 β and the pinning condition (7) implies that  c1 A  Υ g P Γ  Ξ  P Γ is an M-matrix. Hence, there exists a positive definite diagonal matrix Ψ ¼ diagfξ1 ; …; ξN g such that β

½Ψ ðc1 ðA  Υ g P Γ  Ξ  P Γ Þ  ρI nN ÞS o 0: Definition 4. The complex network (3) is said to be synchronized if the trivial solution of system (5) is such that N X i¼1

lim J ei ðt; t 0 ; φi Þ J ¼ 0;

t-1

for any initial data φi A Cð½  τ; 0; Rn Þ. Now, we introduce the following assumptions to study the synchronization of network (1). Assumption 1 (Wang et al. [17]). For each function f ðÞ in network (1), there exists a positive definite diagonal matrix P ¼ diagfp1 ; p2 ; …; pn g, a diagonal matrix Δ ¼ diagfδ1 ; δ2 ; …; δn g and

Choose the Lyapunov–Krasovskii function as 1 Vðt; eðtÞÞ ¼ eðtÞ > Ψ  PeðtÞ 2 Z t eðsÞ > Ψ  QeðsÞ ds þ t  τ 1 ðtÞ

Z þ

t

t  τ 2 ðtÞ

eðsÞ > Ψ  ReðsÞ ds;

where eðtÞ ¼ ðe1 ðtÞ > ; e2 ðtÞ > ; …; eN ðtÞ > Þ > and let ek ðtÞ ¼ ðe1k ðtÞ; g~ k ðxk ðtÞÞ ¼ ðg k ðx1k ðtÞÞ; g k ðx2k ðtÞÞ; …; g k ðxNk ðtÞÞÞ > , e2k ðtÞ; …; eNk ðtÞÞ > , k k ϵ~ g ðx ðtÞÞ ¼ ðϵg;k ðx1k ðtÞÞ; ϵg;k ðx2k ðtÞÞ; …; ϵg;k ðxNk ðtÞÞÞ > ðk ¼ 1; 2; …; nÞ.

92

J. Wang et al. / Neurocomputing 177 (2016) 89–97

Then differentiate V(t) along system (5) to get 8 N < X > _ ξi ei ðtÞ P f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ  f ðt; sðtÞ; sðt  τ1 ðtÞÞÞ V ðt; eðtÞÞ ¼ : i¼1

þ

N X

c1 aij Γ gðxj ðtÞÞ þ

j¼1

N X

c2 bij Γ hðxj ðt  τ2 ðtÞÞÞ  c1 εi Γ ei ðtÞ

j¼1

9 = ;

N X N X

  1 1 ε I N  ðΥ g Þ2 P ΓΘg ÞeðtÞ: þ Ψ AA > Ψ  P ΓΘg þ 2 1  N

þ eðtÞ > Ψ  ReðtÞ ð1  τ_ 2 Þeðt  τ2 ðtÞÞ > Ψ  Reðt  τ2 ðtÞÞ

i¼1j¼1



ξi ei ðtÞ P f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ f ðt; sðtÞ; sðt  τ1 ðtÞÞÞ >

i¼1



 ΔΓ ei ðtÞ þeðtÞ > Ψ  P ΔΓ eðtÞ þ

N X N X

þ

ei ðtÞ > Pbij Γ ðhðxj ðt  τ2 ðtÞÞÞ  hðsðt  τ2 ðtÞÞÞÞ 1

r eðtÞ > ðΨ BB > Ψ  P ΓΘh ÞeðtÞ   1 ε β I N  ðΥ h þ Υ h Þ2 P ΓΘh Þeðt  τ2 ðtÞÞ þeðt  τ2 ðtÞÞ > 2 1  N

