Exponential Cluster Synchronization of Neural Networks with Proportional Delays

Exponential cluster synchronization of neural networks with proportional delays is studied in this paper. Unlike previous constant delay or bounded time delay, we consider the time-varying proportional delay is unbounded, less conservative, and more widely applied. Furthermore, we designed a novel adaptive controller based on Lyapunov function and inequality technique to achieve exponential cluster synchronization for neural networks and by using a unique way of equivalent system we proved the main conclusions. Finally, an example is given to illustrate the effectiveness of our proposed method.


Introduction
In the past most researches focus on the complete synchronization of neural networks, which means that all the nodes of the whole network reach the same synchronization state, and this has achieved significant progress [1][2][3][4][5][6]. However, in many real world networks, it is to complete several functions at the same time, which implies that the network reaches several different states at last, which is a different synchronization type: cluster synchronization. As a particular synchronization phenomenon, cluster synchronization is achieved when the dynamical nodes realize complete synchronization in each subgroup which is called cluster, but no synchronization appears among the different clusters. Owing to the significant application in biological science and communication engineering, many results have been available for cluster synchronization of complex dynamical networks [7][8][9][10]. Ma et al. [9] proposed a coupling scheme with cooperative and competitive weight-couplings to realize cluster synchronization for the connected chaotic networks. In [10], cluster synchronization of complex networks was investigated with centralized adaptive pinning control.
In fact, there exist natural time delays in the operation of systems which are due to the limited communication speed when information transmission exists, extra time required by the sensor to get the measurement information, computation time required for generating the control inputs, and the execution time required for the inputs being acted, and it may cause undesirable dynamic behaviors such as oscillation and instability. Recently, it has been revealed that some types of delayed neural networks (DNNs) exhibit some complicated dynamics and even chaotic behaviors if the parameters and time delays are appropriately chosen. Therefore, dynamic behaviors, especially the cluster synchronization problems of DNNs, have been extensively considered. So far, most studied models of synchronization problem are neural networks (NNs) with constant delays, time-varying and bounded delays, distributed delays, and so on [11][12][13][14][15][16][17][18]. In [17], the authors discuss the synchronization of complex dynamical networks by only designing the adaptive controllers, while we introduce adaptive strategies not only to the coupling strengths but also to the feedback gains. Dahms et al. [18] investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. In addition, delaydependent stability criteria and synchronization control laws are related to the size of delay so that they can be used to design some better networks according to the allowed time delays of networks. So the cluster synchronization of the neural network with time delay is the right direction for further research.
The proportional delay is one of many delay types and is objectively existent. Unlike previous constant delay or bounded time delay, the time-varying proportional delay is unbounded, less conservative, and more widely applied. For example, in Web quality of service (QoS) routing decision, the proportional delay is usually required [19][20][21][22][23]. The proportional delay system as an important mathematical model often rises in some fields such as physics, biology systems, and control theory and it has attracted many scholars' interest [24,25]. If the QoS routing algorithms based on neural networks with proportional delays are proposed, they will be the most suitable algorithms corresponding to the actual situation. Since a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is reasonable that the proportional delays are introduced in the neural networks according to the topology structure and parameters of neural networks. The proportional delay function ( ) = (1 − ) (0 < < 1) is a monotonically increasing function about the time , which will be convenient to control the network's running time according to the time delays of network. So far a few results of dynamic behaviors of DNNs with proportional delays have been reported in [26][27][28][29][30][31]. Zhou [26,30,31] has discussed the global exponential stability, asymptotic stability of cellular neural networks (CNNs), and the research of synchronization with proportional delays, by employing matrix theory and constructing Lyapunov function. Dissipativity of a class of CNNs with proportional delays has been investigated by using the inner product properties in [27]. So far, to the best of the author's knowledge, there are few studies with the cluster synchronization of the neural network with proportional delays.
Motivated by the above discussions, we creatively take the element of the proportional delays into the model of neural networks to achieve the exponential cluster synchronization. As an intersection of multibranches of learning, this model will be closer to the reality and much more pertinence for concrete questions. Whether in the study of delays or in the research of cluster synchronizations, the paper has important referring value. The main novelty of our contribution is as follows.
(1) The cluster synchronization model with proportional delay is considered.
(2) The coupling strength of each edge in the system only depends on its local information.
(3) We use a unique way of equivalent system to prove the main results.
This paper is organized as follows. In Section 2, the models and preliminaries are presented. In Section 3, by the application of Lyapunov function, some inequality analysis technique, and the way of equivalent system, the results can be obtained, which is about the cluster synchronization of the delayed neural network. In Section 4, a numerical example and its simulation are given to illustrate the effectiveness of the proposed method. Conclusions are presented in Section 5.

