# Generalized Synchronization between Two Complex Dynamical Networks with Time-Varying Delay and Nonlinear Coupling

Authors: Qiuxiang Bian; Hongxing Yao
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

The generalized synchronization between two complex networks with nonlinear coupling and time-varying delay is investigated in this paper. The novel adaptive schemes of constructing controller response network are proposed to realize generalized synchronization with the drive network to a given mapping. Two specific examples show and verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

Over the past decade, complex networks have gained a lot of attention in various fields, such as sociology, biology, physical sciences, mathematics, and engineering [1–5]. A complex network is a large number of interconnected nodes, in which each node represents a unit (or element) with certain dynamical system and edge represents the relationship or connection between two units (or elements). Synchronization is one of the most important dynamical properties of dynamical systems, there are different kinds of methods to realize synchronization such as active control [6], feedback control [7], adaptive control [8], impulsive control [9], passive method [10], and so forth. Synchronization of complex networks includes complete synchronization (CS) [11, 12], projective synchronization (PS) [13, 14], phase synchronization [15, 16], generalized synchronization (GS) [17, 18], and so on.As a sort of synchronous behavior, GS is an extension of CS and PS, so GS is more widespread than CS and PS in nature and in technical applications. GS of chaos system has been widely researched. However, most of theoretical results about synchronization of complex networks focus on CS and PS. Especially, due to the complexity of GS, the theoretical results for GS are lacking, but GS of complex networks is attracting special attention; in [17], the author gives a novel definition of GS on networks and a numerical simulation example. Reference [18] applies the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators.Recently, GS of drive-response chaos systems is investigated by the nonlinear control theory in [19]. In this letter, we extend this method to investigate GS between two complex networks, and some criterions for GS are deduced.This letter is organized as follows. In Section2, the definition of GS between the drive-response complex networks is given and some preliminary knowledge, including three assumptions and one lemma is also introduced. By employing the Lyapunov theory and Barbǎlat lemma, some schemes are designed to construct response networks to realize GS with respect to the given nonlinear smooth mapping. In Section 3, two numerical examples are given to demonstrate the effectiveness of the proposed method in Section 2. Finally, conclusions are given in Section 4.