ð12Þ

and after substituting inequalities (11-12) into Eq. (8), we have β V_ ðtÞ r eðtÞ >  ηΨ  I n þ c1 Ψ A  Υ g P Γ  c1 Ψ Ξ  P Γ

c1 ξi aij ei ðtÞ > P Γ gðxj ðtÞÞ

i¼1j¼1 N X N X

ð11Þ

Similarly, we have N X N X

N X

β

ei ðtÞ > Paij Γ ðgðxj ðtÞÞ  gðsðtÞÞÞ r eðtÞ > ðΨ A  Υ g P Γ

i¼1j¼1

þ eðtÞ > Ψ  QeðtÞ  ð1  τ_ 1 Þeðt  τ1 ðtÞÞ > Ψ  Qeðt  τ1 ðtÞÞ

¼

1

c2 ξi bij ei ðtÞ > P Γ hðxj ðt  τ2 ðtÞÞÞ  c1 eðtÞ > Ψ Ξ

i¼1j¼1

 P Γ eðtÞ þ eðtÞ > Ψ  QeðtÞ  ð1  τ_ 1 Þeðt  τ1 ðtÞÞ > Ψ  Qeðt  τ1 ðtÞÞ þeðtÞ > Ψ  ReðtÞ  ð1  τ_ 2 Þeðt  τ2 ðtÞÞ > Ψ  Reðt  τ2 ðtÞÞ: ð8Þ As Assumption 2, we have N X N X

It therefore follows from inequalities (9) and (10) that

ξi ei ðtÞ > Paij Γ ðgðxj ðtÞÞ  gðsðtÞÞÞ

þ Ψ  P ΔΓ þ c1 Ψ AA > Ψ  P ΓΘg   1 ε 1 I N  ðΥ g Þ2 P ΓΘg þ c2 Ψ BB > Ψ  P ΓΘh þ2 1 N   þ Ψ  Q þ Ψ  R eðtÞ þ eðt  τ1 ðtÞÞ > ζΨ  I n   ð1  τ 1 ÞΨ  Q eðt  τ1 ðtÞÞ    1 ε β I N  ðΥ h þ Υ h Þ2 P ΓΘh þ eðt  τ2 ðtÞÞ > 2c2 1  N   ð1  τ 2 ÞΨ  R eðt  τ2 ðtÞÞ:

ð13Þ

i¼1j¼1 n X

¼

k¼1 n X

¼

h i pk γ k ek ðtÞ > Ψ A g~ k ðxk ðtÞÞ  g~ k ðsk ðtÞÞ

βgk pk γ k ek ðtÞ > Ψ Aek ðtÞ

k¼1 n X

h i pk γ k ek ðtÞ > Ψ A ϵ~ kg ðxk ðtÞÞ  ϵ~ kg ðsk ðtÞÞ :

þ

ð9Þ

k¼1

Furthermore, since A is a zero-row-sum matrix and Assumption 2 (by Lemmas 4 and 5), we have n X

h i pk γ k ek ðtÞ > Ψ A ϵ~ kg ðxk ðtÞÞ  ϵ~ kg ðsk ðtÞÞ

Remark 2. Even though the network (3) can be synchronized as β long as the coupling strength satisfies J ð  c1 ðA  Υ g P Γ  Ξ  P Γ ÞÞ 4 ρ (see condition (7), the theoretical value required for the coupling strength is usually much larger than that needed in practice. We shall next use an adaptive approach to find a sharp bound for this coupling strength.

