# Pinning Group Synchronization in Complex Dynamical Networks with Different Groups of Oscillators

Authors: Guangming Deng
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper investigates the pinning control schemes and corresponding criteria for group synchronization in a complex dynamical network with different types of chaotic oscillators. We present the linear pinning and adaptive pinning control schemes to make different groups of oscillators synchronize to their own synchronization states, respectively. The globally asymptotically stable criteria for group synchronization are derived, which indicate that the group synchronization can be realized only by pinning a part of nodes in a general network. Finally, some numerical simulations are provided to verify the theoretical results.


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## Body

## 1. Introduction

Nowadays, complex networks widely exist in nature and society, such as the World Wide Web, the Internet, traffic network, communication network, power network, and social network. Over the past three decades, complex networks have received increasing attention from various fields such as biology, physics, chemistry, engineering, and information [1–3].Synchronization is a kind of typical collective behaviors and basic motion in nature. As a typical network dynamics, the synchronization of complex networks has been intensively studied recently. Different synchronization phenomena have been revealed, such as global synchronization, generalized synchronization, partial synchronization, and cluster synchronization [4–11]. In particular, the group (cluster) synchronization, which means that nodes in the same group achieve the same synchronization state while nodes in different groups achieve different synchronization states, is considered to be significant in biological science, laser technology, and communication engineering [1, 2, 12]. In [13], the authors investigated the group synchronization in complex networks with two clusters of coupled dynamical systems and revealed the group synchronization motions whenever there are connections or not within a cluster. In [14], the authors investigated the group synchronization in delay-coupled dynamical networks with groups of different and identical local dynamics, respectively. In [15], we presented a complex dynamical network model with M groups of different chaotic oscillators and derived some criteria to guarantee the group synchronization for a linearly or adaptively coupled network, respectively, under the “same input” condition. However, in many cases, the “same input” condition is too conservative to satisfy. If the “same input” condition cannot be satisfied, does the network achieve group synchronization? In this paper, we will propose control schemes to solve this problem.As we known now, the real-world complex networks normally have a large number of nodes. Therefore, it is usually difficult to control a complex network by adding the controllers to all nodes. To reduce the number of the controllers, a usual approach, which is called the pinning control, is used to control part of nodes. Wang and Chen [16] and Li et al. [17] proposed some effective methods to pin a complex dynamical network to its equilibrium. In [18], Chen et al. proved that a single controller can pin a coupled complex network to a homogenous solution. Zhou et al. presented a simple approximate formula for estimating the detailed number of pinning nodes and the magitude of the coupling strength for a given general complex dynamical network in [19]. However, the researchers mentioned above focused on the global synchronization of complex dynamical networks by pinning control. In this work, we will use the pinning control schemes to realize the group synchronization in a general complex dynamical network.In this paper, based on the model in [15], we further investigate the group synchronization in a complex network containing M groups of different oscillators. Different from [15], this work mainly focuses on the pinning control schemes and the corresponding criteria to realize the group synchronization in a complex dynamical network. In particular, the network has a general structure rather than the “same input” condition.The rest of the paper is organized as follows. In Section2, the network model of our research and mathematical preliminaries are introduced. In Section 3, we present the linear feedback and adaptive feedback pinning schemes, respectively. The corresponding criteria for the globally asymptotically stable of the group synchronization manifold are derived. In Section 4, some numerical simulations are provided to verify the theoretical results. Finally, a brief summary is given in Section 5.


