# Cluster Synchronization of Time-Varying Delays Coupled Complex Networks

Authors: Shuguo Wang; Hongxing Yao; Mingping Sun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper investigates a new cluster synchronization scheme in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization of the network model. Particularly, the weight configuration matrix is not assumed to be symmetric, irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.


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## Body

## 1. Introduction

Complex networks model is used to describe various interconnected systems of real world, which have become a focal research topic and have drawn much attention from researchers working in different fields; one of the most important reasons is that most practical systems can be modeled by complex dynamical networks. Recently, the research on synchronization and dynamical behavior analysis of complex network systems has become a new and important direction in this field [1–13]; many control approaches have been developed to synchronize complex networks such as feedback control, adaptive control, pinning control, impulsive control, and intermittent control [14–21].Cluster synchronization means that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups [22–25]; Belykh et al. [26] investigated systems of diffusively coupled identical chaotic oscillators; an effective method to determine the possible states of cluster synchronization and ensure their stability is presented. The method, which may find applications in communication engineering and other fields of science and technology, is illustrated through concrete examples of coupled biological cell models. Wu and Lu [27] investigated cluster synchronization in the adaptive complex dynamical networks with nonidentical nodes by a local control method and a novel adaptive strategy for the coupling strengths of the networks. Ma et al. [28] proposed cluster synchronization scheme via dominant intracouplings and common intercluster couplings. Sorrentino and Ott [29] studied local cluster synchronization for bipartite systems, where no intracluster couplings (driving scheme) exist. Chen and Lu [30] investigated global cluster synchronization in networks of two clusters with inter- and intracluster couplings. Belykh et al. [31, 26] studied this problem in 1D and 2D lattices of coupled identical dynamical systems. Lu et al. [32] studied the cluster synchronization of general networks with nonidentical clusters and derived sufficient conditions for achieving local cluster synchronization of networks. Recently, Wang et al. [33] considered the cluster synchronization of dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community by using pinning control schemes. However, there is few theoretical result on the cluster synchronization of nonlinear coupled complex networks with time-varying delays coupling and time-varying delays in nonidentical dynamical nodes.Motivated by the above discussions, this paper investigates cluster synchronization in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization in nonlinear coupled complex dynamical networks with time-varying delays coupling and time-varying delays in dynamic nodes. Particularly the weight configuration matrix is not assumed to be symmetric, irreducible.The paper is organized as follows: the network model is introduced followed by some definitions, lemmas, and hypotheses in Section2. The cluster synchronization of the complex coupled networks is discussed in Section 3. Simulations are obtained in Section 4. Finally, in Section 5 the various conclusions are discussed.