k¼1

r

n X pk γ k

θk

k¼1 n X

ek ðtÞ > Ψ AA > Ψ ek ðtÞ

h i> h i pk γ k θk ϵ~ k ðxk ðtÞÞ  ϵ~ k ðsk ðtÞÞ Q ϵ~ k ðxk ðtÞÞ  ϵ~ k ðsk ðtÞÞ

þ

Let yðtÞ ¼ ðeðtÞ > ; eðt  τ1 ðtÞÞ > ; eðt  τ2 ðtÞÞ > Þ > and recall that Q ¼ ε ðζ =ð1  τ 1 ÞÞI n þ ν1 I n and R ¼ 2c2 λmax ðΨ ÞððN  1Þ=Nð1  τ 2 ÞÞðΥ h þ β 2 _ Υ h Þ P ΓΘh þ ν2 In . Then inequality (13) implies that V ðtÞ r 0 and V_ ðtÞ ¼ 0 if and only if yðtÞ ¼ 03Nn . So the set M ¼ fyðtÞ : yðtÞ ¼ 03Nn g is the largest invariant set contained in the set D ¼ fyðtÞj VðtÞ ¼ 0g for system (5) and thus every solution of system (5) (starting from any initial condition) approaches M as t-1 (according to LaSalle's invariance principle). Therefore J ei ðtÞ J -0, i ¼ 1; 2; …; N and the pinning-controlled network (3) globally asymptotically synchronizes to the isolated node (1).□

k¼1

r

n X pk γ k

θ

k k¼1 n X

ek ðtÞ > Ψ AA > Ψ ek ðtÞ

pk γ k θk



j4i

k¼1

r

n X pk γ k k¼1

X  2 qij ðγ ðxkj ðtÞÞ  γ ðsk ðtÞÞÞ  ðγ ðxki ðtÞÞ  γ ðsk ðtÞÞÞ

θk

>

e ðtÞ Ψ AA Ψ e ðtÞ  2 k

>

k

 γ ðsk ðtÞÞÞ þ ðγ ðxki ðtÞÞ  γ ðsk ðtÞÞÞ n X pk γ k k¼1

 2ε2

θk

pk γ k θk

k¼1

¼

k¼1

þ2

θk

n X k¼1

2



j4i

1

ρ ¼ λmax ð  ηΨ  In þ Ψ  P ΔΓ þc2 Ψ BB > Ψ  P ΓΘh þ where Ψ  Q þ Ψ  RÞ, Q ¼ ðζ =ð1  τ 1 ÞÞIn þ ν1 In , R ¼ 2c2 λmax ðΨ ÞððN 1Þ= Nð1  τ 2 ÞÞP ΓΘh þ ν2 I n , Θh is a positive definite matrices, Ψ is a positive definite diagonal matrix and ν1 and ν2 are positive scales, then the pinning-controlled complex network (3) globally asymptotically synchronizes to the isolated node (2).

 X qij ekj ðtÞ2 þ eki ðtÞ2 j4i

ek ðtÞ > Ψ AA > Ψ ek ðtÞ

ε2k pk γ k θk



 1 k > k e ðtÞ e ðtÞ: 1 N

Corollary 1. Under Assumption 1, if τ1 ðtÞ and τ2 ðtÞ are bounded and continuously differentiable functions such that τ_ 1 ðtÞ o τ 1 o 1 and τ_ 2 ðtÞ o τ 2 o 1 and J ð  c1 ðA  P Γ  Ξ  P Γ ÞÞ 4 ρ;

ek ðtÞ > Ψ AA > Ψ ek ðtÞ

n X

n X pk γ k

X  pk γ k θ k qij ðγ ðxkj ðtÞÞ

k¼1

2

r

n X

To make Theorem 1 more applicative, we give the following corollaries. When hi ðxÞ ¼ x and g i ðxÞ ¼ x are linear functions, system (3) is a linearly-coupled system and we have the following result.

ð10Þ

When hi ðxÞ ¼ x is a linear function and t 1 ðtÞ ¼ τ and c2 ¼ 0 (without time-delay coupling), then system (3) is a linearlycoupled system without time-delay coupling and we have the following result.

J. Wang et al. / Neurocomputing 177 (2016) 89–97

93

Fig. 1. The topological structures of the undirected network with 50 nodes. We assume that aij ¼ aji ¼ 1 if there exists an edge between node i and node j and aij ¼ aji ¼ 0 otherwise in (a); bij ¼ bji ¼ 1 if there exists an edge between node i and node j and bij ¼ bji ¼ 0 otherwise in (b).