## 2. Model Description and Mathematical Preliminaries

Let us consider a complex dynamical network withM groups; each group contains a number of identical dynamical systems, but the individual dynamic system in different groups can be different. Suppose that there are NI nodes in Ith group, and the individual dynamics satisfies x˙iI=FI(xiI). The M-groups network is described by
(1)x˙iI=FI(xiI)+c∑k≠iNI‍aikII(xkI-xiI)+c∑J≠IM‍∑j=1NJ‍aijIJ(xjJ-xiI),hhhhhhhhhhlhhhi=1,2,…,NI,J=1,2,…,M,
where xiI∈RnI is state vector of the ith node in the Ith group. FI is a smooth nonlinear function vector which describes the node’s dynamics of the Ith group. c>0 is the coupling strength among nodes. The matrix AIJ=(aijIJ)∈RNI×NJ represents the connection between Ith and Jth groups, where if node i in Ith cluster and node j in Jth cluster is connected, aijIJ=1; otherwise, aijIJ=0. In particular, matrix AII represents the internal connection in Ith group, while AIJ(I≠J) represents the external connection between the Ith and Jth group.Furthermore, denote the block matrixA=(AIJ), which represents the structure of network (1). The diagonal elements of matrix A satisfy aiiII=-∑I=1M∑j=1NIaijIJ,i=1,2,…,NI. Then A is a N×N symmetric irreducible matrix with zero-row-sum, and Rank(A)=N-1.Symbol Description. The constant I represents Ith group or the set of nodes in Ith group. I~ represents the subset of I containing the nodes with external connections; that is, I~={i:i∈I,∑J≠IM∑j=1NJaijIJ≠0}, and I∖I~={i:i∈I,aijIJ=0 for I≠J}. NI and lI denote the number of nodes of set I (Ith group) and I~, respectively. N is the total number of nodes in network (1). ∥·∥ is the Euclidian 2-norm; that is, ∥x∥=xTx for vector x.Assumption 1 (A1).
Suppose that there exists positive constantɛk such that
(2)∥Fk(z1)-Fk(z2)∥≤ɛk∥z1-z2∥,
where z1,z2 are time-varying vectors.Definition 2.
The set(3)S={(x11,…,xN11,…,x1M,…,xNMM):xiI=sI,hlhhli=1,…,NI,I=1,…,M{(x11,…,xN11,…,x1M,…,xNMM):xiI=sI,}
is called group synchronous manifold of network (1).Here,sI(t) is the solution of the individual dynamic system of the Ith group satisfying s˙I(t)=FI(sI(t)). Our objective is to stable the homogenous trajectory of xiI(i=1,2,…,NI) to sI(t),I=1,2,…,M. sI(t) may be an equilibrium point, a periodic orbit, an aperiodic orbit, or a chaotic orbit in the phase space.The controlled network (1) can be rewritten as
