# Intermittent Control for Cluster-Delay Synchronization in Directed Networks

Authors: Jianbao Zhang; Yi Wang; Zhongjun Ma; Jianlong Qiu; Fawaz Alsaadi
Journal: Complexity
 (2018)

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## Abstract

We investigate cluster-delay synchronization of a directed network possessing cluster structures by designing an intermittent control protocol. Based on Lyapunov stability theory, we proved that synchronization can be realized for oscillators in the same cluster and cluster-delay synchronization can be realized for the whole network. By simplifying the obtained sufficient conditions, we carry out a succinct and utilitarian corollary. In addition, comparative researches are carried out to show the differences and the usefulness of the obtained results with respect to other similar controllers from the recent literature. Finally we provide two numerical examples to show the effectiveness of the control schemes.


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## Body

## 1. Introduction

Chaos theory is an interdisciplinary theory studying the unstable aperiodic behavior of dynamical systems. The most distinguishing characteristic of chaotic systems is the highly sensitive dependence on initial conditions [1], which implies that even if the present determines the future, the approximate present does not approximately determine the future. Therefore, it is difficult to control an unpredictable chaotic system in the long term. However, due to the wide applications of chaos theory, more and more researchers are devoting themselves to studying chaos control theory in many fields of science and engineering.In 1990, it was discovered surprisingly that two chaotic systems started from different initial conditions could synchronize with each other [2], and the great discovery immediately attracted lots of attention and became an important issues of chaos control. Since then, several effective methods have been applied to study synchronization of chaos oscillators. In 1998, the famous master stability function method was proposed to study the local stability of the synchronous state [3], which is based on the calculation of the maximum Lyapunov exponent for the least stable transversal mode of the synchronous manifold and the eigenvalues of the connection matrix. Later, Lyapunov function method was employed to investigate global stability of the synchronous state [4]. Based on the two methods mentioned above, many other surveys have been carried out to explore the mysterious mechanisms of chaos synchronization and chaos control. As the study develops in depth, various kinds of synchronization protocols have been put forward and deeply studied, such as complete synchronization [5], exponential synchronization [6, 7], projective synchronization [8, 9], lag synchronization [10], and cluster synchronization [11–13]. The above-mentioned results only discussed synchronization induced by mutual coupling and the intrinsic structure of the network.During the past decades, many external control strategies have been carried out to synchronize complex networks, such as adaptive control [14, 15], impulsive control [16], sliding mode control [17], pinning control [18–20], sliding mode control [21], and intermittent control [22, 23]. Those control strategies have been widely investigated and used in many network control problems. The primary concern of this paper is cluster synchronization under external control, which has attracted widespread attention [24, 25]. By designing adaptive pinning-control schemes on both coupling strengths and feedback gains, it was shown that a network can realize cluster synchronization under weak coupling strengths and small feedback gains [15]. Later, another feedback controller was designed to realize cluster synchronization under the condition that the topology of each cluster has a directed spanning tree [25, 26]. All in all, great efforts have been devoted to the investigation of cluster synchronization under external control. It has been shown to be an effective method to control a complex network to a desired synchronized state.Up to now, to the best of our knowledge, there are few results concerning cluster-delay synchronization, which is a special type of collective behavior between complete synchronization and cluster synchronization. In our opinion, cluster-delay synchronization implies that the nodes in a complex network are split into several clusters, and all the nodes in the same cluster behave in a synchronous fashion, but nodes in different clusters follow distinct time evolutions with different time delays. It is a new type of collective behavior in complex networks worthy of detailed investigation, and this paper studies cluster-delay synchronization of a complex network via pinning control with intermittent effect. To achieve cluster lag synchronization in community networks, Wu and Fu designed several linear pinning controllers in view of lower cost and more convenient implementation [23]. Recently, motivated by the interesting investigation, we provided some primary theoretical analyses and numerical experiment [27]. Different from the previous results [23, 27], this paper proposes a leader-following system and derives sufficient conditions for cluster-delay synchronization via pinning control with intermittent effect. We first prove that all the oscillators in the same cluster synchronize with each other and then prove that the oscillators in different clusters behave in a synchronous mode but with different time delays. Numerical simulations show that the modified pinning-control scheme works effectively and serves different purposes in practice.The rest of this paper is organized as follows. Section2 introduces some necessary preliminaries and builds a modified clustered network model with an intermittent leader-following controller. Then, both cluster synchronization and cluster-delay synchronization of the network model are investigated through Lyapunov theory in Section 3. Comparative researches with respect to previous controllers are also given there. In Section 4, two examples of numerical simulations are carried out to show the validity of the proposed control schemes. Finally, the main results of this paper are briefly summarized in Section 5.