## 2. GS Theorems between Two Complex Networks with Nonlinear Coupling

### 2.1. Definition and Assumptions

Definition 2.1.
Supposexi=(xi1,xi2,…,xin)T∈Rn,yi=(yi1,yi2,…,yin)T∈Rn,i=1,2,…,N are the state variables of the drive network and the response network, respectively. Given the smooth vector function Φ:Rn→Rn, the drive network and response network are said to achieve GS with respect to Φ. If
(2.1)lim⁡t→∞‖ei(t)‖=0,i=1,2,…,N,
where ei(t)=xi(t)-Φ(yi(t)),i=1,2,…,N, the norm ||·|| of a vector x is defined as ||x||=(xTx)1/2.Remark 2.2.
IfΦ(yi)=yi, then GS is CS in [20]. If Φ(yi)=λyi, then GS is PS in [13, 14].In this paper, we consider a general complex dynamical network with time-varying nonlinear coupling and consisting ofNnonidentical nodes:(2.2)ẋi(t)=fi(xi(t))+∑j=1Ncijh(xj(t-τ(t))),i=1,2,…,N,
where xi=(xi1,xi2,…,xin)T∈Rn,i=1,2,…,N are the state variables of the drive network, fi:Rn→Rn, h:Rn→Rnare smooth nonlinear vector functions, and τ(t) is time-varying delay. C=(cij)N×N is unknown or uncertain coupling matrix; if there is a connection between node i and node j(j≠i), then cij≠0, otherwisecij=0(i≠j),and the diagonal elements of C are defined by(2.3)cii=-∑j=1j≠iNcij.It should be noted that the complex dynamical network model (2.2) is quite general. If fi=f, i=1,2,…,l; gi=g,i=l+1,l+2,…,N, then we can get the following complex dynamical network:(2.4)ẋi(t)=f(xi(t))+∑j=1Ncijh(xj(t-τ(t))),i=1,…,l,ẋi(t)=g(xi(t))+∑j=1Ncijh(xj(t-τ(t))),i=l+1,…,N.
On the other hand, if h(xi)=Axi, with A=(aij)N×N being an inner-coupling constant matrix of the network, then the complex network model (2.2) degenerates into the model of linearly and diffusively coupled network with coupling delays:(2.5)ẋi(t)=fi(xi(t))+∑j=1NcijAxj(t-τ(t)),i=1,2,…,N.
Let(2.6)DΦ(yi)=(∂ϕ1(yi)∂yi1∂ϕ1(yi)∂yi2⋯∂ϕ1(yi)∂yin∂ϕ2(yi)∂yi1∂ϕ2(yi)∂yi2⋯∂ϕ2(yi)∂yin⋯⋯⋯⋯∂ϕn(yi)∂yi1∂ϕn(yi)∂yi2⋯∂ϕn(yi)∂yin)
be the Jacobian matrix of the mapping Φ(yi)=(ϕ1(yi),ϕ2(yi),…,ϕn(yi))T,ϕj(yi)∈R, i=1,2,…,N,  j=1,2,…,n. When matrix DΦ(yi(t)) is reversible, we can give the following controller response network:(2.7)ẏi(t)=DΦ(yi(t))-1[fi(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=1,2,…,N,
where yi=(yi1,yi2,…,yin)T∈Rn,i=1,2,…,N are the state variables of the response network, ui∈Rn,i=1,2,…,N are nonlinear controllers to be designed, and Ĉ=(ĉij)N×N is the estimate of the unknown coupling matrix C=(cij)N×N.Letei(t)=xi(t)-Φ(yi(t)), with the aid of (2.2) and (2.7), the following error network can be obtained:(2.8)ėi(t)=ẋi(t)-[DΦ(yi(t))]ẏi(t)=fi(xi(t))-fi(Φ(yi(t)))+∑j=1Ncijh(xj(t-τ(t)))-∑j=1Nĉijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui=fi(xi(t))-fi(Φ(yi(t)))+∑j=1NcijH(ej(t-τ(t)))-∑j=1Nc̃ijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui,
where
(2.9)H(ej(t))=h(xj(t))-h(Φ(yj(t))),(2.10)c̃ij=ĉij-cij.The following conditions are needed for the solutions of (2.8) to achieve the objective (2.1).Assumption 1.
(A1) Time delayτ(t) is a differential function with 0≤τ(t)≤h, τ̇(t)≤μ<1, where h and μ are positive constants. Obviously, this assumption holds for constant τ(t).Assumption 2.
(A2) Suppose thatfi(·) is bounded and there exists a nonnegative constant α such that
(2.11)‖fi(xi(t))-fi(Φ(yi(t)))‖≤α‖ei(t)‖,i=1,2,…,N.Assumption 3.
(A3) Suppose thath(·) is bounded and there exists a nonnegative constant β such that
(2.12)‖h(xi(t))-h(Φ(yi(t)))‖≤β‖ei(t)‖,i=1,2,…,N.Remark 2.3.
The condition (2.12) is reasonable due to [21, 22]. For example, the Hopfield neural network [23] is described by
(2.13)dxi(t)dt=-xi(t)Ri+∑j=1Nwijfj(xj(t-τij(t)))+Ii,i=1,2,…,N.
Take fj(xj)=(π/2)arctan((π/2)λxj),where λ is positive constant. It is obvious that fj(·) satisfies Assumption 3.Lemma 2.4.
For any vectorsX,Y∈Rn,the following inequality holds
(2.14)2XTY≤XTX+YTY.Next section, we will give some sufficient conditions of complex dynamical networks (2.2) and (2.7) obtaining GS.