(4)x˙iI=FI(xiI)+c∑k≠iNI‍aikII(xkI-xiI)+c∑J≠IM‍∑j=1NJ‍aijIJ(xjJ-xiI)+uiI,hhhhhhhhhhhlhhhhhhi=1,2,…,NI,J=1,2,…,M.Define the synchronous error vectors of theIth group as eiI=xiI-sI(i=1,…,NI) and EI=((e1I)T,…,(eNII)T)T. Then the error equations are given by
(5)e˙iI=FI(xiI)-FI(sI)+c∑k≠iNI‍aikIIekI+c∑J≠IM‍∑j=1NJ‍aijIJejJhhh+c∑J=1M‍∑j=1NJ‍aijIJ(sJ-xiI)+uiI,i=1,2,…,NI,J=1,2,…,M.Fori∈I~, there is
(6)∑J=1M‍∑j=1NJ‍aijIJ(sJ-xiI)=∑J=1M‍∑j=1NJ‍aijIJsJ-∑J=1M‍∑j=1NJ‍aijIJxiI=∑J=1M‍∑j=1NJ‍aijIJsJ.Fori∈I∖I~, then aijIJ=0 for I≠J; there is
(7)∑J=1M‍∑j=1NJ‍aijIJ(sJ-xiI)=∑k=1NI‍aikII(sI-xiI)+∑J≠IM‍∑j=1NJ‍aijIJ(sJ-xiI)=∑k=1NI‍aikII(sI-xiI)=∑J=1M‍∑j=1NJ‍aijIJ(sI-xiI)=0.Then the controlled error system (5) can be rewritten as
(8)e˙iI=FI(xiI)-FI(sI)+c∑k≠iNI‍aikIIekI+c∑J≠IM‍∑j=1NJ‍aijIJejJ+c∑J=1M‍∑j=1NJ‍aijIJsJ+uiI,i∈I~,e˙iI=FI(xiI)-FI(sI)+c∑k≠iNI‍aikIIekI+uiI,hhhhhhhlhhi∈I∖I~,I=1,2,…,M.Obviously, the stability problem of group synchronous manifoldS of the controlled network (1) is equivalent to the stability of system (8) at zero, that is, lim⁡t→+∞∥eiI∥=0(i=1,…,NI,I=1,…,M), which implies that group synchronization is achieved.In order to analyze the group synchronization, the following lemmas are needed.Lemma 3 (see [20]).
Suppose thatF, H are the N×N Hermitian matrices, and the eigenvalues of F, H,F+H are α1≥α1≥⋯≥αN, β1≥β1≥⋯≥βN, and γ1≥γ1≥⋯≥γN, respectively. Then there is
(9)αi+βN≤γi≤αi+β1,1≤i≤N.Lemma 4 (see [18]).
IfG=(gij)N×N is an irreducible matrix with Rank(G)=N-1, satisfying gij=gji≥0 for i≠j, and ∑j=1N‍gij=0(i=1,…,N), then, for any constant ɛ>0, all eigenvalues of the matrix
(10)G~=(g11-ɛg12⋯g1Ng21g22⋯a2N⋮⋮⋱⋮gN1gN2⋯gNN)
are negative.Corollary 5.
Suppose thatG=(gij)N×N satisfies the conditions of Lemma 4. Matrix Δ=diag⁡{ɛ1,…,ɛl,0,…,0} for ɛl>0,1≤l<N. Then all eigenvalues of G^=G-Δ are negative.Proof.
Ifl=1, obviously, the conclusion is established by Lemma 4. If l>1, then G^ can be rewritten as G^=G~-Δ~, where G~ is in the form of (10), Δ~=diag⁡{0,ɛ2,…,ɛl,0,…,0}.
By Lemma4, all eigenvalues of G~ are negative. Denote the eigenvalues of G^ and G~ as λ^1≥λ^2≥⋯≥λ^N and 0>λ~1≥λ~2≥⋯≥λ~N, respectively.
According to Lemma3, one has
(11)λ~i-max⁡1≤i≤l⁡{ɛi}≤λ^i≤λ~i+0<0,1≤i≤N-l+1,λ~N-ɛi≤λ^i≤λ~1-ɛi<0,N-l+1<i≤N.
That is, all eigenvalues ofG^ are negative. Thus the proof is completed.Remark 6.
From the derivation process, the conclusion of Corollary5 is also valid when ɛ1,…,ɛl are in any positions of the diagonal of Δ; that is to say, the result holds without any dependence on the sequence of the diagonal elements of Δ.


