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This paper investigates global synchronization in an array of coupled neural networks with time-varying delays and unbounded distributed delays. In the coupled neural networks, limited transmission efficiency between coupled nodes, which makes the model more practical, is considered. Based on a novel integral inequality and the Lyapunov functional method, sufficient synchronization criteria are derived. The derived synchronization criteria are formulated by linear matrix inequalities (LMIs) and can be easily verified by using Matlab LMI Toolbox. It is displayed that, when some of the transmission efficiencies are limited, the dynamics of the synchronized state are different from those of the isolated node. Furthermore, the transmission efficiency and inner coupling matrices between nodes play important roles in the final synchronized state. The derivative of the time-varying delay can be any given value, and the time-varying delay can be unbounded. The outer-coupling matrices can be symmetric or asymmetric. Numerical simulations are finally given to demonstrate the effectiveness of the theoretical results.

In the past few decades, the problem of chaos synchronization and network synchronization has been extensively studied since its potential engineering applications such as communication, biological systems, and information processing (see [

An array of coupled neural networks, as a special class of complex networks [

Time delays usually exist in neural networks. Some papers concerning synchronization of neural networks have considered various time delays. In [

Motivated by the above analysis, this paper studies the synchronization in an array neural network with both time-varying delays and unbounded distributed delays, under the condition that the transmission efficiencies among nodes are limited. By using a new lemma on infinite integral inequality and the Lyanupov functional method, some synchronization criteria formulated by LMIs are obtained for the considered model. In the obtained synchronization criteria, the time-varying delay studied can be unbounded, and its derivative can be any given value. Especially, when some of the transmission efficiencies are limited (i.e., less than 1), the transmission efficiency and inner coupling matrices between nodes have serious impact on the synchronized state. Results of this paper extend some existing ones. Numerical simulations are finally given to demonstrate the effectiveness of the theoretical results.

The rest of this paper is organized as follows. In Section

An array of coupled neural networks consisting of

The initial condition of (

Model (

We introduce transmission efficiencies between nodes in model (

Based on (

This paper utilizes the following assumptions.

The delay kernel

There exist constant matrices

There are constants

The assumption

When the transmission efficiencies of all the channels are considered and some of them are limited, the final synchronized state is different from that of a single node without coupling. According to

In order to derive our main results, some basic definitions and useful lemmas are needed.

The coupled neural network with limited transmission efficiency (

Let

Let

Let

The following lemma can be easily obtained from [

Let

Suppose

In this section, synchronization criteria formulated by LMIs of the general model (

For

To obtain synchronization criterion in the array of coupled neural networks (

Under assumptions (H_{1})–(H_{3}), if there exist matrices

Consider the following Lyapunov function:

By virtue of Lemma

In view of assumption

From the given condition (

Corresponding to (

For the system (

Under assumptions

Consider the following Lyapunov function:

The rest part of the proof is similar to that of the proof of Theorem

In this paper, the least restriction is imposed on the time-varying delay. The derivative of the time-varying delay can be any given value, and the time-varying delay can be unbounded. However, most of former results are based on either that the derivative of the time-varying delay should be less than 1 [

Synchronization criteria in an array of coupled neural networks with limited transmission efficiency are obtained in Theorem

In this section, one example is provided to illustrate the effectiveness of the results obtained above.

Consider a 2-dimensional neural network with both discrete and unbounded distributed delays as follows:

In the case that the initial condition is chosen as

Chaotic-like trajectory of the system (

Now we consider a coupled neural network consisting of five identical models (

It is easy to check that the activation function

In the simulations, the Runge-Kutta numerical scheme is used to simulate by MATLAB. The initial values of (

Time response of

Error distance of the coupled network (

Figure

Trajectory of the synchronized state of system (

Trajectories of system (

In this paper, a general model of coupled neural networks with time-varying delays and unbounded distributed delays is proposed. Limited transmission efficiency between coupled nodes is considered in the dynamical network model. Based on the integral inequality and the Lyapunov functional method, sufficient conditions in terms of LMIs are derived to guarantee the synchronization of the proposed dynamical network with limited transmission efficiency. The restriction on time-varying delay is the least. The derivative of the time-varying delay can be any given value, and the time-varying delay can be unbounded. Numerical examples are given to verify the effectiveness of the theoretical results. Furthermore, numerical simulations show that, when some of the transmission efficiencies are less than 1, the transmission efficiency and inner coupling matrices between nodes play important roles for the final synchronized state. Since many real-world transmission efficiencies between nodes are usually less than 1, the results of this paper are new and extend some of the existing results.

This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grants nos. 61263020, 61272530, and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, the Scientific Research Fund of Yunnan Province under Grant no. 2010ZC150, and the Scientific Research Fund of Chongqing Normal University under Grants no. 12XLB031 and no. 940115.

_{∞}synchronization and state estimation for discrete time-varying stochastic complex networks over a finite horizon