100

100

90 80

80

x (t),i=1,2,...,50

60 50 40

60

40

i2

xi1(t),i=1,2,...,50

70

20

30 20

0

10 0 −1

0

1

2

3

4

−20 −1

0

1

t

2

3

4

t

Fig. 2. The trajectories of the state variables of xi1 (a) and xi2 (b) for i ¼ 1; 2; …; 50 in system (15) using pinning control.

Corollary 2. Under Assumption 1, if J ð  c1 ðA  P Γ  Ξ  P Γ ÞÞ 4 ρ; where ρ ¼ λmax ð  ηΨ  I n þ Ψ  P ΔΓ þ Ψ  Q Þ, Q ¼ ζ I n þ ν1 I n , Θh is a positive definite matrices, Ψ is a positive definite diagonal matrix and ν1 and ν2 are positive scalars, then the pinning-controlled complex network (3) globally asymptotically synchronizes to the isolated node (2).

coupling strengths by using an adaptive feedback control technique. Let the coupling strengths of the network functions be timevarying (we shall now design the corresponding coupling strength adaptive laws). System (3) can be written as x_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ þ

N X

c1 ðtÞaij Γ ðgðxj ðtÞÞ  gðxi ðtÞÞÞ

j ¼ 1;i a j

Remark 3. Corollary 2 gives a sufficient condition for the synchronization of linearly coupled complex networks using pinning control that is the same as that obtained by Song et al. (Theorem 2 Ref. [27]). We can therefore say that Theorem 1 is a nontrivial extension of the result obtained in [27]. Real-world complex networks have coupling strengths that are usually less than their theoretical values [19]. The following theorem, however, allows us to accommodate for relatively small

þ

N X

c2 bij Γ ðhðxj ðt  τ2 ðtÞÞÞ  hðxi ðt  τ2 ðtÞÞÞÞ þ ui ðtÞ

j ¼ 1;i a j

i ¼ 1; 2; …; N;

ð14Þ

where c1 ðtÞ is an adaptive coupling strength that can be suitably chosen to synchronize system (14). Design a local feedback

94

J. Wang et al. / Neurocomputing 177 (2016) 89–97

controller (  c1 ðtÞεi Γ ðxi ðtÞ sðtÞÞ; ui ðtÞ ¼ 0;

   1 ε β þ eðt  τ2 ðtÞÞ > 2c2 1  I N  ðΥ h þ Υ h Þ2 P ΓΘh N   ð1  τ 2 ÞΨ  R eðt  τ2 ðtÞÞ:

i ¼ 1; 2; …; l; i ¼ l þ 1; l þ 2; …; N:

Theorem 2. Under Assumptions 1 and 2, if τ1 ðtÞ and τ2 ðtÞ are bounded and continuously differentiable functions such that τ_ 1 ðtÞ o τ 1 o 1 and τ_ 2 ðtÞ o τ 2 o 1. Let c1 ðtÞ be an adaptive coupling strength such that c_ 1 ðtÞ ¼ α

N X

ξi ei ðtÞ > Pei ðtÞ ¼ αeðtÞ > Ψ  PeðtÞ

i¼1

for some α 4 0 and initial value c1 ð0Þ Z0 and Ψ is a positive definite diagonal matrix. Then system (14) can be synchronized. Proof. Choose a constant α 4 0 and let ei ðtÞ ¼ xi ðtÞ sðtÞ. Construct the following Lyapunov–Krasovskii function Z t 1 VðtÞ ¼ eðtÞ > Ψ  PeðtÞ þ eðsÞ > Ψ  QeðsÞ ds 2 t  τ1 ðtÞ Z t 1 eðsÞ > Ψ  ReðsÞ dsþ ðc  κ c1 ðtÞÞ2 þ 2ακ t  τ 2 ðtÞ for some constants c and κ that will be determined later. Differentiating V(t) gives ( N dVðtÞ X ¼ ξi ei ðtÞ > P f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ f ðt; sðtÞ; sðt  τ1 ðtÞÞÞ dt i¼1 9 N N = X X c1 ðtÞaij Γ gðxj ðtÞÞ þ c2 bij Γ hðxj ðt  τ2 ðtÞÞÞ  c1 εi Γ ei ðtÞ þ ; j¼1