## 2. Preliminaries

In this section, we make some mathematical preparations for the oscillator network model. Suppose the topology structure of the communication network is represented by a directed graphG={V,E}, which is composed of a set of nodes V={1,…,N} and a set of edges E=V×V. The graph exhibits a clustered structure, which implies that the N nonidentical oscillators are divided into n nonempty subsets called clusters. Let Vk={mk-1+1,…,mk} denote the index set of all the nodes in the kth cluster, where k=1,2,…,n, m0=0, mn=N. For convenience, we define a function φ:{1,…,N}→{1,…,n}, where φ(i)=k implies that the node i∈Vk.Letxi(t)=(xi1(t),…,xid(t))⊤ be the state variable of the ith oscillator, the state equations of the network are given by (1)x˙it=fxit+∑l=1n∑j∈Vlaijxjt+uit,i=1,2,…,N,where t∈[0,+∞) is a continuous time, f(·) is a continuous function that describes the local dynamics of each node, A=(aij)N×N is the coupling matrix with aij≥0 for i≠j, and ∑j=1Naij=0 for i=1,2,…,N. Denoting the state variables of the oscillators in kth cluster as Xk(t)=(xmk-1+1⊤(t),…,xmk⊤(t))⊤, Uk(t)=(umk-1+1⊤(t),…,umk⊤(t))⊤, and F(Xk(t))=(f(xmk-1+1(t))⊤,…,f(xmk(t))⊤)⊤, the network (1) can be rewritten as follows: (2)dXktdt=FXkt+∑l=1nAkl⊗IdXlt+Ukt,k=1,2,…,n,where Akl∈R(mk-mk-1)×(ml-ml-1) are defined by the block matrix (3)A=A11A12⋯A1nA21A22⋯A2n⋯⋯⋯⋯An1An2⋯Ann.For convenience, we decompose matrix A into two matrices as follows:(4)A=A-+A~,where (5)A-=A-kln×n=a-ijN×N=0,φi≠φj;aij,φi=φj,i≠j;-∑j=1,j≠iNa-ij,i=jrepresenting the intracluster couplings, and (6)A~=A~kln×n=a~ijN×N=0,φi=φj,i≠j;aij,φi≠φj;-∑j=1,j≠iNa~ij,i=jrepresenting the intercluster couplings.The dynamics of the virtual leaders in the oscillator network are described by(7)s˙kt=fskt-θktskt-s1t-τk,k=1,2,…,n,where sk(t) is the state variable of the kth virtual leader, τk is the time delay, and (8)θkt=Θk,t∈mT,m+ζT;0,t∈m+ζT,m+1T,where Θk is a positive constant representing the feedback control gain, ζ∈(0,1) is the control width, k=1,2,…,n, m=0,1,2,…. In this paper, we suppose that the time delay τ1=0, which implies that s˙1(t)=f(s1(t)). Different from the previous result on cluster lag synchronization [23], the leader systems in this paper are not isolated nodes s˙k(t)=f(sk(t)). Instead, linear control laws are designed for the leader systems to realize cluster-delay synchronization. This paper aims to make the leader of the kth cluster track the trajectory of the leader of the first cluster with a time delay τk, k=1,…,n.Analyzing systems (2) and (7) comprehensively, we design the control inputs for system (2): (9)Ukt=θktιmk-mk-1⊗skt-s1t-τk-∑l=1nA~kl⊗IdXlt-σkXkt-ιmk-mk-1⊗s1t-τk,where the vector ιmk-mk-1=(1,…,1)⊤∈Rmk-mk-1; the constant σk≥0 is the feedback control gain; T>0 is the control period; k=1,2,…,n.


## 3. Main Results

In this section, we will derive some sufficient conditions for both cluster synchronization and cluster-delay synchronization. Before that, it is necessary to introduce the following assumptions.Assumption 1.
There exists a positive constantδ such that the vector function f satisfies that (10)x-yTfx-fy≤δx-yTx-yfor any x,y∈Rd.It has been checked that many well-known chaotic systems, such as cellular neural networks, Lorenz system, Chen system, Rössler system, and Chua’s circuit, satisfy Assumption1 [28, 29].