### 2.2. Main Results

Theorem 2.5.
Suppose that  (A1)–(A3) hold. Using the following controller:(2.15)ui=DΦ(yi(t))-1diei(t)
and the update laws
(2.16)ḋi=kieiT(t)ei(t),(2.17)ĉ̇ij=δijeiT(t)h(Φ(yj(t-τ(t)))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.2) and  (2.7) will achieve  GS with respect to  Φ.Proof.
Select a Lyapunov-Krasovskii functional candidate as(2.18)V(t,e(t))=∑i=1NeiT(t)ei(t)+∑i=1N∑j=1N1δijc̃ij2+∑i=1N1ki(di-d)2+Nβ21-μ∫t-τ(t)t∑i=1NeiT(ξ)ei(ξ)dξ,
where e(t)=(e1T(t),e2T(t),…,eNT(t))T and d is a positive constant to be determined.
The time derivative ofV along the solution of the error system (2.8) is(2.19)dVdt=∑i=1N2eiT(t)ei̇(t)+∑i=1N∑j=1N2δijc̃ijĉ̇ij+∑i=1N2ki(di-d)di̇+Nβ21-μ∑i=1NeiT(t)ei(t)-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t))=∑i=1N2eiT(t)[fi(xi(t))-fi(Φ(yi(t)))+∑j=1NcijH(ej(t-τ(t)))-∑j=1Nc̃ijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui(t)∑j=1NcijH(ej(t-τ(t)))]+∑i=1N∑j=1N2δijc̃ijĉ̇ij+∑i=1N2ki(di-d)ḋi+Nβ21-μ∑i=1NeiT(t)ei(t)-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t)).
Substituting the controller (2.15) and update laws (2.16)-(2.17) into (2.19) and considering Assumption 2, we obtain
(2.20)dVdt≤∑i=1N(2α-2d+Nβ21-μ)eiT(t)ei(t)+2∑i=1N∑j=1NeiT(t)cijH(ej(t-τ(t)))-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t)).
By Lemma 2.4 and considering Assumptions 1 and 3, we have
(2.21)1-τ̇(t)1-μ≥1,2∑i=1N∑j=1NeiT(t)cijH(ej(t-τ(t)))≤∑i=1N∑j=1Ncij2eiT(t)ei(t)+∑i=1N∑j=1NHT(ej(t-τ(t)))H(ej(t-τ(t)))≤Nmax⁡1≤i,j≤N{cij2}∑i=1NeiT(t)ei(t)+Nβ2∑j=1NejT(t-τ(t))ej(t-τ(t)),
then
(2.22)dVdt≤∑i=1N(2α-2d+Nβ21-μ+Nmax⁡1≤i,j≤N{cij2})eiT(t)ei(t)=-(2d-2α-Nβ21-μ-Nmax⁡1≤i,j≤N{cij2})eT(t)e(t).
Note that we can choose constantd to make dV/dt≤-eT(t)e(t)≤0, thus V is nonincreasing in t≥0. One has V is bounded since 0≤V(t,e(t))≤V(0,e(0)), so limt→+∞V(t,e(t)) exists and(2.23)lim⁡t→+∞∫0teT(s)e(s)ds≤-lim⁡t→+∞∫0tdVdsds=V(0,e(0))-lim⁡t→+∞V(t,e(t)).
From (2.18), we have 0≤eT(t)e(t)≤V(t,e(t)), so eT(t)e(t) is bounded. According to error system (2.8), (d/dt)eT(t)e(t)=2eT(t)ė(t) is bounded for t≥0 due to the boundedness of fi(·) and h(·). From the above, we can see that e(t)∈L2∩L∞ and ė(t)∈L∞. By using another form of Barbǎlat lemma [24], one has limt→+∞eT(t)e(t)=0, so limt→+∞e(t)=0 and the complex dynamical networks (2.2) and (2.7) can obtain generalized synchronization under the controller (2.15) and update laws (2.16)-(2.17). This completes the proof.Remark 2.6.
Iflimt→+∞ė(t) exists,then we can obtain limt→+∞ė(t)=0 for limt→+∞e(t)=0. According to error system (2.8), we have limt→+∞∑j=1Nc̃ijh(Φ(yj(t-τ(t))))=0. When {h(Φ(yj(t-τ(t)))}j=1N are linearly independent on the orbit {yj(t-τ(t))}j=1N of synchronization manifold, limt→+∞c̃ij=0. We can get limt→+∞ĉij=cij,i,j=1,2,…,N; that is, the uncertain coupling matrix C can be successfully estimated using the update laws (2.17).