## 3. Schemes and Criteria

In this section, we present the linear feedback and adaptive feedback pinning schemes, respectively, and derive the corresponding criteria to achieve the group synchronization in network (1).


### 3.1. Linear Feedback Pinning Control for Group Synchronization

For network (1), we add the following linear feedback pinning controllers:
(12)uiI={-cdiIeiI-c∑J=1M‍∑j=1NJ‍aijIJsJ,i∈I~,0,i∈I∖I~.Then the error equation (8) can be rewritten as
(13)e˙iI=FI(xiI)-FI(sI)+c∑J=1M‍∑j=1NJ‍aijIJejJ-cdiIeiI,hldiI>0fori∈I~;diI=0fori∈I∖I~,hhhlhhlhhhi=1,2,…,NI,I=1,2,…,M.Theorem 7.
Suppose thatA1 holds. The group synchronous manifold S of the controlled network (1) under the linear feedback pinning controllers (12) is globally asymptotically stable if the following inequality holds:
(14)min⁡1≤I≤M⁡{c-1ɛI}+λmax⁡(A-D)<0,
where A=(AIJ)N×N, D=diag⁡{d11,…,dl11,0,…,0︸N1,…,d1M,…,dlMM,0,…,0︸NM}.Proof.
Consider the Lyapunov functionV(t)=(1/2)∑I=1M∑i=1NI(eiI)TeiI.
Its time derivative along the trajectory of (13) is
(15)V˙(t)=∑I=1M‍∑i=1NI‍(eiI)T(FI(xiI)-FI(sI))+c∑I=1M‍∑i=1NI‍(eiI)T∑J=1M‍∑j=1NJ‍aijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI≤∑I=1M‍∑i=1NI‍∥(eiI)T∥·∥FI(xiI)-FI(sI)∥+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI≤∑I=1M‍∑i=1NI‍∥(eiI)T∥ɛI∥xiI-sI∥+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI=∑I=1M‍∑i=1NI‍ɛI(eiI)TeiI+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI=ET(ɛIINI⊗IM)E+cETAE-cETDE=ET(ɛIINI⊗IM)E+cETAE-cETDE=cET(P+(A-D))E,
where P=diag⁡{c-1ɛ1,…,c-1ɛ1︸N1,…,c-1ɛM,…,c-1ɛM︸NM}, E=((E1)T,…,(EM)T)T.
According to the Corollary5, A-D is negative definite.
Denote the eigenvalues ofA-D and P+(A-D) as 0=λ1>λ2≥⋯≥λN and γ1≥γ2≥⋯≥γN, respectively. The eigenvalues of P are βj,j=1,2,…,N; there is
(16)min⁡1≤I≤M⁡{c-1ɛI}≤βj≤max⁡1≤I≤M⁡{c-1ɛI}.
According to Lemma3, one has
(17)λN+βj≤γj≤λ1+βj≤λ1+max⁡1≤I≤M⁡{c-1ɛI}=λmax⁡(A-D)+max⁡1≤I≤M⁡{c-1ɛI}<0,hhhhhhhhhhhhhlhhhj=1,2,…,N.
That is,P+(A-D) is negative definite. Then one can obtain lim⁡t→+∞∥eiI∥=0(i=1,…,NI,I=1,…,M), which means that the group synchronous manifold S is globally asymptotically stable. Thus the proof is completed.Remark 8.
From Theorem7, one can obtain the sufficient condition (14) by means of adjusting the coupling strength c. Therefore, the group synchronization can be guaranteed.Remark 9.
In this linear feedback pinning scheme, one only needs to pin a few nodes which have external connections; then the network achieves the group synchronization with large enough coupling strength. In particular, it is only relevant to network connectivity, without any limitation to the network structure. For a cluster network, in general, the connection within a cluster is more than the external connection between clusters. When the network has a cluster property, the higher the clustering, the better the control effect. Since the network with higher clustering means that the number of nodes with external connection is less, thus the less nodes are needed to be pinned. Therefore, the higher the degree of clustering, the more obvious the advantages.


