Cluster Synchronization in Delayed Networks With Adaptive Coupling Strength via Pinning Control

Many previous studies have indicated that pinning control and adaptive feedback control are two of most effective strategies for achieving synchronization. In this paper, we present a mixed method integrating the two control strategies to realize cluster synchronization in delayed networks with directed couplings. Based on Lyapunov stability theory and matrix theory, several criteria on cluster synchronization are obtained. Compared with the previous results on cluster synchronization, the obtained criteria can be applied to delayed network with non-identical nodes, directed couplings, and time-varying coupling strength. Finally, numerical simulations are performed to verify the theoretical results.


INTRODUCTION
In the past decade, more and more attention has been attracted to the problem of complex networks, which could be regarded as lots of dynamical nodes coupled with each other. As the result of the complex coupling effect, many different types of collective dynamical behaviors can be realized, such as complete synchronization [1,2], cluster synchronization [3][4][5], exponential synchronization [6,7], intermittent synchronization [8,9], etc. Cluster synchronization, which is discussed in this paper, implies that the oscillators split into several clusters, and the oscillators in the same cluster synchronize with one another. Many researches have shown that there is a close interplay between cluster synchronization and network topologies. There are also some networks which can not realize a particular type of synchronization only via the inherent coupling effect of oneself. In order to realize cluster synchronization in networks without cluster structures, many various control schemes have been proposed during the past decades, for instance, adaptive feedback control [10,11], pinning control [12,13], intermittent control [14], and so on. The control approaches concerned in this paper are the famous pinning control and adaptive feedback control, both have been intensively investigated recently. First, we make a brief introduction to the background and related work of pinning control. Generally speaking, the node number of each complex network in the real world may be the massive level, and it is hard to control all nodes to realize a particular type of synchronization. In order to reduce the control gain and the number of controllers, pinning control is proposed as a more effective control approach [13]. The advantage of the pinning control approach is that it is only necessary to control a significantly smaller number of local controllers as compared to the randomly pinning scheme. Since the idea of pinning control was first developed, researchers concentrate their efforts on the popularization and application of the control approach [15][16][17][18][19][20]. For instance, by using the Schur complement and Lyapunov stability theory, several pinning schemes were designed to realize lag synchronization between two coupled networks in [15]. It is also shown that a directed network can realize pinning synchronization in [17]. Pinning control scheme was also applied to drive a directed dynamical network with non-identical nodes to cluster synchronization [18]. Similarly, the intermittent pinning-control problem of directed heterogeneous dynamical networks has also been discussed in [19]. Second, it is worth to point out that adaptive control scheme is a feasible approach to avoid larger feedback control gains. Notice that the coupling strength of many real network can not be arbitrarily large, adaptive feedback control should be a valid method for achieving synchronization. During the recent years, researchers have made some great achievements on adaptive feedback control. In [21], cluster synchronization of an adaptive dynamical network with non-identical nodes was discussed.
Motivated by the discussions above, this paper aims to provide several novel criteria on cluster synchronization of complex networks with adaptive coupling strength via pinning control. We carry out a new dynamical network model possessing four characteristics, i.e., delayed dynamics, non-identical nodes, directed topology and time-varying coupling strength. To the best of our knowledge, there is no previous research discussing the problem of cluster synchronization of such a network model. However, the problem mentioned above should be an important issue of complex networks because of its wide applications in practical problems. Then we design proper pinning controllers and adaptive coupling strength, and sufficient conditions are derived to realize cluster synchronization by using Lyapunov stability theory and matrix theory. In order to illustrate the feasibility and validity of the obtained theoretical results, several numerical simulations are presented.
The rest of this paper is designed as follows. The network model is described and some necessary lemmas and assumptions are stated in section 2. Via pinning and adaptive controls, cluster synchronization in a delayed network is discussed and then some sufficient conditions are given in section 3. Numerical examples are obtained in section 4. Finally, in section 5, conclusions are presented.

MODEL DESCRIPTION AND PRELIMINARIES
Consider a network consisting of N nodes, which are divided into m communities C 1 , C 2 , . . . , C m , where 2 ≤ m ≤ N. It is easy to see that there holds ∪ m i=1 C i = {1, 2, · · · , N}. If the ith node belongs to the jth cluster, then we define φ i = j. The function f φ i : R n → R n can be employed to describe the local dynamics of each node in the ith cluster. Now, we consider a delayed dynamic network described by the following equations, where x i (t) = (x i1 (t), x i2 (t), · · · , x in (t)) T ∈ R n is the state variable of node i; f φ i is differentiable and capable of performing abundant dynamical behaviors. If φ i = φ j , i.e., the ith node and the jth node belong to different clusters, then The asymmetric matrix A = (a ij ) N×N describes the topology of the directed network, where a ij ≥ 0, i = j, is the weight of the edge from j to i. Here, the coupling matrix A is supposed At first, we give an assumption about the function f φ i , which has been widely employed in the previous researches on the problem of synchronization in complex networks. Most of the previous researches pointed out that many well-known chaotic systems have been checked to satisfy the following assumption, such as cellular neural networks and Chua's circuit [22].

