Intermittent Control for Cluster-Delay Synchronization in Directed Networks

1School of Information Science and Technology, Linyi University, Linyi 276005, China 2School of Automation, Southeast University, Nanjing 210096, China 3Key Laboratory of Complex Systems and Intelligent Computing in Universities of Shandong (Linyi University), Linyi 276005, China 4School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou 310018, China 5School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China 6Department of Information Technology, King Abdulaziz University, Jeddah 21589, Saudi Arabia


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][12][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 2 Complexity mode control [17], pinning control [18][19][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, clusterdelay 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.Section 2 introduces some necessary preliminaries and builds a modified clustered network model with an intermittent leaderfollowing 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.

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 graph G = {V, E}, which is composed of a set of nodes V = {1, . . ., } and a set of edges E = V × V.The graph exhibits a clustered structure, which implies that the  nonidentical oscillators are divided into  nonempty subsets called clusters.Let V  = { −1 + 1, . . .,   } denote the index set of all the nodes in the th cluster, where  = 1, 2, . . ., ,  0 = 0,   = .For convenience, we define a function  : {1, . . ., } → {1, . . ., }, where () =  implies that the node  ∈ V  .
The dynamics of the virtual leaders in the oscillator network are described by where   () is the state variable of the th virtual leader,   is the time delay, and where Θ  is a positive constant representing the feedback control gain,  ∈ (0, 1) is the control width,  = 1, 2, . . ., ,  = 0, 1, 2, . ... In this paper, we suppose that the time delay  1 = 0, which implies that ṡ 1 () = ( 1 ()).Different from the previous result on cluster lag synchronization [23], the leader systems in this paper are not isolated nodes ṡ  () = (  ()).Instead, linear control laws are designed for the leader systems to realize cluster-delay synchronization.This paper aims to make the leader of the th cluster track the trajectory of the leader of the first cluster with a time delay   ,  = 1, . . ., .

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 constant  such that the vector function  satisfies that for any ,  ∈   .
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 Assumption 1 [28,29].

Cluster Synchronization
Analysis.Now, we first introduce the following definition of both cluster synchronization and cluster-delay synchronization [27].
The oscillator network ( 2)-( 7) is said to realize cluster synchronization, if the synchronization errors satisfy lim The oscillator network ( 2)-( 7) is said to realize cluster-delay synchronization, if the synchronization errors satisfy equality (11) and The preliminaries above, together with Lyapunov function method, bring us to the following theorem for cluster synchronization, which implies that the   −  −1 oscillators in cluster V  to synchronize with each other,  = 1, 2, . . ., .Theorem 3. Suppose that Assumption 1 holds; the oscillator network ( 2)- (7) with the control protocol (9)  Proof.Noticing that   () =   ()−   − −1 ⊗  (), we obtain the error system of the th cluster as follows: Consider the following Lyapunov function: The derivative of  1 () along the trajectories of the error systems ( 13) can be calculated as follows: According to the conditions of Theorem 3 and Lyapunov stability theory, the solutions of the oscillator network satisfy that lim →∞ ‖  ()‖ = 0 for all  = 1, 2, . . ., .Hence, the oscillator network ( 2)-( 7) with the control protocol (9) realizes cluster synchronization.The proof is completed.
Noticing that matrix  represents the intracluster couplings and matrix Ã 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.

Cluster-Delay Synchronization
Analysis.Now, we are in a position to carry out the following theorem on clusterdelay 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 Assumption 1 holds, the oscillator network ( 2)- (7) with the control protocol (9) realizes clusterdelay synchronization if (i) the matrix  − ( +  ⊤ )/2 is negative definite; (ii) there exists a positive constant  such that the matrix Consider the following Lyapunov function: From the oscillator network ( 2)-( 7) with the control protocol (9), it is easy to get the following error system: Calculating the derivative of  2 (  ()), one obtains On the interval  ∈ [, ( + )),  = 0, 1, 2, . .., inequality (20) can be reduced to the following form: Integrating the above inequality over the interval [, ( + )), one has On the interval  ∈ [( + ), ( + 1)), the inequality (20) can be reduced to the following form: which is equivalent to Now, we will prove the following inequality by mathematical induction on parameter : Complexity 5 Firstly, we will show the validity of the base case.In case of parameter  = 0, inequalities ( 22) and ( 24) can be reduced as follows: where It can be concluded from the above two inequalities that inequality (25) holds for  = 0.
Secondly, assuming that inequality ( 25) is correct for  = , we will show its correctness for  =  + 1.In fact, if  =  + 1, inequalities ( 22) and ( 24) can be reduced as follows: where Then, based on the principle of mathematical induction, we declare that inequality (25) holds for  = 0, 1, 2, . ... Combining the monotonic property of the exponential function and the inequality (25), we obtain that 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 Theorem 4 more applicable, we give the following corollary.

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 delays    , which implies that there holds lim →+∞ ‖  () −    ( −    )‖ = 0, where ṡ   ( −    ) =    (   ( −    )),  = 1, 2, . . ., .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 clusterdelay synchronization.The second difference is the design schemes of the controllers with intermittent effect.In [23], Wu and Fu designed  controllers for each oscillator as follows: where the intermittent feedback control gain   () was defined by (8).In this paper, we designed  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.

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

Numerical Example 1.
At first, we consider a directed complex network consisting of 8 nodes separated into three different clusters, the topology of which is shown in Figure 1.Define connectivity matrix  as an asymmetric matrix with zero row-sums, and the off-diagonal elements   = 1 if the th node can receive information from the th node for  ̸ = ; otherwise, define   = 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: (34) It is easy to derive that the nonlinear function  in system (33) satisfies Assumption 1 by choosing the matrix Δ = 5.5685 3 .Then, one can choose  = 5.5685,  1 = 10,  2 = 12,  3 = 14 such that condition (i) of Theorem 4 holds, and choose  = 4.4, Θ 1 = Θ 2 = Θ 3 = 10 such that condition (ii) holds.By taking  = 1, it can be verified that the feedback control width  = 0.557 satisfies condition (iii) of Theorem 4.
Choose the initial conditions randomly; Figure 2 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   () =   () −  () () tend to zero.Therefore, Figure 2 illustrates that cluster-delay synchronization is achieved under the conditions of Theorem 4.
Next, we keep all the parameters unchanged except the feedback control gains decreasing from  1 = 10,  2 = 12,  3 = 14 to  1 =  2 =  3 = 2. Then the conditions of Theorems 3 and 4 cannot be satisfied.The time evolutions of the Chua oscillators  1 (),  2 (),  3 () ( = 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.

Numerical Example 2.
In the following numerical simulations, we consider an undirected network consisting of 100  ⋅2 -coupled Lorenz systems separated into two clusters.The network topology is shown in Figure 4.If there is an undirected link between node  and node  ( ̸ = ), then   =   = 1; otherwise   =   = 0.
The uncoupled Lorenz system ẋ  = (  , ) is described by where  = 10,  = 28,  = 8/3.According to [30], there exists a positive constant  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 100  ⋅2 -coupled Lorenz systems.

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.

Figure 1 :
Figure 1: Topology structure of a directed network consisting of 8 nodes, which are divided into 3 clusters.If there is a directed link from node  to node  ( ̸ = ), then   = 1, otherwise   = 0.

Figure 2 :
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.

Figure 4 :Figure 5 :
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 node  and node  ( ̸ = ), then   =   = 1, otherwise   =   = 0.