# Cluster Anti-Synchronization of Complex Networks with Nonidentical Dynamical Nodes

Authors: Shuguo Wang; Chunyuan He; Hongxing Yao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper investigates a new cluster antisynchronization scheme in the time-varying delays coupled complex dynamical networks with nonidentical nodes. Based on the community structure of the networks, the controllers are designed differently between the nodes in one community that have direct connections to the nodes in other communities and the nodes without direct connections with the nodes in other communities strategy; some sufficient criteria are derived to ensure cluster anti-synchronization of the network model. Particularly, the weight configuration matrix is not assumed to be irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.


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## Body

## 1. Introduction

During the past decade, the research on synchronization and dynamical behavior analysis of complex network systems has become a new and important direction in this field [1–6]; many control approaches and many different synchronization phenomena have been developed, such as  impulsive control, pinning control and complete synchronization, phase synchronization, cluster synchronization, mixed synchronization,  and generalized synchronization, which have been investigated since ten years ago in [7, 8] and references therein.Cluster synchronization means that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups [9, 10]. Wang et al. [11] investigated the cluster synchronization of the dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community, which were considered by using some feedback control schemes. Wu and Lu [12] 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. Qin and Chen [13] investigated the stability of selected cluster synchronization in coupled Josephson equations. Ma et al. [14] showed that the arbitrarily selected cluster synchronization manifolds could be stabilized by constructing a special coupled matrix for connected chaotic networks. Wu et al. [15] investigated the anti-synchronization (AS) problem of two general complex dynamical networks with nondelayed and delayed coupling using pinning adaptive control method. Based on Lyapunov stability theory and Barbalat lemma, a sufficient condition is derived to guarantee the AS between two networks with nondelayed and delayed coupling. However, there is few theoretical result on the cluster anti-synchronization of linearly coupled complex networks with time-varying delays coupling.Motivated by the aforementioned discussions, this paper aims to analyze the cluster anti-synchronization problem for the time-varying delays coupled complex dynamical networks. The main contributions of this paper are threefold: (1) the local dynamics in each community are identical, but those of different communities are nonidentical. (2) For achieving the synchronization, based on the community structure of the networks, the controllers are designed differently between the nodes in one community which have direct connections to the nodes in other communities and the nodes without direct connections with the nodes in other communities strategy; some sufficient criteria are derived to ensure cluster anti-synchronization of the network model. (3) According to Lyapunov stability theory, the sufficient conditions for achieving cluster anti-synchronization are obtained analytically. Compared with some similar designs, our controllers are very simple.The paper is organized as follows: the network model is introduced followed by some definitions, lemmas, and hypotheses in Section2. The cluster anti-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 time-varying delays coupling can be described by(2.1)x˙i(t)=fϕi(t,xi(t))+c∑j=1NaijΓ1xj(t)+c∑j=1NbijΓ2xj(t-ηϕi(t)),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈Rnis the state vector of node i,  c>0 is the coupling strength, andfϕi: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, thenfϕi≠fϕj,  ηϕi(t) is time-varying delay. Γ1∈Rn×n and Γ2∈Rn×n are inner-coupling matrices, for simplicity; we assume that Γ1, Γ2 are diagonal matrices with positive diagonal elements, Γ1=diag(ρ1,ρ2,…,ρn) with ρi≥0, Γ2=diag(θ1,θ2,…,θn) with θi≥0.  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 aij=aji>0, bij=bji>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 matrices are not assumed to be irreducible.When the control inputsvi(t), ui(t)∈Rn (i=1,2,…,N) are introduced, the controlled dynamical network with respect to network (2.1) can be written as
(2.3)x˙i(t)=fϕi(t,xi(t))+c∑j=1NaijΓ1xj(t)+c∑j=1NbijΓ2xj(t-ηϕi(t))+ui(t),ϕi(t)∈J-ϕi,i=1,2,…,N,x˙i(t)=fϕi(t,xi(t))+c∑j=1NaijΓ1xj(t)+c∑j=1NbijΓ2xj(t-ηϕi(t))+vi(t),ϕi(t)∈Jϕi-J-ϕi,i=1,2,…,N,
where Jϕi denotes all the nodes in the ϕi th community and J-ϕi represents the nodes in the ϕi th community, which have direct links with the nodes in other communities.In this paper, 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. The local dynamics of individual nodes in different communities are assumed to be nonidentical, that is, if ϕi≠ϕj, then fϕi≠fϕj. Let sϕi(t) be a solution of an isolated node in the ϕi th community, that is, s˙i(t)=fϕi(t,si(t)), where limt→∞∥si(t)-sj(t)∥≠0 (i≠j) and the set S={s1(t),s2(t),…,sm(t)} is used as the cluster anti-synchronization manifold for network (2.3). Cluster anti-synchronization can be realized if and only if the manifold S is stable, where sk(t) may be an equilibrium point, a periodic orbit, or even a chaotic orbit.Definition 2.1 (see [15]).
The dynamical network (2.1) is said to achieve cluster anti-synchronization (CAS) if
(2.4)lim⁡t→∞‖ei(t)‖=lim⁡t→∞‖xi(t)+sϕi(t)‖=0,i=1,2,…,N.Lemma 2.2.
For any two vectors x and y, a matrixQ>0 with compatible dimensions, one has
(2.5)2xTy≤xTQx+yTQ-1y.Assumption 2.3 (see [15]).
For anyx=(x1,x2,…,xn)T∈Rn, y=(y1,y2,…,yn)T∈Rn, there exists a positive constant L such that, (y-x)T(f(t,y)-f(t,x))≤L(y-x)TΓ(y-x), where Γ is a positive definite matrix. Here x and y are time-varying vectors.Assumption 2.4.
f
(
x
,
t
) is an odd function of x, that is, f(-x,t)=-f(x,t) for arbitrary x∈Rn.Assumption 2.5.
η
ϕ
i
(
t
) is a differential function with 0≤η˙ϕi(t)≤ε≤1. Clearly, this assumption is certainly ensured if the delay ηϕi(t) is constant.