Assumption 1 ([23]). For the vector-valued function
By introducing the control inputs u i (t) ∈ R n (i = 1, 2, . . . , N), the controlled dynamical network with respect to the network (1) can be rewritten aṡ (2) Defining the error variables by e (2) is said to realize cluster synchronization, if the synchronization errors satisfy The following error system can be concluded from the Equations (2) and (3), According to the definition of the diffusive coupling matrix A, denotes the index set consisting of all nodes in the φ i th cluster andḠ φ i represents the index set consisting of all nodes in the φ i th cluster which have direct connections with some nodes in other clusters. In view of this property, the pinning controller is designed as The coupling strengths δ i (t) are constructed by the following adaptive lawṡ where the constant σ i > 0 is the control gain, δ i (0) > 0, the 0 − 1 switching function γ t is defined as follows, T is a positive constant. Here, u i (t) and δ i (t) are different from the corresponding functions of the previous papers, such as [3,12,14,21]. It is easy to see that each node in the network has a different coupling strength. By intuition, the first term in the controller (5) is used to synchronize all nodes in the same cluster, while the second term in the controller (5) is to weaken the influences of the couplings among different clusters. Remark: The controller (5) only works on the nodes which have direct connections with some nodes in other clusters since u i (t) = 0 for i ∈ G φ i −Ḡ φ i . That is to say, only the nodes which have direct connections with some nodes in other clusters need to be pinned. In addition, the coupling strengths δ i (t) are constructed by the adaptive laws. Therefore, the controller (5) is an pinning controller with adaptive strength.

SUFFICIENT CONDITIONS ON REALIZING CLUSTER SYNCHRONIZATION
In this section, the problem of cluster synchronization in delayed networks will be investigated by designing proper pinning controllers and adaptive coupling strength. Based on Lyapunov functional method and matrix theory, the following theorem is established. Theorem 1. Let δ i = δ i (t). Under Assumption 1, if there exists a positive definite matrix Q, such that the matrix H 1 is negative definite for t ≤ T, and the matrix H 2 is negative definite for t > T, where then, by the local control (5) and the corresponding adaptive laws (6), the solutions e 1 (t), e 2 (t), . . . , e N (t) of the system (4) satisfy lim t→+∞ e i (t) = 0, i = 1, 2, . . . , N. That is, cluster synchronization in the network system (2) is realized. Proof. Let Apparently, Then, the derivate of V(t) along the solution of error system (4) under control (5) iṡ Denoting e(t) = (e 1 T (t), e 2 T (t), . . . , e N T (t)) T , we obtain e T (t)(B ⊗ Ŵ)e(t) = e T (t)(B T ⊗ Ŵ)e(t).
Let Q = Ŵ = I n , we obtain the following corollary.
. Under Assumption 1, if Ŵ = I, H 3 is negative definite for t ≤ T, and H 4 is negative definite for t > T, where , then, by the local control (5) and the corresponding adaptive laws (6), the solutions e 1 (t), e 2 (t), . . . , e N (t) of the system (4) satisfy

NUMERICAL SIMULATIONS
In this section, we give a numerical example to verify the theorem given in the previous section.
Consider a directed network consisting of 10 nodes with two communities, the topological structure of which is shown in Figure 1. Choose the node dynamics of the five nodes in the first cluster as the time-delayed Chua's circuit, which is described by the following equations,  v(x i (t − τ )) = 0.3(x i1 (t − τ ), 0, 0) T , i = 1, 2, . . . , 5. Choose the node dynamics of the five nodes in the second cluster as the time-delayed 3-D neural network which is described by the following equations, Matrices V 1 and V 2 are, respectively, defined by Recalling thatḠ φ i denotes the set consisting of all nodes in the φ i th cluster which have direct connections to the nodes in other clusters. It is easy to see thatḠ 1 = {1, 5}, According to the discussion in [18] and some simple calculations, Assumption 1 is satisfied for systems (7) and (8). And then we obtain L = 21. Let It is easy to verify that H 1 < 0 for 0 ≤ t ≤ 2 and H 2 < 0 for t > 2.
. . , 5, j = 6, . . . , 10, where the norm | ·| of vector x is defined as |x| = (x T x) 1/2 . It is clear that the cluster synchronization is achieved if E(t) converges to zero and E ij (t) do not converge to zero as t → +∞.
By the aid of Matlab, we obtain Figure 2, which shows the time evolutions of the component x i1 , i = 1, . . . , 10. In Figure 2, the red lines represent the first five nodes, and the blue lines represent the latter five nodes. Figures 3, 4 show the time evolutions of E(t) and E 58 (t), respectively. It is easy to see that the error E(t) tends to zero, but E 58 doesn't tend to zero when t → +∞. That is to say, 2-cluster synchronization is achieved, and the effectiveness of the theoretic results is verified.

CONCLUSIONS
In this paper, the problem of cluster synchronization in a complex network via pinning control has been investigated, and several novel criteria have been derived by utilizing the matrix theory and the stability analysis technique. The object of this research is a dynamical network possessing four characteristics, delayed dynamics, non-identical nodes, directed topology and time-varying coupling strength. And from the control schemes of this research, we present a novel mixed method, which integrated the pinning control and adaptive feedback control, to design proper pinning controllers and adaptive coupling strengths to realize cluster synchronization in such a dynamical network. For convenience, we also carried out a succinct and utilitarian corollary by simplifying the obtained conditions. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed control methods.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.