In a special caseΦ(yi)=λyi(λis nonzero constant), based on Theorem 2.5, we can construct the following response network(2.24)ẏi(t)=1λ[fi(λyi(t))+∑j=1Nĉijh(λyj(t-τ(t)))]+ui,i=1,2,…,N.Corollary 2.7.
Suppose that  (A1)–(A3) hold. Using the controller(2.25)ui=1λdiei(t)
and the update laws
(2.26)ḋi=kieiT(t)ei(t),ĉ̇ij=δijeiT(t)h(λyj(t-τ(t))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.2) and  (2.24) will obtain PS.To networks (2.4), according to Theorem 2.5, one can construct the following response network:(2.27)ẏi(t)=DΦ(yi(t))-1[f(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=1,…,l,ẏi(t)=DΦ(yi(t))-1[g(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=l+1,…,N
and get the following corollary:Corollary 2.8.
Suppose that  (A1)–(A3) hold. Using the controller(2.28)ui=DΦ(yi(t))-1diei(t)
and the update laws
(2.29)ḋi=kieiT(t)ei(t),ĉ̇ij=δijeiT(t)h(Φ(yj(t-τ(t)))),
where  i,j=1,2,…,N,di is feedback strength,  δij>0,ki>0 are arbitrary constants, then the complex dynamical network networks  (2.4) and  (2.27) will achieve  GS with respect to  Φ.If coupling functionh(xi)=Axi; that is, the network is linearly coupled, then the complex network (2.2) degenerates into (2.5). Note that ∥Aei(t)∥≤∥A∥·∥ei(t)∥,i=1,2,…,N hold. We construct the following response network:(2.30)ẏi(t)=DΦ(yi(t))-1[fi(Φ(yi(t)))+∑j=1NĉijAΦ(yj(t-τ(t)))]+ui,i=1,2,…,N.Corollary 2.9.
Suppose that  (A1) and (A2) hold. Using the controller(2.31)ui=DΦ(yi(t))-1diei(t)
and the update laws
(2.32)ḋi=kieiT(t)ei(t),ĉ̇ij=δijeiT(t)AΦ(yj(t-τ(t))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.5) and  (2.30) will obtain GS.Using different control, we can obtain the following theorem.Theorem 2.10.
Suppose that  (A1) and (A3) hold. Using the following controller:(2.33)ui=DΦ(yi(t))-1[diei(t)+fi(xi(t))-fi(Φ(yi(t)))],
and the update laws
(2.34)ḋi=kieiT(t)ei(t),(2.35)ĉ̇ij=δijeiT(t)h(Φ(yj(t-τ(t)))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.2) and  (2.7) will achieve  GS with respect to  Φ.Proof.
Select the same Lyapunov-Krasovskii function as Theorem2.5, then
(2.36)dVdt=2∑i=1NeiT(t)[fi(xi(t))-fi(Φ(yi(t)))+∑j=1NcijH(ej(t-τ(t)))-∑j=1Nc̃ijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui(t)∑j=1Ncij]+∑i=1N∑j=1N2δijc̃ijĉ̇ij+∑i=1N2ki(di-d)ḋi+Nβ21-μ∑i=1NeiT(t)ei(t)-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t))≤∑i=1N(-2d+Nβ21-μ)eiT(t)ei(t)+∑i=1N∑j=1NeiT(t)cijH(ej(t-τ(t)))-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t)).
The rest of the proof is similar to Theorem 2.5 and omitted here. This completes the proof.Remark 2.11.
According to Remark2.6, when {h(Φ(yj(t-τ(t))))}j=1N are linearly independent on the orbit {yj(t-τ(t))}j=1N of synchronization manifold, we can get limt→+∞ĉij=cij,i,j=1,2,…,N; that is, the uncertain coupling matrix C can be successfully estimated using the updating laws (2.35).Remark 2.12.
Based on Theorem2.10, we can get corollaries corresponding to Corollaries 2.7–2.9.