## 2. Model and Preliminaries

The network with nondelayed and time-varying delays coupling and adaptive coupling strengths can be described by(2.1)ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NaijH1(xj(t))+c∑j=1NbijH2(xj(t-ηϕi(t))),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rn is the state vector of node i;fϕi:Rn→Rn describes the local dynamics of nodes in the ϕith community. For any pair of nodes i and j, if ϕi≠ϕj, that is, nodes i and j belong to different communities, then fϕi≠fϕj·ηϕi(t),τϕi(t), is a time-varying delay. H1(·) and H2(·) are nonlinear functions. c is coupling strength. A=(aij)N×N,B=(bij)N×N are the weight configuration matrices. If there is a connection from node i to node j(j≠i), then the aij>0,bij>0 otherwise, aij=aji=0,bij=bji=0, and the diagonal elements of matrix A,B are defined as
(2.2)aii=-∑j=1,j≠iNaji,bii=-∑j=1,j≠iNbji,i=1,2,…,N.
Particularly, the weight configuration matrix is not assumed to be symmetric, irreducible.When the control inputsui(t)∈Rn and vi(t)∈Rn(i=1,2,…,N) are introduced, the controlled dynamical network with respect to network (2.1) can be written as(2.3)ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NaijH1(xj(t))+c∑j=1NbijH2(xj(t-ηϕi(t)))+ui(t),ϕi(t)∈J̅ϕi,ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NaijH1(xj(t))+c∑j=1NbijH2(xj(t-ηϕi(t)))-vi(t),ϕi(t)∈Jϕi-J̅ϕi,
where Jϕi denotes all the nodes in the ϕith community and J̅ϕi represents the nodes in the ϕith community which have direct links with the nodes in other communities.The study presents the mathematical definition of the cluster synchronization.Let{C1,C2,…,Cm} denote m(2≤m≤N) communities of the networks and ⋃i=1mCi={1,2,…N}. If node i belongs to the jth community, then we denote ϕi=j. We employ fi(·) to represent the local dynamics of all nodes in the ith community. Let si(t) be the solution of the system ṡi(t)=fϕi(t,si(t),si(t-τϕi(t))),(i=1,2,…,m) where limt→∞∥si(t)-sj(t)∥≠0(i≠j); the set S={s1(t),s2(t),…,sm(t)} is used as the cluster synchronization manifold for network (2.3). Cluster synchronization can be realized if and only if the manifold S is stable.Definition 2.1 (see [19]).
The error variables asei(t)=xi(t)-sϕi(t) for i=1,2,…,N, where sϕi(t) satisfies ṡϕi(t)=fϕi(t,sϕi(t),sϕi(t-τϕi(t))).Definition 2.2 (see [19]).
Let{1,2,…,N} be the N nodes of the network and {C1,C2,…,Cm} be the m communities, respectively. A network with m communities is said to realize cluster synchronization if limt→∞ei(t)=0 and limt→∞∥xi(t)-xj(t)∥≠0 for ϕi≠ϕj.Lemma 2.3.
For any two vectorsx and y, a matrix Q>0 with compatible dimensions, one has 2xTy≤xTQx+yTQ-1y.Assumption 2.4.
For the vector valued functionfϕi(t,xi(t),xi(t-τϕi)), assuming that there exist positive constants αϕi>0,γϕi>0 such that f satisfies the semi-Lipschitz condition
(2.4)(xi(t)-yi(t))T(fϕi(t,xi(t),xi(t-τϕi))-fϕi(t,yi(t),yi(t-τϕi)))≤αϕi(xi(t)-yi(t))T(xi(t)-yi(t))+γϕi(xi(t-τϕi)-yi(t-τϕi))T(xi(t-τϕi)-yi(t-τϕi)),
for all x,y∈Rn and τϕi(t)≥0.i=1,2,…,N.Assumption 2.5.
ηϕi(t) and τϕi(t) is a differential function with 0≤η̇ϕi(t)≤ε≤1 and 0≤τ̇ϕi(t)≤ε≤1. Clearly, this assumption is certainly ensured if the delay ηϕi(t) and τϕi(t) is constant.Assumption 2.6 (34) (Global Lipschitz Condition).
Suppose that there exist nonnegative constantsϑ,β, for all t∈R+, such that for any time-varying vectors x(t),y(t)∈Rn(2.5)‖H1(x)-H1(y)‖≤ϑ‖x-y‖,‖H2(x)-H2(y)‖≤β‖x-y‖,
where ∥∥ denotes the 2-norm throughout the paper.