### 3.1. Cluster Synchronization Analysis

Now, we first introduce the following definition of both cluster synchronization and cluster-delay synchronization [27].Definition 2.
Define the synchronization errorsei(t)=xi(t)-sφ(i)(t), Ek(t)=Xk(t)-ιmk-mk-1⊗sk(t), E(t)=(E1⊤(t),…,En⊤(t))⊤, ekτ(t)=sk(t)-s1(t-τk), Eτ(t)=(e1τ⊤(t),…,enτ⊤(t))⊤. The oscillator network (2)–(7) is said to realize cluster synchronization, if the synchronization errors satisfy (11)limt→∞⁡∑i=1Neit=limt→∞⁡∑k=1nEkt=limt→∞⁡Et=0.The oscillator network (2)–(7) is said to realize cluster-delay synchronization, if the synchronization errors satisfy equality (11) and (12)limt→∞⁡∑k=2nekτt=limt→∞⁡Eτt=0.The preliminaries above, together with Lyapunov function method, bring us to the following theorem for cluster synchronization, which implies that themk-mk-1 oscillators in cluster Vk to synchronize with each other, k=1,2,…,d.Theorem 3.
Suppose that Assumption1 holds; the oscillator network (2)–(7) with the control protocol (9) realizes cluster synchronization if the matrix D-(A-+A-⊤)/2 is negative definite, where D=diag{(δ-σ1)Im1-m0,…,(δ-σn)Imn-mn-1}.Proof.
Noticing thatEk(t)=Xk(t)-ιmk-mk-1⊗sk(t), we obtain the error system of the kth cluster as follows: (13)dEktdt=dXktdt-ιmk-mk-1⊗dsktdt=FXkt-ιmk-mk-1⊗fskt+∑l=1nA-kl⊗IdXlt-σkXkt-ιmk-mk-1⊗s1t-τk=FXkt-Fιmk-mk-1⊗skt+∑l=1nA-kl⊗IdElt-σkEkt.Consider the following Lyapunov function: (14)V1t=12∑k=1nEk⊤tEkt.The derivative of V1(t) along the trajectories of the error systems (13) can be calculated as follows:(15)dV1tdt=∑k=1nEk⊤tdEktdt=∑k=1nEk⊤tFXkt-Fιmk-mk-1⊗skt+∑l=1nA-kl⊗IdElt-σkEkt≤∑k=1nδ-σkEk⊤tEkt+∑l=1nEk⊤tA-kl⊗IdElt=E⊤tD-A-+A-⊤2⊗IdEt.According to the conditions of Theorem 3 and Lyapunov stability theory, the solutions of the oscillator network satisfy that limt→∞⁡Ek(t)=0 for all k=1,2,…,n. Hence, the oscillator network (2)–(7) with the control protocol (9) realizes cluster synchronization. The proof is completed.Noticing that matrixA- represents the intracluster couplings and matrix A~ represents the intercluster couplings, one gets that the results of Theorem 3 is irrelevant to the intercluster couplings. In other words, cluster synchronization can be guaranteed by the intracluster couplings of each cluster, and the intercluster couplings can be chosen arbitrarily.