## 2.1. Definition and Assumptions

Definition 2.1.
Supposexi=(xi1,xi2,…,xin)T∈Rn,yi=(yi1,yi2,…,yin)T∈Rn,i=1,2,…,N are the state variables of the drive network and the response network, respectively. Given the smooth vector function Φ:Rn→Rn, the drive network and response network are said to achieve GS with respect to Φ. If
(2.1)lim⁡t→∞‖ei(t)‖=0,i=1,2,…,N,
where ei(t)=xi(t)-Φ(yi(t)),i=1,2,…,N, the norm ||·|| of a vector x is defined as ||x||=(xTx)1/2.Remark 2.2.
IfΦ(yi)=yi, then GS is CS in [20]. If Φ(yi)=λyi, then GS is PS in [13, 14].In this paper, we consider a general complex dynamical network with time-varying nonlinear coupling and consisting ofNnonidentical nodes:(2.2)ẋi(t)=fi(xi(t))+∑j=1Ncijh(xj(t-τ(t))),i=1,2,…,N,
where xi=(xi1,xi2,…,xin)T∈Rn,i=1,2,…,N are the state variables of the drive network, fi:Rn→Rn, h:Rn→Rnare smooth nonlinear vector functions, and τ(t) is time-varying delay. C=(cij)N×N is unknown or uncertain coupling matrix; if there is a connection between node i and node j(j≠i), then cij≠0, otherwisecij=0(i≠j),and the diagonal elements of C are defined by(2.3)cii=-∑j=1j≠iNcij.It should be noted that the complex dynamical network model (2.2) is quite general. If fi=f, i=1,2,…,l; gi=g,i=l+1,l+2,…,N, then we can get the following complex dynamical network:(2.4)ẋi(t)=f(xi(t))+∑j=1Ncijh(xj(t-τ(t))),i=1,…,l,ẋi(t)=g(xi(t))+∑j=1Ncijh(xj(t-τ(t))),i=l+1,…,N.
On the other hand, if h(xi)=Axi, with A=(aij)N×N being an inner-coupling constant matrix of the network, then the complex network model (2.2) degenerates into the model of linearly and diffusively coupled network with coupling delays:(2.5)ẋi(t)=fi(xi(t))+∑j=1NcijAxj(t-τ(t)),i=1,2,…,N.
Let(2.6)DΦ(yi)=(∂ϕ1(yi)∂yi1∂ϕ1(yi)∂yi2⋯∂ϕ1(yi)∂yin∂ϕ2(yi)∂yi1∂ϕ2(yi)∂yi2⋯∂ϕ2(yi)∂yin⋯⋯⋯⋯∂ϕn(yi)∂yi1∂ϕn(yi)∂yi2⋯∂ϕn(yi)∂yin)
be the Jacobian matrix of the mapping Φ(yi)=(ϕ1(yi),ϕ2(yi),…,ϕn(yi))T,ϕj(yi)∈R, i=1,2,…,N,  j=1,2,…,n. When matrix DΦ(yi(t)) is reversible, we can give the following controller response network:(2.7)ẏi(t)=DΦ(yi(t))-1[fi(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=1,2,…,N,
where yi=(yi1,yi2,…,yin)T∈Rn,i=1,2,…,N are the state variables of the response network, ui∈Rn,i=1,2,…,N are nonlinear controllers to be designed, and Ĉ=(ĉij)N×N is the estimate of the unknown coupling matrix C=(cij)N×N.Letei(t)=xi(t)-Φ(yi(t)), with the aid of (2.2) and (2.7), the following error network can be obtained:(2.8)ėi(t)=ẋi(t)-[DΦ(yi(t))]ẏi(t)=fi(xi(t))-fi(Φ(yi(t)))+∑j=1Ncijh(xj(t-τ(t)))-∑j=1Nĉijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui=fi(xi(t))-fi(Φ(yi(t)))+∑j=1NcijH(ej(t-τ(t)))-∑j=1Nc̃ijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui,
where
(2.9)H(ej(t))=h(xj(t))-h(Φ(yj(t))),(2.10)c̃ij=ĉij-cij.The following conditions are needed for the solutions of (2.8) to achieve the objective (2.1).Assumption 1.
(A1) Time delayτ(t) is a differential function with 0≤τ(t)≤h, τ̇(t)≤μ<1, where h and μ are positive constants. Obviously, this assumption holds for constant τ(t).Assumption 2.
(A2) Suppose thatfi(·) is bounded and there exists a nonnegative constant α such that
(2.11)‖fi(xi(t))-fi(Φ(yi(t)))‖≤α‖ei(t)‖,i=1,2,…,N.Assumption 3.
(A3) Suppose thath(·) is bounded and there exists a nonnegative constant β such that
(2.12)‖h(xi(t))-h(Φ(yi(t)))‖≤β‖ei(t)‖,i=1,2,…,N.Remark 2.3.
The condition (2.12) is reasonable due to [21, 22]. For example, the Hopfield neural network [23] is described by
(2.13)dxi(t)dt=-xi(t)Ri+∑j=1Nwijfj(xj(t-τij(t)))+Ii,i=1,2,…,N.
Take fj(xj)=(π/2)arctan((π/2)λxj),where λ is positive constant. It is obvious that fj(·) satisfies Assumption 3.Lemma 2.4.
For any vectorsX,Y∈Rn,the following inequality holds
(2.14)2XTY≤XTX+YTY.Next section, we will give some sufficient conditions of complex dynamical networks (2.2) and (2.7) obtaining GS.


