#### Abstract

Global synchronization analysis for complex networks with coupling delay is investigated. Firstly the constant time delay is analyzed and then the case for time-varying delay is considered. Sufficient conditions for network synchronization are given based on Lyapunov functional, linear matrix inequality, and Kronecker product technique. The unknown variables in the sufficient conditions are fewer than those in the recent reference. Moreover, for the time-varying delay case, we find that the conditions are dependent on the bounds of both time delay and its derivative, and the derivative of the time-varying delay can be any value in the bounds. Finally, numerical examples are given to validate the effectiveness of the obtained results.

#### 1. Introduction

Dynamical recurrent neural networks are used extensively in classification of patterns, associative memories, optimization [1–3], and so on. The networks are composed of a large number of highly interconnected dynamical units and exhibit very complicated dynamics. Therefore, theory analysis of complex networks has become a focal research field and attracted a great deal of attention.

Recently, it is found that synchronization is one of the most important dynamical properties of complex networks and has been extensively investigated in different ways [4–19]. In [5], Lü and Chen considered a dynamical network and gave the sufficient conditions for local synchronization. Because of the network traffic congestions as well as the finite speed of signal transmission over the links, time delays occur commonly in complex networks. Therefore, Zhou and Chen [7] and Gao et al. [8] improved the models with no delays, the case with constant coupling delays is considered, and they also analyzed the synchronization problem [9, 10]. However, in many real-world networks time delay is varying; therefore time varying coupling delays are considered in [11–13]. By using free-weighting matrices, some synchronization criteria for general complex dynamical networks with time-varying delays are proposed. However, the computation is huge, and there are a large amount of variables in the condition. Moreover, in order to derive the synchronization condition, in some papers [7–13], the time-varying delay is usually confined to (lower bound of the delay is zero), and the derivative is restricted to less than 1. Therefore, how to improve the system performance by removing the redundant variables and reducing computation still remains unsolved.

Motivated by the mentioned work, we study the synchronization problems for general complex networks with time constant coupling delays and interval time-varying delays. Using different Lyapunov functions, the synchronization conditions derived turn out to be less conservative, and the addressed systems contain some models as their special cases; the more effective mathematical techniques are employed to reduce the conservatism.

The remainder of this paper is organized as follows. In Section 2, the investigated systems are formulated and some lemmas and notations are given. In Section 3, the conditions for synchronization are derived. In Section 4, two numerical examples are presented to demonstrate the effectiveness and the advantage of the proposed method. Finally, conclusions are drawn in Section 5.

#### 2. Problem Formulations

In this paper, denotes the set of real matrixes, and denotes that matrix is positive semidefinite, while denotes that matrix is positive definite. We use to denote a block-diagonal matrix; stands for . The notation denotes the Kronecker product of matrices and ; represents the -dimensional identity matrix. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

We use to denote the state of coupling nodes, , and then the dynamic neural networks (DNNs) of general form can be described by in which is the state vector of the th network at time . The functions are sufficiently smooth nonlinear vector fields, and is the external input vector. , , and are coefficient matrixes, and are inner-coupling matrixes, and represents the outer-coupling connections. The constant represents the coupling strength; and represent the time constant delay and time-varying delay, respectively.

For the networks (1), we have the following assumptions.

*Assumption (H1)*. is the interval time-varying delay satisfying

*Assumption (H2)*. The outer-coupling configuration matrices of the network satisfy

*Assumption (H3)*. There exist constants , , and the functions satisfy
We denote

Based on Assumption (H2), system (1) can be rewritten as the following form:

The initial conditions of (6) are given by . , , where . The initial conditions of (7) are given by . , , where , and in order to simplify we set .

*Remark 1. *The constants , in Assumption (H3) are allowed to be any value. Then, most of the previous results in similar networks are just special cases of this assumption, which means that the activation functions are more general than those of other works.

Considering the sign of Kronecker product, models (6) and (7) can be rewritten as
with , , and .

*Definition 2 (see [14]). *Dynamical networks (6) are said to be global asymptotic synchronization, if for any , any . There exists such that , where , , and denotes the Euclidean norm.

*Definition 3 (see [15]). *Dynamical networks (7) are said to be global exponential synchronization, if for any initial conditions , there exist and such that , in which and denotes the Euclidean norm.

Lemma 4. *Let denote the notation of Kronecker product; then one has the following conclusions:*(1)(2)(3)

Lemma 5 (see [16]). *Let , , , , , with , and then .*

Lemma 6 (see [17]). *For any constant matrix , , a scalar functional , and a vector function , the following inequality holds .*

#### 3. Main Results

##### 3.1. Synchronization Condition for Constant Time Delay

In this section, we state and investigate the global asymptotic synchronization for system (6).

Theorem 7. *Suppose Assumptions (H1)–(H3) hold, if there exist three symmetric positive definite matrices , , and three positive diagonal matrices , , , such that the LMIs hold for each , in . Consider**where , .*

Then system (8) is global asymptotic synchronization.