### 3.2. Adaptive Feedback Pinning Control for Group Synchronization

In order to avoid estimating the feedback control gain, in this section, we present an adaptive feedback pining scheme to realize the group synchronization in network (1).For network (1), we add the adaptive feedback pinning controllers as follows:
(18)uiI={-cdiI(t)eiI-c∑J=1M‍∑j=1NJ‍aijIJsJ,d˙iI(t)=cd(eiI)TeiI,d>0,fori∈I~;0,fori∈I∖I~.Then the error equation (8) can be rewritten as
(19)e˙iI=FI(xiI)-FI(sI)+c∑J=1M‍∑j=1NJ‍aijIJejJ-cdiIeiI,d˙iI=cd(eiI)TeiIfori∈I~;diI=0fori∈I∖I~,i=1,2,…,NI,I=1,2,…,M.Theorem 10.
Suppose thatA1 holds. The group synchronous manifold S of the controlled network (1) under the adaptive feedback pinning controllers (18) is globally asymptotically stable if the following inequality holds:
(20)min⁡1≤I≤M⁡{c-1ɛI}+λmax⁡(A-D~)<0,
where A=(AIJ)N×N,D~=diag⁡{d,…,d︸l1,0,…,0︸N1-l1,…,d,…,d︸lM,0,…,0︸NM-lM}.Proof.
Consider the Lyapunov function(21)W(t)=12∑I=1M‍∑i=1NI‍(eiI)TeiI+12d∑I=1M‍∑i=1NI‍(diI-d)2.
Its time derivative along the trajectory of (19) is
(22)W˙=∑I=1M‍∑i=1NI‍(eiI)Te˙iI+1d∑I=1M‍∑i=1NI‍(diI-d)d˙iI=∑I=1M‍∑i=1NI‍(eiI)T(FI(xiI)-FI(sI))+c∑I=1M‍∑i=1NI‍(eiI)T∑J=1M‍∑j=1NJ‍aijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI+c∑I=1M‍∑i=1NI‍(diI-d)(eiI)TeiI≤∑I=1M‍∑i=1NI‍(eiI)TɛieiI+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdeiI=cET(P+(A-D~))E.
The rest is similar to the proof of Theorem7; one has lim⁡t→+∞∥eiI∥=0(i=1,…,NI,I=1,…,M), which means that the group synchronous manifold S is globally asymptotically stable. Thus the proof is completed.Remark 11.
In Theorem10, the adaptive constant d can be large enough such that inequality (20) holds. Therefore, the group synchronization can be easily guaranteed.


## 3.1. Linear Feedback Pinning Control for Group Synchronization

For network (1), we add the following linear feedback pinning controllers:
(12)uiI={-cdiIeiI-c∑J=1M‍∑j=1NJ‍aijIJsJ,i∈I~,0,i∈I∖I~.Then the error equation (8) can be rewritten as
(13)e˙iI=FI(xiI)-FI(sI)+c∑J=1M‍∑j=1NJ‍aijIJejJ-cdiIeiI,hldiI>0fori∈I~;diI=0fori∈I∖I~,hhhlhhlhhhi=1,2,…,NI,I=1,2,…,M.Theorem 7.
Suppose thatA1 holds. The group synchronous manifold S of the controlled network (1) under the linear feedback pinning controllers (12) is globally asymptotically stable if the following inequality holds:
(14)min⁡1≤I≤M⁡{c-1ɛI}+λmax⁡(A-D)<0,
where A=(AIJ)N×N, D=diag⁡{d11,…,dl11,0,…,0︸N1,…,d1M,…,dlMM,0,…,0︸NM}.Proof.
Consider the Lyapunov functionV(t)=(1/2)∑I=1M∑i=1NI(eiI)TeiI.
Its time derivative along the trajectory of (13) is
(15)V˙(t)=∑I=1M‍∑i=1NI‍(eiI)T(FI(xiI)-FI(sI))+c∑I=1M‍∑i=1NI‍(eiI)T∑J=1M‍∑j=1NJ‍aijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI≤∑I=1M‍∑i=1NI‍∥(eiI)T∥·∥FI(xiI)-FI(sI)∥+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI≤∑I=1M‍∑i=1NI‍∥(eiI)T∥ɛI∥xiI-sI∥+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI=∑I=1M‍∑i=1NI‍ɛI(eiI)TeiI+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI=ET(ɛIINI⊗IM)E+cETAE-cETDE=ET(ɛIINI⊗IM)E+cETAE-cETDE=cET(P+(A-D))E,
where P=diag⁡{c-1ɛ1,…,c-1ɛ1︸N1,…,c-1ɛM,…,c-1ɛM︸NM}, E=((E1)T,…,(EM)T)T.
According to the Corollary5, A-D is negative definite.
Denote the eigenvalues ofA-D and P+(A-D) as 0=λ1>λ2≥⋯≥λN and γ1≥γ2≥⋯≥γN, respectively. The eigenvalues of P are βj,j=1,2,…,N; there is
(16)min⁡1≤I≤M⁡{c-1ɛI}≤βj≤max⁡1≤I≤M⁡{c-1ɛI}.
According to Lemma3, one has
(17)λN+βj≤γj≤λ1+βj≤λ1+max⁡1≤I≤M⁡{c-1ɛI}=λmax⁡(A-D)+max⁡1≤I≤M⁡{c-1ɛI}<0,hhhhhhhhhhhhhlhhhj=1,2,…,N.
That is,P+(A-D) is negative definite. Then one can obtain lim⁡t→+∞∥eiI∥=0(i=1,…,NI,I=1,…,M), which means that the group synchronous manifold S is globally asymptotically stable. Thus the proof is completed.Remark 8.
From Theorem7, one can obtain the sufficient condition (14) by means of adjusting the coupling strength c. Therefore, the group synchronization can be guaranteed.Remark 9.
In this linear feedback pinning scheme, one only needs to pin a few nodes which have external connections; then the network achieves the group synchronization with large enough coupling strength. In particular, it is only relevant to network connectivity, without any limitation to the network structure. For a cluster network, in general, the connection within a cluster is more than the external connection between clusters. When the network has a cluster property, the higher the clustering, the better the control effect. Since the network with higher clustering means that the number of nodes with external connection is less, thus the less nodes are needed to be pinned. Therefore, the higher the degree of clustering, the more obvious the advantages.