## 2.2. Main Results

Theorem 2.5.
Suppose that  (A1)–(A3) hold. Using the following controller:(2.15)ui=DΦ(yi(t))-1diei(t)
and the update laws
(2.16)ḋi=kieiT(t)ei(t),(2.17)ĉ̇ij=δijeiT(t)h(Φ(yj(t-τ(t)))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.2) and  (2.7) will achieve  GS with respect to  Φ.Proof.
Select a Lyapunov-Krasovskii functional candidate as(2.18)V(t,e(t))=∑i=1NeiT(t)ei(t)+∑i=1N∑j=1N1δijc̃ij2+∑i=1N1ki(di-d)2+Nβ21-μ∫t-τ(t)t∑i=1NeiT(ξ)ei(ξ)dξ,
where e(t)=(e1T(t),e2T(t),…,eNT(t))T and d is a positive constant to be determined.
The time derivative ofV along the solution of the error system (2.8) is(2.19)dVdt=∑i=1N2eiT(t)ei̇(t)+∑i=1N∑j=1N2δijc̃ijĉ̇ij+∑i=1N2ki(di-d)di̇+Nβ21-μ∑i=1NeiT(t)ei(t)-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t))=∑i=1N2eiT(t)[fi(xi(t))-fi(Φ(yi(t)))+∑j=1NcijH(ej(t-τ(t)))-∑j=1Nc̃ijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui(t)∑j=1NcijH(ej(t-τ(t)))]+∑i=1N∑j=1N2δijc̃ijĉ̇ij+∑i=1N2ki(di-d)ḋi+Nβ21-μ∑i=1NeiT(t)ei(t)-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t)).
Substituting the controller (2.15) and update laws (2.16)-(2.17) into (2.19) and considering Assumption 2, we obtain
(2.20)dVdt≤∑i=1N(2α-2d+Nβ21-μ)eiT(t)ei(t)+2∑i=1N∑j=1NeiT(t)cijH(ej(t-τ(t)))-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t)).
By Lemma 2.4 and considering Assumptions 1 and 3, we have
(2.21)1-τ̇(t)1-μ≥1,2∑i=1N∑j=1NeiT(t)cijH(ej(t-τ(t)))≤∑i=1N∑j=1Ncij2eiT(t)ei(t)+∑i=1N∑j=1NHT(ej(t-τ(t)))H(ej(t-τ(t)))≤Nmax⁡1≤i,j≤N{cij2}∑i=1NeiT(t)ei(t)+Nβ2∑j=1NejT(t-τ(t))ej(t-τ(t)),
then
(2.22)dVdt≤∑i=1N(2α-2d+Nβ21-μ+Nmax⁡1≤i,j≤N{cij2})eiT(t)ei(t)=-(2d-2α-Nβ21-μ-Nmax⁡1≤i,j≤N{cij2})eT(t)e(t).
Note that we can choose constantd to make dV/dt≤-eT(t)e(t)≤0, thus V is nonincreasing in t≥0. One has V is bounded since 0≤V(t,e(t))≤V(0,e(0)), so limt→+∞V(t,e(t)) exists and(2.23)lim⁡t→+∞∫0teT(s)e(s)ds≤-lim⁡t→+∞∫0tdVdsds=V(0,e(0))-lim⁡t→+∞V(t,e(t)).
From (2.18), we have 0≤eT(t)e(t)≤V(t,e(t)), so eT(t)e(t) is bounded. According to error system (2.8), (d/dt)eT(t)e(t)=2eT(t)ė(t) is bounded for t≥0 due to the boundedness of fi(·) and h(·). From the above, we can see that e(t)∈L2∩L∞ and ė(t)∈L∞. By using another form of Barbǎlat lemma [24], one has limt→+∞eT(t)e(t)=0, so limt→+∞e(t)=0 and the complex dynamical networks (2.2) and (2.7) can obtain generalized synchronization under the controller (2.15) and update laws (2.16)-(2.17). This completes the proof.Remark 2.6.
Iflimt→+∞ė(t) exists,then we can obtain limt→+∞ė(t)=0 for limt→+∞e(t)=0. According to error system (2.8), we have limt→+∞∑j=1Nc̃ijh(Φ(yj(t-τ(t))))=0. When {h(Φ(yj(t-τ(t)))}j=1N are linearly independent on the orbit {yj(t-τ(t))}j=1N of synchronization manifold, limt→+∞c̃ij=0. We can get limt→+∞ĉij=cij,i,j=1,2,…,N; that is, the uncertain coupling matrix C can be successfully estimated using the update laws (2.17).
