Journal of Applied Mathematics

Volume 2014 (2014), Article ID 623249, 12 pages

http://dx.doi.org/10.1155/2014/623249

Research Article

New Global Synchronization Analysis for Complex Networks with Coupling Delay

Jinfang Zhang,¹ Yuanhua Qiao,¹ Jun Miao,² Lijuan Duan,³ and Yanjun Zeng⁴

¹College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

²Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China

³College of Computer Science, Beijing University of Technology, Beijing 100124, China

⁴College of Life Science and Bioengineering, Beijing University of Technology, Beijing 100124, China

Received 27 October 2013; Accepted 21 December 2013; Published 12 March 2014

Academic Editor: Qiankun Song

Copyright © 2014 Jinfang Zhang 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.

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].

Table 1: Admissible delay upper bound for different .

Methods
Theorem in [12]
Theorem in [10]
Our result

1.345
1.632
1.666

0.950
0.998
1.000

0.731
0.732
0.898

The number of variables
More than 11
7
6

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 .

Figure 1: The chaotic trajectories of system (39).

Figure 2: The synchronized state of system (39).

Figure 3: The synchronous error of system (39).

Figure 4: The synchronized trajectory of , .

Figure 5: The synchronized trajectory of , .

Figure 6: The synchronized trajectory of , .

5. Conclusion

In this paper, global synchronization problem is investigated for general complex networks with coupling delays. Two novel theorems are obtained about synchronization of the coupled systems by employing Lyapunov-Krasovskii functional and convex combination techniques. We find that the derivative of the time-varying delay can be any given value, although most of the former results are based on assumption that the derivative of the time-varying delay should be less than one. The condition obtained in this paper is expressed in the form of linear matrix inequalities, which have less variables and are easy to be computed and checked by resorting to Matlab LMI toolbox. The proposed network model may shed some new lights on the synchronization with one delay coupling. Furthermore, there are abundant dynamical behaviors in arrays of coupled systems with different coupling schemes and they deserve to be further studied in the future. Finally, two examples are given to illustrate the usefulness of the derived methods by the simulation results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of the paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61070149, 61175115, 61370113, and 61272320), the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (Grant no. CIT&TCD201304035), and Jing-Hua Talents Project of Beijing University of Technology (Grant no. 2014-JH-L06).

References

J. Wang and Z. Xu, “New study on neural networks: the essential order of approximation,” Neural Networks, vol. 23, no. 5, pp. 618–624, 2010.

G. Wang, J. Cao, L. Wang, and J. Franklin, “Global dissipativity of stochastic neural networks with time delay,” Journal of the Franklin Institute, vol. 346, no. 8, pp. 794–807, 2009.

H. Zhang, Z. Wang, and D. Liu, “Global asymptotic stability of recurrent neural networks with multiple time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, no. 5, pp. 855–873, 2008.

Z. Yuan, J. Cai, and M. Lin, “Global synchronization in complex networks with adaptive coupling,” Mathematical Problems in Engineering, vol. 2010, Article ID 826721, 10 pages, 2010.

J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,” IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 841–846, 2005.

C. Li and G. Chen, “Synchronization in general complex dynamical networks with coupling delays,” Physica A, vol. 343, no. 1–4, pp. 263–278, 2004.

J. Zhou and T. Chen, “Synchronization in general complex delayed dynamical networks,” IEEE Transactions on Circuits and Systems I, vol. 53, no. 3, pp. 733–744, 2006.

H. Gao, J. Lam, and G. Chen, “New criteria for synchronization stability of general complex dynamical networks with coupling delays,” Physics Letters A, vol. 360, no. 2, pp. 263–273, 2006.

W. Guo, F. Austin, and S. Chen, “Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 6, pp. 1631–1616, 2010.

D. Gong, H. Zhang, Z. Wang, and B. Huang, “New global synchronization analysis for complex networks with coupling delay based on a useful inequality,” Neural Computing and Applications, vol. 22, no. 2, pp. 205–210, 2011.

K. Li, S. Guan, X. Gong, and C.-H. Lai, “Synchronization stability of general complex dynamical networks with time-varying delays,” Physics Letters A, vol. 372, no. 48, pp. 7133–7139, 2008.

H. Li and D. Yue, “Synchronization stability of general complex dynamical networks with time-varying delays: a piecewise analysis method,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 149–158, 2009.

Z. Fei, H. Gao, and W. X. Zheng, “New synchronization stability of complex networks with an interval time-varying coupling delay,” IEEE Transactions on Circuits and Systems II, vol. 56, no. 6, pp. 499–503, 2009.

W. He and J. Cao, “Global synchronization in arrays of coupled networks with one single time-varying delay coupling,” Physics Letters A, vol. 373, no. 31, pp. 2682–2694, 2009.

Q. Song, “Synchronization analysis of coupled connected neural networks with mixed time delays,” Neurocomputing, vol. 72, no. 16–18, pp. 3907–3914, 2009.

Y. Wang, Z. Wang, and J. Liang, “A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances,” Physics Letters A, vol. 372, no. 39, pp. 6066–6073, 2008.

H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009.

T. Li, T. Wang, A.-G. Song, and S.-M. Fei, “Exponential synchronization for arrays of coupled neural networks with time-delay couplings,” International Journal of Control, Automation and Systems, vol. 9, no. 1, pp. 187–196, 2011.

L. O. Chua, “The genesis of Chua’s circuit,” Archiv für Elektronik und Ubertragung-stechnik, vol. 46, pp. 250–257, 1992.