β

Hence, by choosing κ and c such that Ψ A  Υ g P Γ  Ψ Ξ    1 ε P Γ þ κΨ  P Γ þ Ψ AA > Ψ  P ΓΘg r0 and 2 1  1=N I N  ðΥ g Þ2 1 > P ΓΘg  ηΨ  I n  cΨ  P Γ þ Ψ  P ΔΓ þc2 Ψ BB Ψ  P ΓΘh þ Ψ  Q þ Ψ  R r 0, where Q ¼ ðζ =ð1  τ 1 ÞÞIn þ ν1 In , R ¼ 2c2 λmax ðΨ ε β ÞððN  1Þ= Nð1  τ 2 ÞÞðΥ h þ Υ h Þ2 P ΓΘh þ ν2 I n , and ν1 and ν2 are positive scalars, we have m X dVðtÞ r η ei ðtÞ > ei ðtÞ o0 dt i¼1

and so xi ðtÞ-sðtÞ and c_ ðtÞ-0 (by LaSalle's invariance principle). Therefore c(t) converges to some final coupling strength c0.□

4. Numerical simulation In this section, we present some numerical simulation results that validate the theorems of the previous section. 450 400 350

j¼1

300

þ eðtÞ > Ψ  QeðtÞ ð1  τ_ 1 Þeðt  τ1 ðtÞÞ > Ψ  Qeðt  τ1 ðtÞÞ þ eðtÞ Ψ  ReðtÞ  ð1  τ_ 2 Þeðt  τ2 ðtÞÞ Ψ  Reðt  τ2 ðtÞÞ >

>

E(t)

 ceðtÞ > Ψ  P Γ eðtÞ þ κ c1 ðtÞeðtÞ > Ψ  P Γ eðtÞ

250 200

and so, by Theorem 1, we have β V_ ðtÞ r eðtÞ > c1 ðtÞΨ A  Υ g P Γ  c1 ðtÞΨ Ξ  P Γ þ κ c1 ðtÞΨ  P Γ   1 1 ε I N  ðΥ g Þ2 P ΓΘg þ c1 ðtÞΨ AA > Ψ  P ΓΘg þ 2 1  N  ηΨ  I n cΨ  P Γ þ Ψ  P ΔΓ  1 þ c2 Ψ BB > Ψ  P ΓΘh þ Ψ  Q þ Ψ  R eðtÞ   þ eðt  τ1 ðtÞÞ > ζΨ  I n  ð1  τ 1 ÞΨ  Q eðt  τ1 ðtÞÞ

150 100 50 0 −1

0

1

2

3

4

t Fig. 4. The synchronization process in system (15) using pinning control.

70

20

60 0

50

ei2(t),i=1,2,...,50

i1

e (t),i=1,2,...,50

40 30 20 10

−20

−40

−60

0 −10

−80

−20 −30 −1

0

1

2 t

3

4

−100 −1

0

1

2 t

Fig. 3. The time-evolution of ei1 (a) and ei2 (b) for i ¼ 1; 2; …; 50 in system (15) using pinning control.

3

4

J. Wang et al. / Neurocomputing 177 (2016) 89–97

95

100

100 90

80

80

60 xi2(t),i=1,2,...,50

i1

x (t),i=1,2,...,50

70 60 50 40

40

20

30 20

0

10 0 −1

0

1

2

3

−20 −1

4

0

1

t

2

3

4

t

Fig. 5. The trajectories of the state variables of xi1 (a) and xi2 (b) for i ¼ 1; 2; …; 50 in system (15) using adaptive and pinning control.