## 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)e˙i(t)=f~ϕi(t,xi(t))+c∑j=1NaijΓ1ej(t)+c∑j=1NbijΓ2ej(t-ηϕi(t))-c∑i=1NaijΓ1sϕi(t)-c∑i=1NbijΓ2sϕi(t-ηϕi(t))+ui(t),ϕi(t)∈J-ϕi,i=1,2,…,N,e˙i(t)=f~ϕi(t,xi(t))+c∑j=1NaijΓ1ej(t)+c∑j=1NbijΓ2ej(t-ηϕi(t))+vi(t),ϕi(t)∈Jϕi-J-ϕi,i=1,2,…,N,
where f~ϕi(t,ei(t))=fϕi(t,xi(t))+fϕi(t,sϕi(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=1NaijΓ1sϕi(t)+c∑i=1NbijΓ2sϕi(t-ηϕi(t))=0,i∈Jϕi-J-ϕi.
On the basis of this property, for achieving cluster anti-synchronization, we design the controllers as follows:
(3.3)ui(t)=c∑i=1NaijΓ1sϕi(t)+c∑i=1NbijΓ2sϕi(t-ηϕi(t))-diei(t),i∈J-ϕi,vi(t)=-diei(t),i∈Jϕi-J-ϕi,
where d˙i=kieiT(t)ei(t), di are the feedback strength and ki  are arbitrary positive constants.It is easy to see that the synchronization of the controlled complex network (2.1) is achieved if the zero solution of the error system (3.1) is globally asymptotically stable, which is ensured by the following theorem.Lete~j(t)=(e1j(t),e2j(t),…,eNj(t))T, D=diag(d1,…,dN)T; then we have the following result.Theorem 3.1.
Suppose Assumptions2.3–2.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold:
(3.4)λmax⁡(Γ)L+cλmax⁡(Q)+12cλmax⁡(PPT)+c2(1-ε)<d,
then the system (2.3) is cluster anti-synchronization, where d is sufficiently large positive constant to be determined.Proof.
From Assumptions2.3 and 2.4, we get
(3.5)(y+x)T(f(t,y)+f(t,x))=(y-(-x))T(f(t,y)-f(t,-x))≤L(y+x)TΓ(y+x)
for x,y∈Rn.
Construct the following Lyapunov functional:(3.6)V(t)=12∑i=1NeiT(t)ei(t)+c2(1-ε)∑i=1N∫t-ηϕi(t)teiT(ξ)ei(ξ)dξ+12∑i=1N(di-d)2ki.
Calculating the derivative of V(t), we have
(3.7)V˙(t)=∑i=1NeiT(t)[f~ϕi(t,ei(t))+c∑j=1NaijΓ1ej(t)+c∑j=1NbijΓ2ej(t-ηϕi(t))-diei(t)]+c2(1-ε)∑i=1NeiT(t)ei(t)-c(1-η˙ϕi(t)2(1-ε))∑i=1NeiT(t-ηϕi(t))ei(t-ηϕi(t))+∑i=1N(di-d)eiT(t)ei(t).
By the Assumptions2.3–2.5, we have
(3.8)V˙(t)≤∑i=1NLeiT(t)Γei(t)+c∑i=1N∑j=1NaijeiT(t)Γ1ej(t)+c∑i=1N∑j=1NbijeiT(t)Γ2ej(t-ηϕi(t))+c2(1-ε)∑i=1NeiT(t)ei(t)-c2∑i=1NeiT(t-ηϕi(t))ei(t-ηϕi(t))-∑i=1NdeiT(t)ei(t)≤∑i=1NLeiT(t)Γei(t)+ceT(A⊗Γ1)e+ceT(B⊗Γ2)e(t-ηϕi(t))+c2(1-ε)∑i=1NeiT(t)ei(t)-c2eT(t-ηϕi(t))e(t-ηϕi(t))-deT(t)e(t).
Lete(t)=(e1T(t),e2T(t),…,eNT(t))T∈RnN,  Q=(A⊗Γ1),  P=(B⊗Γ2), where ⊗  represents the Kronecker product. Then
(3.9)V˙(t)≤λmax⁡(Γ)LeT(t)e(t)+ceT(t)Qe(t)+ceT(t)Pe(t-ηϕi(t))+c2(1-ε)eT(t)e(t)-c2eT(t-ηϕi(t))e(t-ηϕi(t))-deT(t)e(t).
By Lemma 2.2, we have
(3.10)V˙(t)≤λmax⁡(Γ)LeT(t)e(t)+ceT(t)Qe(t)+c2eT(t)PPTe(t)+c2(1-ε)eT(t)e(t)-deT(t)e(t)≤(λmax⁡(Γ)L+cλmax⁡(Q)+12cλmax⁡(PPT)+c2(1-ε)-d)eT(t)e(t).
Therefore, if we have λmax(Γ)L+cλmax(Q)+(1/2)cλmax(PPT)+c/2(1-ε)<d, then
(3.11)V˙(t)≤-eT(t)e(t).
Chooseλmax(Γ)L+cλmax(Q)+(1/2)cλmax(PPT)+c/2(1-ε)+1<d. Thus, one obtains V˙(t)≤-eT(t)e(t)≤0. It is obvious that the largest invariant set contained in set E={V˙(t)=0}={ei(t)=0,i=1,2,…,N} is Q~={ei(t)=0,di(t)=d,i=1,2,…,N}. Based on LaSalle invariance principle, starting with any initial values of the error dynamical system, the trajectory asymptotically converges to the largest invariant Q~ which implies that limt→∞∥ei(t)∥=0 for i=1,2,…,N. Therefore, cluster anti-synchronization in the network (2.3) is achieved under the controllers (3.3). This completes the proof.Corollary 3.2.
When A  =  0, network (2.1) is translated into
(3.12)x˙i(t)=fϕi(t,xi(t))+c∑j=1NbijΓ2xj(t-ηϕi(t)),i=1,2,…,N.
We design the controllers, as follows, then the complex networks can also achieve synchronization, where(3.13)ui(t)=c∑i=1NbijΓ2sϕi(t-ηϕi(t))-diei(t),i∈J-ϕi,vi(t)=-diei(t),i∈Jϕi-J-ϕi.Corollary 3.3.
WhenB=0, network (2.1) is translated into
(3.14)x˙i(t)=fϕi(t,xi(t))+c∑j=1NaijΓ1xj(t),i=1,2,…,N.
We design the controllers, as follows, then the complex networks can also achieve synchronization, where(3.15)ui(t)=c∑i=1NaijΓ1sϕi(t)-diei(t),i∈J-ϕi,vi(t)=-diei(t),i∈Jϕi-J-ϕi.