In a special caseΦ(yi)=λyi(λis nonzero constant), based on Theorem 2.5, we can construct the following response network(2.24)ẏi(t)=1λ[fi(λyi(t))+∑j=1Nĉijh(λyj(t-τ(t)))]+ui,i=1,2,…,N.Corollary 2.7.
Suppose that  (A1)–(A3) hold. Using the controller(2.25)ui=1λdiei(t)
and the update laws
(2.26)ḋi=kieiT(t)ei(t),ĉ̇ij=δijeiT(t)h(λyj(t-τ(t))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.2) and  (2.24) will obtain PS.To networks (2.4), according to Theorem 2.5, one can construct the following response network:(2.27)ẏi(t)=DΦ(yi(t))-1[f(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=1,…,l,ẏi(t)=DΦ(yi(t))-1[g(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=l+1,…,N
and get the following corollary:Corollary 2.8.
Suppose that  (A1)–(A3) hold. Using the controller(2.28)ui=DΦ(yi(t))-1diei(t)
and the update laws
(2.29)ḋi=kieiT(t)ei(t),ĉ̇ij=δijeiT(t)h(Φ(yj(t-τ(t)))),
where  i,j=1,2,…,N,di is feedback strength,  δij>0,ki>0 are arbitrary constants, then the complex dynamical network networks  (2.4) and  (2.27) will achieve  GS with respect to  Φ.If coupling functionh(xi)=Axi; that is, the network is linearly coupled, then the complex network (2.2) degenerates into (2.5). Note that ∥Aei(t)∥≤∥A∥·∥ei(t)∥,i=1,2,…,N hold. We construct the following response network:(2.30)ẏi(t)=DΦ(yi(t))-1[fi(Φ(yi(t)))+∑j=1NĉijAΦ(yj(t-τ(t)))]+ui,i=1,2,…,N.Corollary 2.9.
Suppose that  (A1) and (A2) hold. Using the controller(2.31)ui=DΦ(yi(t))-1diei(t)
and the update laws
(2.32)ḋi=kieiT(t)ei(t),ĉ̇ij=δijeiT(t)AΦ(yj(t-τ(t))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.5) and  (2.30) will obtain GS.Using different control, we can obtain the following theorem.Theorem 2.10.
Suppose that  (A1) and (A3) hold. Using the following controller:(2.33)ui=DΦ(yi(t))-1[diei(t)+fi(xi(t))-fi(Φ(yi(t)))],
and the update laws
(2.34)ḋi=kieiT(t)ei(t),(2.35)ĉ̇ij=δijeiT(t)h(Φ(yj(t-τ(t)))),
where  i,j=1,2,…,N,di is feedback strength, and  δij>0,ki>0 are arbitrary constants, then the complex dynamical networks  (2.2) and  (2.7) will achieve  GS with respect to  Φ.Proof.
Select the same Lyapunov-Krasovskii function as Theorem2.5, then
(2.36)dVdt=2∑i=1NeiT(t)[fi(xi(t))-fi(Φ(yi(t)))+∑j=1NcijH(ej(t-τ(t)))-∑j=1Nc̃ijh(Φ(yj(t-τ(t))))-DΦ(yi(t))ui(t)∑j=1Ncij]+∑i=1N∑j=1N2δijc̃ijĉ̇ij+∑i=1N2ki(di-d)ḋi+Nβ21-μ∑i=1NeiT(t)ei(t)-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t))≤∑i=1N(-2d+Nβ21-μ)eiT(t)ei(t)+∑i=1N∑j=1NeiT(t)cijH(ej(t-τ(t)))-1-τ̇(t)1-μNβ2∑i=1NeiT(t-τ(t))ei(t-τ(t)).
The rest of the proof is similar to Theorem 2.5 and omitted here. This completes the proof.Remark 2.11.
According to Remark2.6, when {h(Φ(yj(t-τ(t))))}j=1N are linearly independent on the orbit {yj(t-τ(t))}j=1N of synchronization manifold, we can get limt→+∞ĉij=cij,i,j=1,2,…,N; that is, the uncertain coupling matrix C can be successfully estimated using the updating laws (2.35).Remark 2.12.
Based on Theorem2.10, we can get corollaries corresponding to Corollaries 2.7–2.9.