70

20

60 0

50

−20 ei2(t),i=1,2,...,50

i1

e (t),i=1,2,...,50

40 30 20 10

−40

−60

0 −10

−80

−20 −30 −1

0

1

2

3

−100 −1

4

0

1

t

2

3

4

t

Fig. 6. The time-evolution of ei1 (a) and ei2 (b) for i ¼ 1; 2; …; 50 in system (15) using adaptive and pinning control.

450

"

400 350

E(t)

300

Consider the isolated 2D chaotic delayed neural network # #

#

" " s_ 1 ðtÞ 1 0 s1 ðtÞ 2  0:1 tanhðs1 ðtÞÞ ¼ þ s_ 2 ðtÞ tanhðs2 ðtÞÞ 5 4:5 0 1 s2 ðtÞ #

"  1:5  0:1 tanhðs1 ðt  τ1 ðtÞÞÞ ; þ tanhðs2 ðt  τ1 ðtÞÞÞ  0:2 4

where τ1 ðtÞ ¼ 1. Taking P ¼ diagf1; 1g and Δ ¼ diagf5; 11:5g, we have η ¼ 0:15 and ζ ¼ 3:25 so that condition (6) is satisfied. Now consider a network with 50 nodes with a topological structure that is shown in Fig. 1

250 200 150

x_ i ðtÞ ¼ f ðt; xi ðtÞ; xi ðt  τ1 ðtÞÞÞ þ c1

100

aij Γ gðxj ðtÞÞ

j¼1

50 0 −1

50 X

þ c2 0

1

2

3

4

t Fig. 7. The synchronization process of system (15) using adaptive and pinning control.

50 X

bij Γ hðxj ðt  τ2 ðtÞÞÞ c1 εi Γ ðxi ðtÞ  sðtÞÞ; i ¼ 1; 2; …; 50;

j¼1

ð15Þ where τ2 ðtÞ ¼ 0:1et =ð1 þ et Þ, g j ðxÞ ¼ 25x þ sin ðxÞ and hj ðxÞ ¼ 5x þ cos ðxÞ for j ¼ 1; 2, Γ ¼ I 2 .

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J. Wang et al. / Neurocomputing 177 (2016) 89–97

30

Acknowledgment This work was supported by the National Science Foundation of China under Grant No. 61273220, 61373087, 61472257 and 61401283. The authors are grateful to the reviewers and editors for their valuable comments and suggestions to improve the presentation of this paper.

c1(t)

20

References

10

0

0

0.5

1

1.5

2 t

2.5

3

3.5

4

Fig. 8. The adaptive coupling strength c1 ðtÞ of the pinning-controlled complex network.

Computations then yield τ 1 ¼ 0 and τ 2 ¼ 0:1 for i ¼ 1; 2; …; 50 β β ε ε and j ¼ 1; 2, Υ g ¼ 25I 2 , Υ h ¼ 5I 2 , Υ g ¼ I 2 , Υ h ¼ I 2 . Let l¼ 7 and the control strength εi ¼ 100 for i ¼ 1; 2; …; 7. Then the solutions of inequalities (7) are (by using the Matlab LMI toolbox): c1 ¼ 10, c2 ¼ 0.01, ρ ¼ 180:3202 Θh ¼ 1:9082I 2 , Θg ¼ 41:9134I 2 , Q ¼ 3:251I 2 , R¼ 1.497, ν1 ¼ 0:001, ν2 ¼ 0:001 and J ¼ 183:1517. The initial conditions for this simulation are xij ðt 0 Þ ¼ randij
cos ðt 0 Þ for some random integer randij in ½0; 100 for i ¼ 1; 2; …; 50, j¼1,2 and sðt 0 Þ ¼ ½30 cos ðt 0 Þ; 2 cos ðt 0 Þ > for all t 0 A ½  1; 0 and the trajectories of the states are shown in Fig. 2. Fig. 3 shows the time evolution of the synchronization errors with pinning control. Measuring the quality of the synchronization process by EðtÞ ¼

