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.
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 Section 2. 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 byẋi(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
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 inputs ui(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 asẋi(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 as ei(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 vectors x and y, a matrix Q>0 with compatible dimensions, one has 2xTy≤xTQx+yTQ-1y.
Assumption 2.4.
For the vector valued function fϕi(t,xi(t),xi(t-τϕi)), assuming that there exist positive constants αϕi>0,γϕi>0 such that f satisfies the semi-Lipschitz condition
(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 constants ϑ,β, for all t∈R+, such that for any time-varying vectors x(t),y(t)∈Rn‖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 dynamics sϕi(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 asėi(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 havec∑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: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 assumptions 2.4–2.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold:
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:
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
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
≤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
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
≤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
V̇(t)≤0.
Theorem 3.1 is proved completely.
We can conclude that, for any initial values, the solutions x1(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.
When A=0, network (2.1) is translated into
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
ui(t)={-c∑i=1NbijH2(sϕi(t-ηϕi(t)))-diei(t),i∈J̅ϕi,vi(t)=diei(t),i∈Jϕi-J̅ϕi.
Corollary 3.3.
When B=0, network (2.1) is translated into
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
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: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 choose h11(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),D1=[-10100284000-83],D2=[0-1-110.2000-1.2],D3=[-2.1691001-110-19.53-0.1636].
Taking the weight configuration coupling matricesA=B=[-210001-1-2100001-2100001-2100001-2110001-2].
The following quantities are utilized to measure the process of cluster synchronizationE(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.
Time evolution of the synchronization errors E(t).
Time evolution of the synchronization errors E12(t).
Time evolution of the synchronization errors E13(t).
Time evolution of the synchronization errors E23(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.
Acknowledgments
This research is partially supported by the National Nature Science Foundation of China (no. 70871056) and by the Six Talents Peak Foundation of Jiangsu Province.
CaoJ.LiP.WangW.Global synchronization in arrays of delayed neural networks with constant and delayed coupling200635343183252-s2.0-3364587679410.1016/j.physleta.2005.12.092ChenM.Some simple synchronization criteria for complex dynamical networks20065311118511892-s2.0-3454714276810.1109/TCSII.2006.882363GaoH.LamJ.ChenG.New criteria for synchronization stability of general complex dynamical networks with coupling delays200636022632732-s2.0-3375046449310.1016/j.physleta.2006.08.033HeY.WangQ.-G.ZhengW.-X.Global robust stability for delayed neural networks with polytopic type uncertainties20052651349135410.1016/j.chaos.2005.04.0052149320ZBL1083.34535ZhouJ.XiangL.LiuZ.Global synchronization in general complex delayed dynamical networks and its applications2007385272974210.1016/j.physa.2007.07.0062584888HanX. P.LuJ.-A.ChenG. R.Nonlinear integral synchronization of ring networks200855480881810.1016/j.camwa.2007.04.0372387465ZBL1161.34025XuS. Y.YangY.Synchronization for a class of complex dynamical networks with time-delay20091483230323810.1016/j.cnsns.2008.12.0222502399ZBL1221.34205JiaZ.WangH.WangJ.LiY.The weighted identification of a bipartite-graph complex dynamical network based on adaptive synchronization20102711071102-s2.0-77950887203WangL.DaiH.-P.DongH.ShenY.-H.SunY.-X.Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays2008372203632363910.1016/j.physleta.2008.02.0102413862ZBL1220.90041JiD. H.ParkJ. H.YooW. J.WonS. C.LeeS. M.Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay201037410121812272-s2.0-7444909030310.1016/j.physleta.2010.01.005YuanK.Robust synchronization in arrays of coupled networks with delay and mixed coupling2009724–6102610312-s2.0-5814947522710.1016/j.neucom.2008.04.018LiuT.ZhaoJ.HillD. J.Synchronization of complex delayed dynamical networks with nonlinearly coupled nodes20094031506151910.1016/j.chaos.2007.09.0752526366ZBL1197.34092GuoW. L.AustinF.ChenS. H.Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling20101561631163910.1016/j.cnsns.2009.06.0162576789ZBL1221.34213DeLellisP.diBernardoM.GarofaloF.Novel decentralized adaptive strategies for the synchronization of complex networks20094551312131810.1016/j.automatica.2009.01.0012531611ZhengS.WangS.DongG.BiQ.Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling201217128429110.1016/j.cnsns.2010.11.029PorfiriM.di BernardoM.Criteria for global pinning-controllability of complex networks200844123100310610.1016/j.automatica.2008.05.0062531411ZBL1153.93329SorrentinoF.BernardoM. D.GarofaloF.ChenG. R.Controllability of complex networks via pinning200775462-s2.0-3414712448510.1103/PhysRevE.75.046103046103XiaW.CaoJ.Pinning synchronization of delayed dynamical networks via periodically intermittent control2009191810.1063/1.30719332513764013120CaiS.LiuZ.XuF.ShenJ.Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit2009373423846385410.1016/j.physleta.2009.07.0812552542KhadraA.LiuX. Z.ShenX.Impulsively synchronizing chaotic systems with delay and applications to secure communication20054191491150210.1016/j.automatica.2005.04.0122161112ZBL1086.93051SunW.HuT.ChenZ.ChenS.XiaoL.Impulsive synchronization of a general nonlinear coupled complex network201116114501450710.1016/j.cnsns.2011.02.0322806762ZBL1222.93102RabinovichM. I.TorresJ. J.VaronaP.HuertaR.WeidmanP.Origin of coherent structures in a discrete chaotic medium1999602R1130R11332-s2.0-000157461410.1103/PhysRevE.60.R1130ZanetteD. H.MikhailovA. S.Mutual synchronization in ensembles of globally coupled neural networks19985818728752-s2.0-003211257310.1103/PhysRevE.58.872BelykhI.BelykhV.NevidinK.HaslerM.Persistent clusters in lattices of coupled nonidentical chaotic systems200313116517810.1063/1.15142021964970ZBL1080.37525BelykhV. N.BelykhI. V.HaslerM.NevidinK. V.Cluster synchronization in three-dimensional lattices of diffusively coupled oscillators200313475577910.1142/S02181274030069231980766ZBL1056.37088BelykhN. V.BelykhI. V.MosekildeE.Cluster synchronization modes in an ensemble of coupled chaotic oscillators2001633, part 22-s2.0-0035276596036216WuX.LuH.Cluster synchronization in the adaptive complex dynamical networks via a novel approach201137514155915652-s2.0-7995220223010.1016/j.physleta.2011.02.052MaZ.LiuZ.ZhangG.A new method to realize cluster synchronization in connected chaotic networks2006162910.1063/1.21849482243805023103ZBL1146.37330SorrentinoF.OttE.Network synchronization of groups20077651010.1103/PhysRevE.76.0561142495364056114ChenL.LuJ.Cluster synchronization in a complex dynamical network with two nonidentical clusters2008211203310.1007/s11424-008-9063-42377222ZBL1172.93002BelykhV. N.BelykhI. V.HaslerM.Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems2000625633263452-s2.0-003431718310.1103/PhysRevE.62.6332LuW. L.LiuB.ChenT.Cluster synchronization in networks of distinct groups of maps20107722572642-s2.0-7814928540710.1140/epjb/e2010-00202-7WangK.FuX.LiK.Cluster synchronization in community networks with nonidentical nodes20091921010.1063/1.31257142548747023106LiuH.ChenJ.LuJ. A.CaoM.Generalized synchronization in complex dynamical networks via adaptive couplings20103898175917702-s2.0-7484908459210.1016/j.physa.2009.12.035