## 3.2. Adaptive Feedback Pinning Control for Group Synchronization

In order to avoid estimating the feedback control gain, in this section, we present an adaptive feedback pining scheme to realize the group synchronization in network (1).For network (1), we add the adaptive feedback pinning controllers as follows:
(18)uiI={-cdiI(t)eiI-c∑J=1M‍∑j=1NJ‍aijIJsJ,d˙iI(t)=cd(eiI)TeiI,d>0,fori∈I~;0,fori∈I∖I~.Then the error equation (8) can be rewritten as
(19)e˙iI=FI(xiI)-FI(sI)+c∑J=1M‍∑j=1NJ‍aijIJejJ-cdiIeiI,d˙iI=cd(eiI)TeiIfori∈I~;diI=0fori∈I∖I~,i=1,2,…,NI,I=1,2,…,M.Theorem 10.
Suppose thatA1 holds. The group synchronous manifold S of the controlled network (1) under the adaptive feedback pinning controllers (18) is globally asymptotically stable if the following inequality holds:
(20)min⁡1≤I≤M⁡{c-1ɛI}+λmax⁡(A-D~)<0,
where A=(AIJ)N×N,D~=diag⁡{d,…,d︸l1,0,…,0︸N1-l1,…,d,…,d︸lM,0,…,0︸NM-lM}.Proof.
Consider the Lyapunov function(21)W(t)=12∑I=1M‍∑i=1NI‍(eiI)TeiI+12d∑I=1M‍∑i=1NI‍(diI-d)2.
Its time derivative along the trajectory of (19) is
(22)W˙=∑I=1M‍∑i=1NI‍(eiI)Te˙iI+1d∑I=1M‍∑i=1NI‍(diI-d)d˙iI=∑I=1M‍∑i=1NI‍(eiI)T(FI(xiI)-FI(sI))+c∑I=1M‍∑i=1NI‍(eiI)T∑J=1M‍∑j=1NJ‍aijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdiIeiI+c∑I=1M‍∑i=1NI‍(diI-d)(eiI)TeiI≤∑I=1M‍∑i=1NI‍(eiI)TɛieiI+c∑I=1M‍∑i=1NI‍∑J=1M‍∑j=1NJ‍(eiI)TaijIJejJ-c∑I=1M‍∑i=1NI‍(eiI)TdeiI=cET(P+(A-D~))E.
The rest is similar to the proof of Theorem7; one has lim⁡t→+∞∥eiI∥=0(i=1,…,NI,I=1,…,M), which means that the group synchronous manifold S is globally asymptotically stable. Thus the proof is completed.Remark 11.
In Theorem10, the adaptive constant d can be large enough such that inequality (20) holds. Therefore, the group synchronization can be easily guaranteed.