Model Description and Preliminaries
Consider a delayed neural network (DNN) with proportional delay of coupled identical nodes, with each node being an -dimensional dynamical system, which can be described by the following equations [26]: where ≥ 2 denotes the number of nodes (neurons) in the network; = ( ) × and = ( ) × are the corresponding weight and delayed weight matrices; ( ( )) = ( ( 1 , 2 , 3 , . . . , )) ∈ R denotes a nonlinear activation function of nodes; = ( ) × is the coupling matrix between nodes. If there is a connection between node and node at time , In this paper, the network is connected in the sense that there are no isolated clusters in the network, which means that the matrix is symmetric and irreducible [10]. According to the characteristics of the matrix , system network (1) is easily turned to the following further simplified equations: Assume the whole nodes are divided into clusters. Without loss of generality, the sets of subscripts of these clusters are 1 = {1, 2, 3, . . . , 1 }, 2 = { 1 + 1, 1 + 2, . . . , 1 + 2 }, . . . , The network is considered to realize Mathematical Problems in Engineering 3 cluster synchronization, when the states of nodes achieve the following states [32]: where ( ) = ( 1 ( ), 2 ( ), . . . , ( )) is the desired state of node at time and satisfieṡ In order to make the network (1) realize cluster synchronization, an input control is added in (1) which is designed as follows: According to the characteristics of the matrix , the input (5) is easily turned to the following further simplified equation: where the coupling strength ( ) = ( 1 , 2 , . . . , ) between nodes adopts the following local adaptive strategy: where > 0 is the adaptive gain.
Before starting to discuss the cluster synchronization, we make the following assumption.

Remark 3.
The translations are to convert the desired cluster synchronization state to another equivalent control system, which is convenient to study the cluster synchronization. Namely, the state when the original system realizes the cluster synchronization means that the control system and the original system realized synchronization. (18) and (19) is said to be exponentially synchronized if there exist constants ≥ 1 and > 0 such that

Definition 4. System
where the constant is defined as the exponential synchronization rate.
Thus the goal of this paper is further equivalently turned into the proof of the dynamic equations of synchronization error to achieve an exponential stabilization.
Proof. According to the definition ofḣ ( ) and Assumption 1, we get Consider the following function: From (24) to (26) Now construct the following positive Lyapunov functional as follows: where ≥ 0, > 1; is the edges' set of the network; ℎ is a constant.
Then the differential of ( ) iṡ According to (27), we can geṫ By the conditions of the theorems | | + | | + − 1 < | |, and, for any edge ( , ) ∈ , there exists the value ℎ which is bigger than or equal to the corresponding edge strength ( ), and we geṫ which means ( ) ≤ (0), for ≥ 0. Then, by (25) and (28), one has It follows from (28) that The proof is completed. (18) and (19) is asymptotically synchronized if the following conditions are satisfied: (1) the exponential synchronization rate is zero; that is, = 0 and > 1;

Remark 6. System
Thus, control system (4) and original system (1) are considered to be exponentially synchronized, and the exponential synchronization rate is less than because of 0 < ln < ; that is, ln / < 1, as ≥ 1.
The proof is omitted.

