Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 523424, 10 pages

http://dx.doi.org/10.1155/2015/523424

## Exponential Cluster Synchronization of Neural Networks with Proportional Delays

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 15 January 2015; Revised 9 March 2015; Accepted 24 March 2015

Academic Editor: Kun Liu

Copyright © 2015 Nian Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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 [1–6]. 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 [7–10]. 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 [11–18]. 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, delay-dependent 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 [19–23]. 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 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 [26–31]. 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.

#### 2. 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 denotes the number of nodes (neurons) in the network; is the state variable of node ; is a diagonal matrix with positive entries. and are the corresponding weight and delayed weight matrices; denotes a nonlinear activation function of nodes; is the coupling matrix between nodes. If there is a connection between node and node at time , then ; otherwise . The diagonal item of the matrix is . 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]. are proportional delay coefficient and satisfy , , and in which corresponds to the time delay, and , as , . Furthermore, the neural networks described in (1) possess initial conditions of are constants, and . is inner matrix, and .

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 , + , where . The network is considered to realize cluster synchronization, when the states of nodes achieve the following states [32]:where is the desired state of node at time and satisfies

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 between nodes adopts the following local adaptive strategy:where is the adaptive gain.

Before starting to discuss the cluster synchronization, we make the following assumption.

*Assumption 1. *The activation function satisfies that , where and is positive constant.

Let us consider transformation defined by [27]

*Case 1. *When , then and ; that is,Let , ; then (8) is written asAccording to the above various formulas one getsThat is to say,According to (8), we haveUsing (8) and (13) in (12) one gets

*Case 2. *When , we havewhere . Thus, the initial function associated with system (12) is given byThe analysis is completed.

*Remark 2. *Conversely, let , ; by (8), (16) and (14) can be turned to (1), which means that the transformation is reversible and equivalent.

Above all, letSystem (1) with the input control (5) can be equivalently turned to the following neural network with a fixed delay, which can be described aswhere , , , and , , is constant continuous function.

*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.

*Definition 4. *System (18) and (19) is said to be exponentially synchronized if there exist constants and such thatwhere the constant is defined as the exponential synchronization rate.

The goal of this paper is to prove that system (1) with adaptive control (5) can achieve the desired cluster synchronization. Now it is equivalently turned to prove that system (18) and (19) can meet the exponential synchronization. In order to prove that it satisfies the exponential synchronization, the following dynamic equations of synchronization error can be obtained:where , and ; ; , .

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.

#### 3. Main Results

Theorem 5. *Assume that Assumption 1 holds. If is true, system (18) and (19) can achieve exponential stabilization with the synchronization rate , where ; ; .*

*Proof. *According to the definition of and Assumption 1, we getConsider the following function:From (24) to (26) one getsNow construct the following positive Lyapunov functional as follows:where , ; is the edges’ set of the network; is a constant.

Then the differential of isAccording to (27), we can getAccording to Assumption 1 and (25), we can getBy the conditions of the theorems , and, for any edge , there exists the value which is bigger than or equal to the corresponding edge strength , and we getwhich means , for . Then, by (25) and (28), one hasIt follows from (28) thatwhere .

All told, one getswhere , .

The proof is completed.

*Remark 6. *System (18) and (19) is asymptotically synchronized if the following conditions are satisfied: the exponential synchronization rate is zero; that is, and ; then , ; .

According to the foregoing contents , , one getsLet ; then and ; let ; then and . Thus, it follows from (36) thatLet ; the following inequality is obtained:Thus, control system (4) and original system (1) are considered to be exponentially synchronized, and the exponential synchronization rate is less than because of ; that is, , as .

Thus, the following theorem is derived.

*Inference 1*. Assume that Assumption 1 holds. If is true, system (1) with the input control (5) can achieve an exponential cluster synchronization for the desired state, with the exponential rate being , where ; ; .

The proof is omitted.

#### 4. Illustrative Examples

In the following simulation, the whole neural network is divided into three clusters, which means that . 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:where the three clusters are , , and , respectively.

Consider the following delayed neural networks (DCNNs):where , , and the parameters are

The system meets Assumption 1 and Theorem 5 with . The chaotic behavior of system (40) with the initial condition , , has already been reported in the case of [33] (see Figure 1).