### 3.2. Cluster-Delay Synchronization Analysis

Now, we are in a position to carry out the following theorem on cluster-delay synchronization, which implies that the oscillators in the same cluster behave in a synchronous fashion, but oscillators in different clusters follow distinct time evolutions with different time delays.Theorem 4.
Suppose that Assumption1 holds, the oscillator network (2)–(7) with the control protocol (9) realizes cluster-delay synchronization if (i)
the matrixD-(A-+A-⊤)/2 is negative definite;(ii)
there exists a positive constantα such that the matrix (δ+α)In-Θ is negative semidefinite, where Θ=diag{Θ1,Θ2,…,Θn};(iii)
the constantδ-δζ-αζ≤0.Proof.
According to Theorem3 and condition (i) of Theorem 4, it is easy to prove that (16)limt→∞⁡∑i=1Neit=limt→∞⁡∑k=1nEkt=limt→∞⁡Et=0.Now, we will prove that (17)limt→∞⁡∑k=2nekτt=limt→∞⁡Eτt=0.
Consider the following Lyapunov function:(18)V2Eτt=12∑k=1nekτ⊤tekτt=12Eτ⊤tEτt.From the oscillator network (2)–(7) with the control protocol (9), it is easy to get the following error system:(19)dekτtdt=dsktdt-ds1t-τkdt=fskt-fs1t-τk-θktekτt,k=1,2,…,n.Calculating the derivative of V2(Eτ(t)), one obtains (20)dV2Eτtdt=∑k=1nekτ⊤tdekτtdt=∑k=1nekτ⊤tfskt-fs1t-τk-θktekτt≤∑k=1nδekτ⊤tekτt-∑k=1nθktekτ⊤tekτt.On the interval t∈[mT,(m+ζ)T), m=0,1,2,…, inequality (20) can be reduced to the following form:(21)dV2Eτtdt≤∑k=1nδekτ⊤tekτt-∑k=1nΘkekτ⊤tekτt=Eτ⊤tδ+αIn-Θ⊗IdEτt-αEτ⊤tEτt≤-2αV2Eτt.Integrating the above inequality over the interval [mT,(m+ζ)T), one has (22)V2Eτt≤V2Eτ⊤mTe-2αt-mT.On the interval t∈[(m+ζ)T,(m+1)T), the inequality (20) can be reduced to the following form: (23)dV2Eτtdt≤∑k=1nδekτ⊤ekτ≤2δV2Eτt,which is equivalent to (24)V2Eτt≤V2Eτm+ζTe2δt-m+ζT.
Now, we will prove the following inequality by mathematical induction on parameterm: (25)V2Eτt≤V2Eτ0e2mδ+α1-ζT-2αt,mT≤t<m+ζT;V2Eτt≤V2Eτ0e2δt-2m+1δ+αζT,m+ζT≤t<m+1T;V2Eτm+1T≤V2Eτ0e2m+1δ-δζ-αζT.
Firstly, we will show the validity of the base case. In case of parameterm=0, inequalities (22) and (24) can be reduced as follows:(26)V2Eτt≤V2Eτ0e-2αt,0≤t<ζT,V2Eτt≤V2EτζTe2δt-ζT≤V2Eτ0e2δt-2α+δζT,ζT≤t<T,where (27)V2EτζT≤V2Eτ0e-2αζT,V2EτT≤V2Eτ0e2δT-2α+δζT.It can be concluded from the above two inequalities that inequality (25) holds for m=0.
Secondly, assuming that inequality (25) is correct for m=k, we will show its correctness for m=k+1. In fact, if m=k+1, inequalities (22) and (24) can be reduced as follows:(28)V2Eτt≤V2Eτ⊤k+1Te-2αt-k+1T≤V2Eτ0e2k+1δ+α1-ζT-2αt,k+1T≤t<k+1+ζT,V2Eτt≤V2Eτk+1+ζTe2δt-k+1+ζT≤V2Eτ0e2δt-2k+2δ+αζT,k+1+ζT≤t<k+2T,where (29)V2Eτk+1+ζT≤V2Eτ0e2k+1δ-δζ-αζT-2αζT,V2Eτk+2T≤V2Eτ0e2k+2δ-δζ-αζT.Then, based on the principle of mathematical induction, we declare that inequality (25) holds for m=0,1,2,….
Combining the monotonic property of the exponential function and the inequality (25), we obtain that (30)V2Eτt≤V2Eτ0e2mδ-δζ-αζT.Noticing condition (iii) of Theorem 4, we can derive that cluster synchronization of the controlled network (2)–(7) is achieved. Hence the proof is completed.To make Theorem4 more applicable, we give the following corollary.Corollary 5.
Suppose that Assumption1 holds, the oscillator network (2)–(7) with the control protocol (9) realizes cluster-delay synchronization if (i)
the constantsδ-σk-λ-k<0, where λ-k are the eigenvalues of the symmetric matrix (A-+A-⊤)/2, k=1,2,…,n;(ii)
there exists a positive constantα satisfying that (31)1ζ-1δ≤α≤Θk-δ,k=1,2,…,n.It is worth noting that condition (ii) can be simplified into an inequality≤Θkζ, k=1,2,…,n. The proof of this corollary is not particularly difficult and will not be given here.


### 3.3. Comparative Studies with Previous Results

In [23], cluster lag synchronization of the undirected networks (1) has been studied by using the intermittent pinning-control method. Enlightened by the design schemes of the controllers with intermittent effect, we proposed the leader-following system (2)–(7) and designed the intermittent pinning controller (9) to realize cluster-delay synchronization.In order to verify the usefulness of the obtained controller with respect to the previous controllers, we carry out some comparative studies to show the differences from two aspects. The first difference is the definition of cluster lag synchronization with respect to the time delaysτφi, which implies that there holds limt→+∞xi(t)-sφi(t-τφi)=0, where s˙φi(t-τφi)=fφi(sφi(t-τφi)), i=1,2,…,N. In this paper, we proposed the definition of cluster-delay synchronization (Definition 2) in two steps and developed a series of sufficient conditions for both cluster synchronization and cluster-delay synchronization. The second difference is the design schemes of the controllers with intermittent effect. In [23], Wu and Fu designed N controllers for each oscillator as follows: (32)uit=-εθitxit-st-τφi-ε∑j=1NcijΓst-τφi,i∈V-φi0,i∈Vφi-V-φi,where the intermittent feedback control gain θi(t) was defined by (8). In this paper, we designed n controllers for each cluster in (9) and simplified the complexity of the previous design schemes to some extent.Based on the aforementioned comparison and analysis, we show the characteristics and advantages of the proposed method with respect to the previous controllers. In our view, the proposed method might serve different purposes in practice.