## 3. Main Results

In this section, a control scheme is developed to synchronize a delayed complex network with nonidentical nodes to any smooth dynamicssϕi(t). Let synchronization errors ei(t)=xi(t)-sϕi(t) for i=1,2,…,N, according to system (2.1), the error dynamical system can be derived as(3.1)ėi(t)=f̃ϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1Naij[H1(xj(t))-H1(sϕi(t))]+c∑j=1Nbij[H2(xj(t-ηϕi(t)))-H2(sϕi(t-ηϕi(t)))]+∑i=1NaijH1(sϕi(t))+∑i=1NbijH2(sϕi(t-ηϕi(t)))+ui(t),ϕi(t)∈J̅ϕi,ėi(t)=f̃ϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1Naij[H1(xj(t))-H1(sϕi(t))]+c∑j=1Nbij[H2(xj(t-ηϕi(t)))-H2(sϕi(t-ηϕi(t)))]-vi(t),ϕi(t)∈Jϕi-J̅ϕi,
where f̃ϕi(t,xi(t),xi(t-τϕi(t)))=fϕi(t,xi(t),xi(t-τϕi(t)))-fϕi(t,sϕi(t),sϕi(t-τϕi(t))) for i=1,2,…,N.According to the diffusive coupling condition (2.2) of the matrix A,B we have(3.2)c∑i=1NaijH1(sϕi(t))+c∑i=1NbijH2(sϕi(t-ηϕi(t)))=0,i∈Jϕi-J̅ϕi.
On the basis of this property, for achieving cluster synchronization, we design controllers as follows:(3.3)ui(t)={-c∑i=1NaijH1(sϕi(t))-c∑i=1NbijH2(sϕi(t-ηϕi(t)))-diei(t),i∈J̅ϕi,vi(t)=diei(t),i∈Jϕi-J̅ϕi,
where ḋi=kieiT(t)ei(t).Theorem 3.1.
Suppose assumptions2.4–2.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold:
(3.4)α+ϑcλmax⁡(Q)+12β2c2λmax⁡(PPT)+11-ε(γ+12)<d,
where α=max(αϕ1,αϕ2,…,αϕm),γ=max(γϕ1,γϕ2,…,γϕm). Then, the systems (2.3) is cluster synchronization.Proof.
Construct the following Lyapunov functional:(3.5)V(t)=12∑i=1NeiT(t)ei(t)+γ1-ε∫t-τϕi(t)t∑i=1NeiT(θ)ei(θ)dθ+12(1-ε)∫t-ηϕi(t)t∑i=1NeiT(θ)ei(θ)dθ+12∑i=1N(di-d)2ki.
Calculating the derivative of V(t), we have
(3.6)V̇(t)=∑i=1NeiT(t)ėi(t)+11-ε(γ+12)∑i=1NeiT(t)ei(t)-γ(1-τ̇ϕi(t))1-ε∑i=1NeiT(t-τϕi(t))ei(t-τϕi(t))-1-η̇ϕi(t)2(1-ε)∑i=1NeiT(t-ηϕi(t))ei(t-ηϕi(t))+∑i=1N(di-d)eiT(t)ei(t)=∑i=1NeiT(t){f̃ϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1Naij[H1(xj(t))-H1(sϕi(t))]+c∑j=1Nbij[H2(xj(t-ηϕi(t)))-H2(sϕi(t-ηϕi(t)))]-diei(t)}+11-ε(γ+12)∑i=1NeiT(t)ei(t)-γ(1-τ̇ϕi(t))1-ε∑i=1NeiT(t-τϕi(t))ei(t-τϕi(t))-1-η̇ϕi(t)2(1-ε)∑i=1NeiT(t-ηϕi(t))ei(t-ηϕi(t))+∑i=1N(di-d)eiT(t)ei(t).
By assumptions 2.4–2.6, we obtain
(3.7)≤α∑i=1NeiT(t)ei(t)+γ∑i=1NeiT(t-τϕi(t))ei(t-τϕi(t))+ϑc∑i=1NeiT(t)∑j=1Naijej(t)+βc∑i=1NeiT(t)∑j=1Nbijej(t-ηϕi(t))-di∑i=1NeiT(t)ei(t)+11-ε(γ+12)∑i=1NeiT(t)ei(t)-γ∑i=1NeiT(t-τϕi(t))ei(t-τϕi(t))-12∑i=1NeiT(t-ηϕi(t))ei(t-ηϕi(t))+∑i=1N(di-d)eiT(t)ei(t)≤α∑i=1NeiT(t)ei(t)+ϑceT(A⊗I)e+βceT(B⊗I)e(t-ηϕi(t))+11-ε(γ+12)∑i=1NeiT(t)ei(t)-12∑i=1NeiT(t-ηϕi(t))ei(t-ηϕi(t))-deT(t)e(t).
Let e(t)=(e1T(t),e2T(t),…,eNT(t))T∈RnN,Q=(A⊗I),P=(B⊗I), where ⊗ represents the Kronecker product. Then
(3.8)V̇(t)≤αeT(t)e(t)+ϑceT(t)Qe(t)+βceT(t)Pe(t-ηϕi(t))+11-ε(γ+12)eT(t)e(t)-12eT(t-ηϕi(t))e(t-ηϕi(t))-deT(t)e(t).
By the Lemma 2.3, we have
(3.9)≤αeT(t)e(t)+ϑceT(t)Qe(t)+12(βc)2eT(t)PPTe(t)+11-ε(γ+12)eT(t)e(t)-deT(t)e(t)≤(α+ϑcλmax⁡(Q)+12β2c2λmax⁡(PPT)+11-ε(γ+12)-d)eT(t)e(t).
Therefore, if we have α+ϑcλmax(Q)+(1/2)β2c2λmax(PPT)+(1/(1-ε))(γ+(1/2))<d then
(3.10)V̇(t)≤0.
Theorem 3.1 is proved completely.
We can conclude that, for any initial values, the solutionsx1(t),x2(t),…,xN(t) of the system (2.3) satisfy limt→∞∑k=1m∑i∈Ck∥xi(t)-sk(t)∥=0, that is, we get the global stability of the cluster synchronization manifold S. Therefore, cluster synchronization in the network (2.3) is achieved under the local controllers (3.3). This completes the proof.Corollary 3.2.
WhenA=0, network (2.1) is translated into
(3.11)ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NbijH2(xj(t-ηϕi(t))),i=1,2,…,N.
We design the controllers, as follows, then the complex networks can also achieve synchronization, where(3.12)ui(t)={-c∑i=1NbijH2(sϕi(t-ηϕi(t)))-diei(t),i∈J̅ϕi,vi(t)=diei(t),i∈Jϕi-J̅ϕi.Corollary 3.3.
WhenB=0, network (2.1) is translated into
(3.13)ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NaijH1(xj(t)),i=1,2,…,N.
We design the controllers, as follows, then the complex networks can also achieve synchronization, where(3.14)ui(t)={-c∑i=1NaijH1(sϕi(t))-diei(t),i∈J̅ϕi,vi(t)=diei(t),i∈Jϕi-J̅ϕi,