## 4. Numerical Simulations

In this section, an illustrative example is provided to verify the above theoretical analysis.We consider a network (see Figure1) which consists of three groups with 4 Lorenz chaotic oscillators [20] as the x-group, 5 Chua oscillators [21] as the y-group, and 6 Chen oscillators [22] as the z-group, respectively. The Lorenz, Chua, and Chen chaotic attractors are displayed in Figure 2.Figure 1
The topology connection of network (23).The chaotic attractors of Lorenz (a), Chua (b), and Chen (c).
(a)
(b)
(c)The network as an example of the controlled network (1) is described by
(23)x˙i=F1(xi)+c∑j=14‍aijxj+c∑j=15‍bij(yj-xi)hhhh+c∑j=16‍hij(zj-xi)+ui1,1≤i≤4,y˙i=F2(yi)+c∑j=15‍a¯ijyj+c∑j=14‍b¯ij(xj-yi)hhhh+c∑j=16‍h¯ij(zj-yi)+ui2,1≤i≤5,z˙i=F3(zi)+c∑j=16‍a~ijzjhhhh+c∑j=14‍b~ij(xj-zi)+c∑j=15‍h~ij(yj-zi)+ui3,hhhhhhhhhhlhhhhhhhhhhhhhhhhh1≤i≤6,
where F1(xi)=(10(xi2-xi1),28xi1-xi2-xi1xi3,xi1xi2-8/3xi3)T; F2(yi)=(10(yi2-yi1)-f(yi1),yi1-yi2+yi3,-14.7yi2)T, f(v)=m0v+0.5(m1-m0)(|x+1|-|x-1|),m0=-0.68,m1=-1.27; F3(zi)=(35(zi2-zi1),-7zi1+28zi2-zi1zi3,zi1zi2-8/3zi3)T. The controllers uiI are taken as (12) and (18), respectively.In the simulation, we take the initial valuesxi(0)=(-5+0.5i,0.5i,5+0.5i)T(1≤i≤4), yi(0)=(-5+0.5i,0.5i,5+0.5i,10+0.5i)T(1≤i≤5), zi(0)=(-5+0.5i,0.5i,5+0.5i)T(1≤i≤6), and d(0)=1.Figures3 and 4 display the numerical simulation results of network (23) under the linear feedback and adaptive feedback pinning control, respectively. They both show that a set of nodes belonging to each group synchronizes to the same behavior; that is, the group synchronization has been achieved.Group synchronization of network (23) with linear feedback pinning controllers (12). (a) The synchrony states xi(1≤i≤4) of Lorenz oscillators in x-group; (b) the synchrony states yi(1≤i≤5) of Chua oscillators in y-group; (c) the synchrony states zi(1≤i≤6) of Chen oscillators in z-group.
(a)
(b)
(c)Group synchronization of network (23) with adaptive feedback pinning controllers (18). (a) The synchrony states xi(1≤i≤4) of Lorenz oscillators in x-group; (b) the synchrony states yi(1≤i≤5) of Chua oscillators in y-group; (c) the synchrony states zi(1≤i≤6) of Chen oscillators in z-group.
(a)
(b)
(c)Remark 12.
By the above theoretical analysis and instance, it is concluded that the pinning control strategy can be applied to large networks with tens of thousands of nodes based on the accurate network topology. In fact, this control strategy depends on the structure information rather than the size of the network.


## 5. Conclusions

In this paper, we have investigated the group synchronization in a complex dynamical network consisting of different groups of coupled chaotic oscillators. The linear feedback and adaptive feedback pinning schemes are presented, respectively. The corresponding sufficient conditions are derived to ensure each group synchronizes to the same states. Our study shows that the global asymptotic stability of the group synchronous manifold can be guaranteed by controlling a few nodes which have external connections. For the large networks with obvious cluster property, our pinning control schemes can save much cost in practical.



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*Source: 864193-2014-03-04.xml*