## 3.1. Cluster Synchronization Analysis

Now, we first introduce the following definition of both cluster synchronization and cluster-delay synchronization [27].Definition 2.
Define the synchronization errorsei(t)=xi(t)-sφ(i)(t), Ek(t)=Xk(t)-ιmk-mk-1⊗sk(t), E(t)=(E1⊤(t),…,En⊤(t))⊤, ekτ(t)=sk(t)-s1(t-τk), Eτ(t)=(e1τ⊤(t),…,enτ⊤(t))⊤. The oscillator network (2)–(7) is said to realize cluster synchronization, if the synchronization errors satisfy (11)limt→∞⁡∑i=1Neit=limt→∞⁡∑k=1nEkt=limt→∞⁡Et=0.The oscillator network (2)–(7) is said to realize cluster-delay synchronization, if the synchronization errors satisfy equality (11) and (12)limt→∞⁡∑k=2nekτt=limt→∞⁡Eτt=0.The preliminaries above, together with Lyapunov function method, bring us to the following theorem for cluster synchronization, which implies that themk-mk-1 oscillators in cluster Vk to synchronize with each other, k=1,2,…,d.Theorem 3.
Suppose that Assumption1 holds; the oscillator network (2)–(7) with the control protocol (9) realizes cluster synchronization if the matrix D-(A-+A-⊤)/2 is negative definite, where D=diag{(δ-σ1)Im1-m0,…,(δ-σn)Imn-mn-1}.Proof.
Noticing thatEk(t)=Xk(t)-ιmk-mk-1⊗sk(t), we obtain the error system of the kth cluster as follows: (13)dEktdt=dXktdt-ιmk-mk-1⊗dsktdt=FXkt-ιmk-mk-1⊗fskt+∑l=1nA-kl⊗IdXlt-σkXkt-ιmk-mk-1⊗s1t-τk=FXkt-Fιmk-mk-1⊗skt+∑l=1nA-kl⊗IdElt-σkEkt.Consider the following Lyapunov function: (14)V1t=12∑k=1nEk⊤tEkt.The derivative of V1(t) along the trajectories of the error systems (13) can be calculated as follows:(15)dV1tdt=∑k=1nEk⊤tdEktdt=∑k=1nEk⊤tFXkt-Fιmk-mk-1⊗skt+∑l=1nA-kl⊗IdElt-σkEkt≤∑k=1nδ-σkEk⊤tEkt+∑l=1nEk⊤tA-kl⊗IdElt=E⊤tD-A-+A-⊤2⊗IdEt.According to the conditions of Theorem 3 and Lyapunov stability theory, the solutions of the oscillator network satisfy that limt→∞⁡Ek(t)=0 for all k=1,2,…,n. Hence, the oscillator network (2)–(7) with the control protocol (9) realizes cluster synchronization. The proof is completed.Noticing that matrixA- represents the intracluster couplings and matrix A~ represents the intercluster couplings, one gets that the results of Theorem 3 is irrelevant to the intercluster couplings. In other words, cluster synchronization can be guaranteed by the intracluster couplings of each cluster, and the intercluster couplings can be chosen arbitrarily.