50 1 X J x ðtÞ  sðtÞ J 2 50 i ¼ 1 i

in Fig. 4, it is clear that synchronization is achieved. The above simulations show that large coupling and control strengths (c1 and εi) are needed to meet condition (7). Now use the same parameters with α ¼ 0:0001 to simulate the controlled dynamic network (14) with adaptive coupling strength c1 ðtÞ. The trajectories of the states are shown in Fig. 5 and Fig. 6 shows the time evolution of the synchronization errors with pinning control. Figs. 7 and 8 show the time evolution of E(t) and c1 ðtÞ (note that c1 ðtÞ approaches 28.169) for the pinned complex network by adaptive control, respectively. It is clear that synchronization is achieved. (Note that the final coupling strength is smaller than its theoretical value.)

5. Conclusion In this paper, we investigated the synchronization problem for complex networks with non-delayed and time-varying delayed couplings. Specifically, we achieved global synchronization by applying a pinning control scheme to a small fraction of the nodes using the M-matrix strategies. Noting the big difference in the coupling strength magnitudes between theoretical and real-world systems, we also rigorously proved an adaptive feedback control technique that can be used to synchronize any real-world network. Finally, we considered some numerical examples that illustrate the theoretical analysis.

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Jingyi Wang received the B.S. degree in Information Management and Information System from Northwest Normal University, Lanzhou, China and the M.S. degree in Applied Mathematics from Shenzhen University, China, in 2009 and 2012, respectively. Currently, he is working toward the Ph.D. degree in the College of Information Engineering, Shenzhen University, China. His research interests include multi-agent systems, complex networks and nonlinear dynamics.

Jianwen Feng was born in Hubei, China in 1964. He received the B.S. degree in Mathematics/Applied Mathematics from Hubei Normal University, Huangshi, China, in 1986, and the M.S. and Ph.D. degrees in Mathematics/Applied Mathematics from Wuhan University, Wuhan, China, in 1995, and 2001, respectively. From 1986 to 1998, he was a faculty member in Yunyang Normal College, Shiyan, China. Since 2001, he has worked in the College of Mathematics and Computational Science, Shenzhen University, Shenzhen. And he is currently a Professor of Applied Mathematics in Shenzhen University. From 2009 to 2010, he was a Visiting Research Fellow and a Visiting Professor with the Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. He has authored and co-authored more than 30 refereed international journal papers and has been the reviewer for several international journals. His research interests include nonlinear systems, control theory and applications, complex networks, stability theory and applied mathematics.

Chen Xu received the B.Sc. and M.Sc. degrees from Xidian University in 1986 and 1989, respectively, and the Ph.D. degree from Xi'an Jiaotong University in 1992. He joined the Shenzhen University, Shenzhen, China in 1992 and currently is a Professor. From September 1999 to January 2000, he was a research fellow with the Kansai University, Japan. From August 2002 to August 2003, he was a research fellow with the University of Hawaii, USA. His research interests are image processing, intelligent computing and wavelet analysis.

97 Yi Zhao was born in Xinjiang, China, in 1980. He received the B.S. degree in Mathematics and Applied Mathematics from Fudan University, Shanghai, China, in 2003, and received the Ph.D. degree in Mathematics from City University of Hong Kong, Hong Kong, in 2007. Now he is a Lecturer in the college of mathematics and computational science, Shenzhen University, Guangdong, China. His main research interests are complex networks, control theory and applications and asymptotic theory.

Jiqiang Feng received the B.Sc. degree from Yantai Normal College in 2005, and received the M.Sc. and Ph. D. degrees from Shenzhen University in 2008 and 2011, respectively. He joined the Shenzhen University, Shenzhen, China in 2011 and currently is a Lecturer. His research interests are swarm optimization, fuzzy theory and image processing.