## 4. Illustrative Examples

In this section, a numerical example will be given to demonstrate the validity of the synchronization criteria obtained in the previous sections. Considering the following network:(4.1)ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NaijH1(xj(t))+c∑j=1NbijH2(xj(t-ηϕi(t)))+ui(t),ϕi(t)∈J̅ϕi,ẋi(t)=fϕi(t,xi(t),xi(t-τϕi(t)))+c∑j=1NaijH1(xj(t))+c∑j=1NbijH2(xj(t-ηϕi(t)))-vi(t),ϕi(t)∈Jϕi-J̅ϕi,i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),xi3(t))T, f1(t,xi(t),xi(t-τ1(t)))=D1xi(t)+h11(xi(t))+h12(xi(t-τ1(t))), f2(t,xi(t),xi(t-τ2(t)))=D2xi(t)+h21(xi(t))+h22(xi(t-τ2(t)))+V, f3(t,xi(t),xi(t-τ3(t)))=D3xi(t)+h31(xi(t))+h32(xi(t-τ3(t))). k1=k2=⋯=kN=10, c=1, H1(x)=sinx, H2(x)=cosx.In simulation, we chooseh11(xi)=(0,-xi1xi3,xi1xi2)T, h12(xi)=(0,5xi2,0)T, h21(xi)=(0,0,xi1xi3)T, h22(xi)=(xi1,0,0)T, V=[0,0,0.2]T, h31(xi)=(3.247(|xi1+1|-|xi1-1|),0,0)T, h32(xi)=(0,0,-3.906sin(0.5xi1))T, τ1(t)=et/(1+et), τ2(t)=2et/(1+et), τ3(t)=0.5et/(1+et),(4.2)D1=[-10100284000-83],D2=[0-1-110.2000-1.2],D3=[-2.1691001-110-19.53-0.1636].
Taking the weight configuration coupling matrices(4.3)A=B=[-210001-1-2100001-2100001-2100001-2110001-2].
The following quantities are utilized to measure the process of cluster synchronization(4.4)E(t)=∑i=1N‖xi(t)-sϕi(t)‖,E12(t)=‖xu(t)-xv(t)‖,u∈C1,v∈C2,E13(t)=‖xu(t)-xv(t)‖,u∈C1,v∈C3,E23(t)=‖xu(t)-xv(t)‖,u∈C2,v∈C3,
where E(t) is the error of cluster synchronization for this controlled network (2.2); E12(t), E13(t), and E23(t) are the errors between two communities; cluster synchronization is achieved if the synchronization error E(t) converges to zero and E12(t),E13(t) and E23(t) do not as t→∞. Simulation results are given in Figures 1, 2, 3, and 4. From the Figures 1–4, we see the time evolution of the synchronization errors. The numerical results show that Theorem 3.1 is effective.Figure 1
Time evolution of the synchronization errorsE(t).Figure 2
Time evolution of the synchronization errorsE12(t).Figure 3
Time evolution of the synchronization errorsE13(t).Figure 4
Time evolution of the synchronization errorsE23(t).


## 5. Conclusions

The problems of cluster synchronization and adaptive feedback controller for the nonlinear coupled complex networks are investigated. The weight configuration matrix is not assumed to be symmetric, irreducible. It is shown that cluster synchronization can be realized via adaptive feedback controller. The study showed that the use of simple control law helps to derive sufficient criteria which ensure that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups is derived. Particularly the synchronization criteria are independent of time delay. The developed techniques are applied three complex community networks which are synchronized to different chaotic trajectories. Finally, the numerical simulations were performed to verify the effectiveness of the theoretical results.



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