## 3. Illustrative Numerical Examples

In this section, two groups of drive and response networks are concretely presented to demonstrate the effectiveness of the proposed method in the previous section.It is well known that the unified chaotic system [25] is described by(3.1)(ẋ1ẋ2ẋ3)=(-(25β+10)(x1-x2)-x1x3+(28-35β)x1+(29β-1)x2x1x2-(β+8)3x3)(3.2)=(-10(x1-x2)-x1x3+28x1-x2x1x2-83x3)+(-25(x1-x2)-35x1+29x2-13x3)β=F(x)+G(x)β,
which is chaotic if β∈[0,1]. Obviously, system (3.2) is the original Lorenz system for β=0 while system (3.2) belongs to the original Chen system for β=1. In fact, system (3.2) bridges the gap between the Lorenz system and Chen system.The unified new chaotic system [26] can be described as(3.3)(ẋ1ẋ2ẋ3)=(ax1-x2x3bx2+x1x3cx3+13x1x2)=g(x).
It is chaotic when a=5.0,b=-10.0, and c=-3.8.In the following, we will take these two chaotic systems as node dynamics to validate the effectiveness of Theorems2.5 and 2.10. To do that, we first verify that function f(x)=F(x)+G(x)β(β∈[0,1]) satisfies Assumption 2.Since the attractor is confined to a bounded region, there exists a constantM>0, satisfying for ally=(y1,y2,y3),z=(z1,z2,z3)∈R3,||y||≤M,||z||≤M; therefore,(3.4)‖f(y)-f(z)‖2=(25β+10)2[(y2-y1)-(z2-z1)]2+[z1z3-y1y3+(28-35β)(y1-z1)+(29β-1)(y2-z2)]2+[y1y2-z1z2-(β+8)3(y3-z3)]2=(25β+10)2[(y2-z2)-(y1-z1)]2+[z1(z3-y3)+(-y3+28-35β)(y1-z1)+(29β-1)(y2-z2)]2+[y1(y2-z2)+z2(y1-z1)-(β+8)3(y3-z3)]2≤352[2(y2-z2)2+2(y1-z1)2]+3M2(y3-z3)2+6(282+M2)(y1-z1)2+3×282(y2-z2)2+3M2(y2-z2)2+3M2(y1-z1)2+9(y3-z3)2≤(2×352+6×282+9M2)‖y-z‖2.
Thus, function f(x)=F(x)+G(x)β(β∈[0,1]) satisfies Assumption 2. By the same process, we can obtain that function g(x) satisfies Assumption 2, too.Example 3.1.
In this subsection, we consider a weighted complex dynamical network with coupling delay consisting of 3 Lorenz systems and 2 new chaotic systems (3.3). The entire networked system is given as
(3.5)ẋi(t)=F(xi(t))+∑j=15cijh(xj(t-τ(t))),i=1,2,3,ẋi(t)=g(xi(t))+∑j=15cijh(xj(t-τ(t))),i=4,5,
where xi(t)=(xi1(t),xi2(t),xi3(t))T,i=1,2,…,5.τ(t)=0.1, the weight configuration matrix
(3.6)C=(cij)5×5=(-513101-210031-400100-210001-1).
The coupling functions are h(xj(t))=(sin(xj1(t)),arctan(xj2(t)),arctan(xj3(t)))T,j=1,2,…,5.
LetΦ(yi)=(yi1+yi2,2yi2,2yi3)T,then DΦ(yi)=(110020002),i=1,2,…,5.
Since (A1)–(A3) hold, therefore, according to Theorem2.5, we can use the following response network:(3.7)ẏi(t)=DΦ(yi(t))-1[F(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=1,2,3,ẏi(t)=DΦ(yi(t))-1[g(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=4,5.
The controller and update laws are given by (2.15)–(2.17). The initial values are given as follows:
ĉij(0)=3,δij=1,di(0)=1,ki=1,xi(0)=(12+i×0.1,15+i×0.1,30+i×0.15)T,x̂i(0)=(5+i×0.1,7.5+i×0.1,15+i×0.1),i,j=1,2,…,5. Figure 1 shows GS errors ∥ei(t)∥, i=1,2,…,5. One can see that all nodes’ errors converge to zero. Some elements of matrix Ĉ are displayed in Figure 2. The numerical results show that this adaptive scheme is effective and we can get limt→∞ĉij=cij,i,j=1,2,…,5.GS errors of model (3.5) and (3.7) with respect to Φ(y)=(y1+y2,2y2,2y3)T.
(a)(b)(c)(d)(e)Estimation of topology of model (3.5).Example 3.2.
In the following simulation, we choose a weighted complex dynamical network with coupling delay consisting of 5 unified chaotic systems. The entire networked system is given as(3.8)ẋi(t)=fi(xi(t))+∑j=15cijh(xj(t-τ(t))),i=1,2,…,5,
where xi(t)=(xi1(t),xi2(t),xi3(t))T,fi(x)=F(x)βi+G(x),βi=0.1×(i-1),i=1,2,…,5. We assume τ(t)=0.3,h(xj(t))=(arctan(xj1(t)),arctan(xj2(t)),arctan(xj3(t)))T,j=1,2,…,5.C is the same as that in model (3.5).
LetΦ(yi)=(yi1+yi2,2yi2,yi33+yi3)T,DΦ(yi)=(110020003yi32+1),i=1,2,…,5.
According to Theorem2.10, the response network is given by(3.9)ẏi(t)=DΦ(yi(t))-1[fi(Φ(yi(t)))+∑j=1Nĉijh(Φ(yj(t-τ(t))))]+ui,i=1,2,…,5.
The controller and update laws are given by (2.33)–(2.35). The initial values are given as follows: ĉij(0)=6,δij=1,di(0)=1,ki=1,xi(0)=(12,15,30)T,x̂i(0)=(5,7.5,3)T,i,j=1,2,…,5. Figures 3 and 4 show the GS errors ∥ei(t)∥, i=1,2,…,5 and some elements of matrix Ĉ, respectively. The results illustrate that this scheme is effective and we can get limt→+∞ĉij=cij,i,j=1,2,…,5.GS errors of model (3.8) and (3.9) with respect to Φ(y)=(y1+y2,2y2,y33+y3)T.
(a)(b)(c)(d)(e)Estimation of topology of model (3.8).


## 4. Conclusion

In this paper, we have investigated GS between two complex networks with different node dynamics and proposed some new GS schemes via nonlinear control using Lyapunov theory and Barbǎlat lemma. Our results generalize CS of complex dynamical network with linear coupling and without delay in [20] to GS of complex dynamical network with nonidentical nodes and time-varying delay nonlinear coupling. Numerical examples are provided to verify the effectiveness of the theoretical results. This work extends the study of GS.



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