## 3.2. Cluster-Delay Synchronization Analysis

Now, we are in a position to carry out the following theorem on cluster-delay synchronization, which implies that the oscillators in the same cluster behave in a synchronous fashion, but oscillators in different clusters follow distinct time evolutions with different time delays.Theorem 4.
Suppose that Assumption1 holds, the oscillator network (2)–(7) with the control protocol (9) realizes cluster-delay synchronization if (i)
the matrixD-(A-+A-⊤)/2 is negative definite;(ii)
there exists a positive constantα such that the matrix (δ+α)In-Θ is negative semidefinite, where Θ=diag{Θ1,Θ2,…,Θn};(iii)
the constantδ-δζ-αζ≤0.Proof.
According to Theorem3 and condition (i) of Theorem 4, it is easy to prove that (16)limt→∞⁡∑i=1Neit=limt→∞⁡∑k=1nEkt=limt→∞⁡Et=0.Now, we will prove that (17)limt→∞⁡∑k=2nekτt=limt→∞⁡Eτt=0.
Consider the following Lyapunov function:(18)V2Eτt=12∑k=1nekτ⊤tekτt=12Eτ⊤tEτt.From the oscillator network (2)–(7) with the control protocol (9), it is easy to get the following error system:(19)dekτtdt=dsktdt-ds1t-τkdt=fskt-fs1t-τk-θktekτt,k=1,2,…,n.Calculating the derivative of V2(Eτ(t)), one obtains (20)dV2Eτtdt=∑k=1nekτ⊤tdekτtdt=∑k=1nekτ⊤tfskt-fs1t-τk-θktekτt≤∑k=1nδekτ⊤tekτt-∑k=1nθktekτ⊤tekτt.On the interval t∈[mT,(m+ζ)T), m=0,1,2,…, inequality (20) can be reduced to the following form:(21)dV2Eτtdt≤∑k=1nδekτ⊤tekτt-∑k=1nΘkekτ⊤tekτt=Eτ⊤tδ+αIn-Θ⊗IdEτt-αEτ⊤tEτt≤-2αV2Eτt.Integrating the above inequality over the interval [mT,(m+ζ)T), one has (22)V2Eτt≤V2Eτ⊤mTe-2αt-mT.On the interval t∈[(m+ζ)T,(m+1)T), the inequality (20) can be reduced to the following form: (23)dV2Eτtdt≤∑k=1nδekτ⊤ekτ≤2δV2Eτt,which is equivalent to (24)V2Eτt≤V2Eτm+ζTe2δt-m+ζT.
Now, we will prove the following inequality by mathematical induction on parameterm: (25)V2Eτt≤V2Eτ0e2mδ+α1-ζT-2αt,mT≤t<m+ζT;V2Eτt≤V2Eτ0e2δt-2m+1δ+αζT,m+ζT≤t<m+1T;V2Eτm+1T≤V2Eτ0e2m+1δ-δζ-αζT.
Firstly, we will show the validity of the base case. In case of parameterm=0, inequalities (22) and (24) can be reduced as follows:(26)V2Eτt≤V2Eτ0e-2αt,0≤t<ζT,V2Eτt≤V2EτζTe2δt-ζT≤V2Eτ0e2δt-2α+δζT,ζT≤t<T,where (27)V2EτζT≤V2Eτ0e-2αζT,V2EτT≤V2Eτ0e2δT-2α+δζT.It can be concluded from the above two inequalities that inequality (25) holds for m=0.
Secondly, assuming that inequality (25) is correct for m=k, we will show its correctness for m=k+1. In fact, if m=k+1, inequalities (22) and (24) can be reduced as follows:(28)V2Eτt≤V2Eτ⊤k+1Te-2αt-k+1T≤V2Eτ0e2k+1δ+α1-ζT-2αt,k+1T≤t<k+1+ζT,V2Eτt≤V2Eτk+1+ζTe2δt-k+1+ζT≤V2Eτ0e2δt-2k+2δ+αζT,k+1+ζT≤t<k+2T,where (29)V2Eτk+1+ζT≤V2Eτ0e2k+1δ-δζ-αζT-2αζT,V2Eτk+2T≤V2Eτ0e2k+2δ-δζ-αζT.Then, based on the principle of mathematical induction, we declare that inequality (25) holds for m=0,1,2,….
Combining the monotonic property of the exponential function and the inequality (25), we obtain that (30)V2Eτt≤V2Eτ0e2mδ-δζ-αζT.Noticing condition (iii) of Theorem 4, we can derive that cluster synchronization of the controlled network (2)–(7) is achieved. Hence the proof is completed.To make Theorem4 more applicable, we give the following corollary.Corollary 5.
Suppose that Assumption1 holds, the oscillator network (2)–(7) with the control protocol (9) realizes cluster-delay synchronization if (i)
the constantsδ-σk-λ-k<0, where λ-k are the eigenvalues of the symmetric matrix (A-+A-⊤)/2, k=1,2,…,n;(ii)
there exists a positive constantα satisfying that (31)1ζ-1δ≤α≤Θk-δ,k=1,2,…,n.It is worth noting that condition (ii) can be simplified into an inequality≤Θkζ, k=1,2,…,n. The proof of this corollary is not particularly difficult and will not be given here.


## 3.3. Comparative Studies with Previous Results

In [23], cluster lag synchronization of the undirected networks (1) has been studied by using the intermittent pinning-control method. Enlightened by the design schemes of the controllers with intermittent effect, we proposed the leader-following system (2)–(7) and designed the intermittent pinning controller (9) to realize cluster-delay synchronization.In order to verify the usefulness of the obtained controller with respect to the previous controllers, we carry out some comparative studies to show the differences from two aspects. The first difference is the definition of cluster lag synchronization with respect to the time delaysτφi, which implies that there holds limt→+∞xi(t)-sφi(t-τφi)=0, where s˙φi(t-τφi)=fφi(sφi(t-τφi)), i=1,2,…,N. In this paper, we proposed the definition of cluster-delay synchronization (Definition 2) in two steps and developed a series of sufficient conditions for both cluster synchronization and cluster-delay synchronization. The second difference is the design schemes of the controllers with intermittent effect. In [23], Wu and Fu designed N controllers for each oscillator as follows: (32)uit=-εθitxit-st-τφi-ε∑j=1NcijΓst-τφi,i∈V-φi0,i∈Vφi-V-φi,where the intermittent feedback control gain θi(t) was defined by (8). In this paper, we designed n controllers for each cluster in (9) and simplified the complexity of the previous design schemes to some extent.Based on the aforementioned comparison and analysis, we show the characteristics and advantages of the proposed method with respect to the previous controllers. In our view, the proposed method might serve different purposes in practice.


## 4. Numerical Simulations

In this section, we carry out some numerical simulations to illustrate the effectiveness of the theoretical results obtained in this paper.