Illustrative Examples
In the following simulation, the whole neural network is divided into three clusters, which means that = 3. The desired state ( ) of node at time is chosen as the average states of all the nodes in the same cluster at time as follows:

Mathematical Problems in Engineering
where the three clusters are 1 , 2 , and 3 , respectively. Consider the following delayed neural networks (DCNNs): (41) The system meets Assumption 1 and Theorem 5 with = 1. The chaotic behavior of system (40) with the initial condition In the simulation experiment, a BA scale-free neural network is needed, which is constructed with = 0 = 5 and the neural network size = 50. The construction method of the BA scale-free neural network can refer to [34]. Suppose the whole network will reach three clusters with the cluster size being 14, 8, and 28, respectively. The initial states 1 ( ) and 2 ( ), = 1, 2, . . . , 50, are randomly distributed between [−1, 1] and [ −5, 5].
As shown in Figure 2 one red curve, and the remaining 28 blue curves changed into one blue curve; that is, the 50 state lines of 1 ( ), ( = 1, 2, . . . , 50), in system gradually merged into 3 lines, which means that each node of the neural network realizes the desired cluster synchronization. It reflects that the number of the nodes in the cluster influences the rate of the cluster synchronization. Then in Figure 2(b), for > 1.8 s, it can still be observed that 14 black curves changed into one black curve, 8 red curves changed into one red curve, and the remaining 28 blue curves changed into one blue curve; that is, the 50 state lines of 1 ( ), ( = 1, 2, . . . , 50), in system gradually merged into 3 lines, which means that each node of the neural network realizes the desired cluster synchronization. It reflects that the number of the nodes in the cluster influences the rate of the cluster synchronization. So the conclusion is that the larger the cluster in the delay systems, the slower the rate of the cluster synchronization with the same way of the coupling mode. Figure 3 depicts the synchronization error of the state variables between system (42) and the desired state, which makes it easy to see that the exponential convergence rate of the red curves within all the curves of the 3 cluster's nodes is relatively quick (14 were black, 8 red, and 24 blue classification). And it proves that from another aspect the number of nodes affects the rate of cluster synchronization. Furthermore the shape of the curves indicates that system (42) realizes the cluster synchronization with the exponential synchronization rules, and it is feasible to further study the cluster synchronization of the neural system with proportional delay through the exponential analysis of the synchronization.
In Figure 4, it can be seen that when > 2s, the edge coupling strength ( ) between node and node is kept constant in all its dimensions. Combined with Figure 2, it is found that when the neural network realizes cluster synchronization, the edge coupling strengths are fixed, which implies that the left side of (7) will be zero. According to Figure 4, it is shown that most edge coupling strengths are  Figure 3: The synchronization error between system (42) and the system which the desired state is equivalently turned into. less than 0.20, and only a small amount of edge coupling strengths is distributed in the interval [0.2, 0.255]. If we do not use the adaptive coupling strength in the neural network with the proportional delay, the network will realize the cluster synchronization only when all the edge coupling strengths are greater than 0.20, which means that a lot of unnecessary coupling strengths are wasted. All in all, it is easy to design coupling intensity for each edge by using the adaptive coupling strength, in order to achieve the desired cluster synchronization with faster cluster synchronization rate, which can be implemented in real networks.

Conclusions
In this paper, an exponential cluster synchronization control frame of the neural networks with time delay has been established. Different from the prior works, delays in the paper are proportional delays which are unbounded and time-varying, and the coupling strength of each edge in the system only depends on its local information. By constructing the appropriate Lyapunov function and using the inequality technique, several sufficient conditions were obtained to make the neural network with the proportional delay realize cluster synchronization. Finally, a numerical simulation has been given to verify the theoretical results. Further the example indicates that the cluster synchronization rate is related to the clusters size. This strategy may be extended to study the effects of the rate of the neural system with proportional delay in order to realize cluster synchronization. Furthermore we can also extend the current result to the Networked Control Systems (NCS) under scheduling protocol [35][36][37]. Based on this paper, we can provide a stability certificate that takes into account the network imperfections: communication delays (proportional delays), scheduling protocols, and quantization in order to offer several practical advantages: reduced costs, ease of installation and maintenance, and increased flexibility.