## 4. Illustrative Examples

In this section, several numerical examples are provided to illustrate the proposed synchronization methods. The nodes dynamics are the following well-known modified Chua’s circuit [16] with different system parameters. Considering the following network:
(4.1)x˙i(t)=fϕi(t,xi(t))+c∑j=1NaijΓ1xj(t)+c∑j=1NbijΓ2xj(t-ηϕi(t))+ui(t),ϕi(t)∈J-ϕi,i=1,2,…,N,x˙i(t)=fϕi(t,xi(t))+c∑j=1NaijΓ1xj(t)+c∑j=1NbijΓ2xj(t-ηϕi(t))+vi(t),ϕi(t)∈Jϕi-J-ϕi,i=1,2,…,N,
where xi(t)=(x1i(t),x2i(t),x3i(t))T, f1(t,xi(t))=D1xi(t)+h1(xi(t)), f2(t,xi(t))=D2xi(t)+h2(xi(t)), f3(t,xi(t))=D3xi(t)+h3(xi(t)),  h1(xi)=h2(xi)=h3(xi)=((-20/7)xi13,0,0)T, Γ1=Γ2=Diag{1,1,1},
(4.2)η1(t)=0.1et1+et,η2(t)=2et1+et,η3(t)=0.8et1+et,D1=[-1071001-110-120],D2=[-1071001-110-130],D3=[-1071001-110-140],c=0.01,d=10,A=B=[-211-211-1-3012110-3110111-4101110-4111000-2].
Simulation results are given in Figures 1, 2, and 3. Cluster anti-synchronization is achieved by the controller (3.3).The trajectories of nodes in the first community.
(a)
(b)
(c)The trajectories of nodes in the second community.
(a)
(b)
(c)The trajectories of nodes in the third community.
(a)
(b)
(c)The following quantities are utilized to measure the process of cluster anti-synchronization(4.3)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 anti-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 4, 5, 6, and 7.Figure 4
Time evolution of the synchronization errorE(t).Figure 5
Time evolution of the synchronization errorE12(t).Figure 6
Time evolution of the synchronization errorE23(t).Figure 7
Time evolution of the synchronization errorE13(t).


## 5. Conclusions

The cluster anti-synchronization in community networks has been studied in this paper, based on the community structure of the networks. Particularly, weight configuration matrix is not assumed to be irreducible. Some simple and useful criteria are derived by constructing an effective control scheme. The synchronization criteria are independent of time delay. Finally, the developed techniques are applied in three complex community networks. The numerical simulations are performed to verify the effectiveness of the theoretical results.



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