### 4.1. Numerical Example 1

At first, we consider a directed complex network consisting of8 nodes separated into three different clusters, the topology of which is shown in Figure 1.Figure 1
Topology structure of a directed network consisting of 8 nodes, which are divided into3 clusters. If there is a directed link from node j to node i (j≠i), then aij=1, otherwise aij=0.Define connectivity matrixA as an asymmetric matrix with zero row-sums, and the off-diagonal elements aij=1 if the jth node can receive information from the ith node for j≠i; otherwise, define aij=0. Choose the node dynamics of the network as the well-known Chua oscillators; then the oscillator network (2)–(7) with the control protocol (9) can be rewritten as follows: (33)x˙i=-Dxi+Tgxi+∑j=18aijxj+uit,uit=θφitsφit-s1t-τφi-∑j=18a~ijxj-σφixit-s1t-τφi,s˙φit=fsφit-θφitsφit-s1t-τφi,where xi∈R3, D=I3, g(xi)=(g(xi1),g(xi2),g(xi3))⊤, i=1,2,…,8, g(s)=(|s+1|-|s-1|)/2, and (34)T=1.25-3.2-3.2-3.21.1-4.4-3.24.41.0.It is easy to derive that the nonlinear functionf in system (33) satisfies Assumption 1 by choosing the matrix Δ=5.5685I3. Then, one can choose δ=5.5685, σ1=10, σ2=12, σ3=14 such that condition (i) of Theorem 4 holds, and choose α=4.4, Θ1=Θ2=Θ3=10 such that condition (ii) holds. By taking T=1, it can be verified that the feedback control width ζ=0.557 satisfies condition (iii) of Theorem 4.Choose the initial conditions randomly; Figure2 is plotted to show the evolutions of the state variable of the 8 nodes and the cluster errors. From (a), (b), and (c), one can see that the evolutions of the nodes in the same cluster synchronize with each other. And (d) shows more clearly that the cluster errors ei(t)=xi(t)-sφ(i)(t) tend to zero. Therefore, Figure 2 illustrates that cluster-delay synchronization is achieved under the conditions of Theorem 4.Figure 2
Cluster-delay synchronization of the network (33). (a), (b), and (c) show the time evolutions of the state variable of the 8 nodes splitting into three different clusters, and (d) describes the time evolutions of the cluster errors.
(a)
(b)
(c)
(d)Next, we keep all the parameters unchanged except the feedback control gains decreasing fromσ1=10, σ2=12, σ3=14 to σ1=σ2=σ3=2. Then the conditions of Theorems 3 and 4 cannot be satisfied. The time evolutions of the Chua oscillators xi1(t), xi2(t), xi3(t)  (i=1,2,…,8) in the network (33) are plotted in Figure 3. One can see clearly that the evolutions of nodes 7 and 8 do not synchronize with each other though they are in the same cluster. Therefore, neither cluster synchronization nor cluster-delay synchronization is achieved.Figure 3
Time evolutions of the Chua oscillatorsxi1(t), xi2(t), xi3(t)  (i=1,2,…,8) in the network (33) with the feedback control gains σ1=σ2=σ3=2.


### 4.2. Numerical Example 2

In the following numerical simulations, we consider an undirected network consisting of 100x·2-coupled Lorenz systems separated into two clusters. The network topology is shown in Figure 4. If there is an undirected link between node j and node i (j≠i), then aij=aji=1; otherwise aij=aji=0.Figure 4
Topology structure of the undirected network consisting of 100 nodes, which are separated into two equal clusters. If there is an undirected link between nodej and node i (j≠i), then aij=aji=1, otherwise aij=aji=0.The uncoupled Lorenz systemx˙i=f(xi,t) is described by(35)x˙i1=σxi2-xi1,x˙i2=rxi1-xi2-xi1xi3,x˙i3=-bxi3+xi1xi2,where σ=10, r=28, b=8/3. According to [30], there exists a positive constant δ such that Assumption 1 holds. Then, one can choose appropriate values for other parameters in the network (2)–(7) with the controller (9) such that the conditions of Theorem 4 are satisfied. The time evolutions of the 100 Lorenz systems are shown in Figure 5. As indicated by Figure 5, the first fifty nodes fall into the same cluster, and the rest of the nodes fall into another cluster. All the nodes in the same cluster behave in the same synchronous fashion, but nodes in different clusters follow distinct time evolutions with time delays. Therefore, cluster-delay synchronization has been realized in the network consisting of 100x·2-coupled Lorenz systems.Figure 5
Time evolutions of thex⋅2-coupled Lorenz systems xi1(t), xi2(t), xi3(t)  (i=1,2,…,100) in the network (2)–(7) with the controller (9).