*Proof. *Choose a Lyapunov-Krasovskii functional as

Now by directly computing along the trajectory of system (8), we have

In view of Lemma 6, we have

Noting the facts that and , using Lemmas 4 and 5, we have

On the other hand, it is easy to see from the formulation of (8) that the following equation also holds for any matrices :

For convenience, let , , , and .

For any diagonal matrices , , and , from Assumption (H3), we have

Adding up (14)–(16) from both sides, we have
where is from (10), and

Following Theorem 7, we have , and then we have for all . Therefore, system (8) is global asymptotic synchronization. Then end the proof.

Corollary 8. *Suppose Assumptions (H1)–(H3) hold and in the system (8), if there exist three symmetric positive definite matrices , , and three positive diagonal matrices , , , such that the LMIs holds for each , in . Consider**where , .*

Then system (8) is global asymptotic synchronization.

The proof is obvious from Theorem 7 and we omit the details.

##### 3.2. Time-Varying Delay Synchronization Condition

In this section, by utilizing the improved techniques used in [18], we obtain the following global exponential synchronization criterion for system (7).

Theorem 9. *Suppose Assumptions (H1)–(H3) hold, if there exist five symmetric positive definite matrices , , , , for each in and four positive diagonal matrices , , , , such that the LMIs holds for each , in . Consider
**
where*

Then system (9) is global exponential synchronization.

*Proof. *We construct the Lyapunov-Krasovskii functional as follows:

Calculating the time derivative of along the trajectories of system (9), from Lemmas 4 and 5, we have

Using Lemma 4, we have

On the other hand, based on the approach in [18], and using , then we have

And in order to combine some of the terms in (26), letting , we have

For any , diagonal matrices , , , and , and from Assumption (H3), the following inequality holds:

Adding up (24)–(28) from both sides, we have
where are given in (20), (21), and (22), respectively, and and . Consider

Following Theorem 9, we have , and there exists a positive constant that satisfied , such that

Furthermore, based on the proof in [12], there exist two positive scalars and , such that
for . By Definition 3, therefore, system (9) is global exponential synchronization. Then end the proof.

*Remark 10. *In [10–13], the authors studied the synchronization of an array of linearly coupled networks with constant coupling delay or time-varying coupling delay, and the derivative of the time-varying delay is confined to be less than 1. We remove this restrictiveness and the derivative of the time-varying delay can be any value.

*Remark 11. *During the estimation, is separated into two parts as follows: , and we estimate each part, respectively, ignoring the direct estimate for , and then we have more accurate estimate than those from the reference [12, 13].

When in Assumption (H1), we have the following corollary from Theorem 9.

Corollary 12. *Suppose Assumptions (H1)–(H3) hold, if there exist four symmetric positive definite matrices , , , for each in and four positive diagonal matrices , , , such that the LMIs holds for each , in . Consider
*

Then system (9) is global exponential synchronization. The proof is obvious from Theorem 9 and we omit the details. When in Assumption (H1) is not differentiable or is unknown, we have the following corollary from Theorem 9.

Corollary 13. *Suppose Assumptions (H1)–(H3) hold, if there exist four symmetric positive definite matrices , , , for each in and four positive diagonal matrices , , , , such that the LMIs holds for each , in . Consider
**
where**With
**
then system (9) is global exponential synchronization.*

*Remark 14. *Theorem 9 can be applied to both slow and fast time-varying delays only if is known. But when is not differentiable or is unknown, Theorem 9 fails to work; however, Corollary 13 can check the synchronization of system (9) instead.

#### 4. Numerical Examples

In this section, two examples are provided to illustrate the effectiveness of the conclusion.

*Example 1. *Consider a lower-dimensional network model with 5 nodes, where each node is a simple three-dimensional stable linear system. To simplify, we assume that , , , and the inner-coupling matrix is , and the outer-coupling matrix is defined as

We compare the admissible delay upper bounds for and the number of variables with those from other references; it is clear from Table 1 that the delay is a little larger and the number of variables is less than those from [10, 12].

*Example 2. *We consider the following DNNs:

Let us see the synchronized states of a chaotic system: Chua’s circuit [19].

The dynamics of Chua’s circuit is
where

Here we choose Chua’s circuit (39) as the uncoupled system. Figure 1 illustrates the chaotic trajectories of system (39). The coupling time-varying delay and the coupling strength , then we consider dynamic networks consisting of three linearly coupled identical DNNs with couplings as
for . Let
, , and .

By calculating we have , , and , and the activation functions satisfy Assumption (H3). If the system reaches synchronization, we have the following synchronized state equation:
By calculating other variables in Theorem 9 using Matlab toolbox, we realize the system synchronization, and the total error is given as . Figures 2 and 3 show the synchronized state of system (39) and the synchronous error, and Figures 4, 5, and 6 show the curves of state variables , , and with the initial value randomly chosen from .