## 4.1. Numerical Example 1

At first, we consider a directed complex network consisting of8 nodes separated into three different clusters, the topology of which is shown in Figure 1.Figure 1
Topology structure of a directed network consisting of 8 nodes, which are divided into3 clusters. If there is a directed link from node j to node i (j≠i), then aij=1, otherwise aij=0.Define connectivity matrixA as an asymmetric matrix with zero row-sums, and the off-diagonal elements aij=1 if the jth node can receive information from the ith node for j≠i; otherwise, define aij=0. Choose the node dynamics of the network as the well-known Chua oscillators; then the oscillator network (2)–(7) with the control protocol (9) can be rewritten as follows: (33)x˙i=-Dxi+Tgxi+∑j=18aijxj+uit,uit=θφitsφit-s1t-τφi-∑j=18a~ijxj-σφixit-s1t-τφi,s˙φit=fsφit-θφitsφit-s1t-τφi,where xi∈R3, D=I3, g(xi)=(g(xi1),g(xi2),g(xi3))⊤, i=1,2,…,8, g(s)=(|s+1|-|s-1|)/2, and (34)T=1.25-3.2-3.2-3.21.1-4.4-3.24.41.0.It is easy to derive that the nonlinear functionf in system (33) satisfies Assumption 1 by choosing the matrix Δ=5.5685I3. Then, one can choose δ=5.5685, σ1=10, σ2=12, σ3=14 such that condition (i) of Theorem 4 holds, and choose α=4.4, Θ1=Θ2=Θ3=10 such that condition (ii) holds. By taking T=1, it can be verified that the feedback control width ζ=0.557 satisfies condition (iii) of Theorem 4.Choose the initial conditions randomly; Figure2 is plotted to show the evolutions of the state variable of the 8 nodes and the cluster errors. From (a), (b), and (c), one can see that the evolutions of the nodes in the same cluster synchronize with each other. And (d) shows more clearly that the cluster errors ei(t)=xi(t)-sφ(i)(t) tend to zero. Therefore, Figure 2 illustrates that cluster-delay synchronization is achieved under the conditions of Theorem 4.Figure 2
Cluster-delay synchronization of the network (33). (a), (b), and (c) show the time evolutions of the state variable of the 8 nodes splitting into three different clusters, and (d) describes the time evolutions of the cluster errors.
(a)
(b)
(c)
(d)Next, we keep all the parameters unchanged except the feedback control gains decreasing fromσ1=10, σ2=12, σ3=14 to σ1=σ2=σ3=2. Then the conditions of Theorems 3 and 4 cannot be satisfied. The time evolutions of the Chua oscillators xi1(t), xi2(t), xi3(t)  (i=1,2,…,8) in the network (33) are plotted in Figure 3. One can see clearly that the evolutions of nodes 7 and 8 do not synchronize with each other though they are in the same cluster. Therefore, neither cluster synchronization nor cluster-delay synchronization is achieved.Figure 3
Time evolutions of the Chua oscillatorsxi1(t), xi2(t), xi3(t)  (i=1,2,…,8) in the network (33) with the feedback control gains σ1=σ2=σ3=2.


## 4.2. Numerical Example 2

In the following numerical simulations, we consider an undirected network consisting of 100x·2-coupled Lorenz systems separated into two clusters. The network topology is shown in Figure 4. If there is an undirected link between node j and node i (j≠i), then aij=aji=1; otherwise aij=aji=0.Figure 4
Topology structure of the undirected network consisting of 100 nodes, which are separated into two equal clusters. If there is an undirected link between nodej and node i (j≠i), then aij=aji=1, otherwise aij=aji=0.The uncoupled Lorenz systemx˙i=f(xi,t) is described by(35)x˙i1=σxi2-xi1,x˙i2=rxi1-xi2-xi1xi3,x˙i3=-bxi3+xi1xi2,where σ=10, r=28, b=8/3. According to [30], there exists a positive constant δ such that Assumption 1 holds. Then, one can choose appropriate values for other parameters in the network (2)–(7) with the controller (9) such that the conditions of Theorem 4 are satisfied. The time evolutions of the 100 Lorenz systems are shown in Figure 5. As indicated by Figure 5, the first fifty nodes fall into the same cluster, and the rest of the nodes fall into another cluster. All the nodes in the same cluster behave in the same synchronous fashion, but nodes in different clusters follow distinct time evolutions with time delays. Therefore, cluster-delay synchronization has been realized in the network consisting of 100x·2-coupled Lorenz systems.Figure 5
Time evolutions of thex⋅2-coupled Lorenz systems xi1(t), xi2(t), xi3(t)  (i=1,2,…,100) in the network (2)–(7) with the controller (9).


## 5. Conclusions

In this paper, the cluster-delay synchronization problem of a directed network with cluster structure has been discussed. First, a control protocol with intermittent effect has been presented to realize cluster synchronization via periodically intermittent control. The proposed control methods can be applied to discuss many real-world networks with intermittent effect. Second, we extended the criterion on cluster synchronization to the cluster-delay synchronization problem, which implies that the oscillators in the following clusters track the trajectory of those in the leader clusters with different time delays. Finally, we presented two delayed dynamical networks as illustrative examples and carry out some simulated results to show the feasibility of the proposed control methods.



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*Source: 1069839-2018-02-18.xml*