Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 397250, 11 pages

doi:10.1155/2012/397250

Research Article

Impulsive Synchronization of Multilinks Delayed Coupled Complex Networks with Perturb Effects

Jun Li,1,2 Bing Li,³ Yuhua Xu,4,5 Wuneng Zhou,⁶ and Jian'an Fang⁶

¹Shiyan Vocational and Technical College, Hubei, Shiyan 442000, China

²School of Economics, Huazhong University of Science and Technology, Hubei 430074, China

³NOSTA, The Ministry of Science and Technology of China, GPO Box 2143, Beijing 100045, China

⁴Computer School of Wuhan University, Wuhan 430079, China

⁵Department of Mathematics and Finance, Yunyang Teachers' College, Hubei 442000, China

⁶College of Information Science and Technology, Donghua University, Shanghai 201620, China

Received 28 August 2012; Revised 30 September 2012; Accepted 1 October 2012

Academic Editor: Xinzhu Meng

Copyright © 2012 Jun Li 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.

This paper investigates impulsive synchronization of multilinks delayed coupled complex networks with perturb effects. Based on the comparison theory of impulsive differential system, a novel synchronization criterion is derived and an impulsive controller is designed simultaneously. Finally, numerical simulations demonstrate the effectiveness of the proposed synchronization criteria.

1. Introduction

In the past few decades, the problem of control and synchronization of complex dynamical networks has been extensively investigated in various fields of science and engineering due to its many potential practical applications [1–8]. One important consideration in practical networks is the existence of time delays because obstructions to the transmission of signals are inevitable in a biological neural network, in an epidemiological model, in a communications network, or in an electrical power grid. Since recently, there are many studies on dynamical networks with time delays [9–16]. Moreover, the multidelayed coupling consists of providing more information about the dynamics in nodes to the other nodes in the network, such as the transportation network; we all know that the transmission speed is different among highway network railway network, and airline network. So we can use timedelay to describe these networks [17]. In [17], the authors studied synchronization of a class of timedelayed complex dynamical networks with multilinks, and this model is suitable to investigate and simulate more realistic complex networks, so we should pay attention to this network with multilinks. Moreover, in [9–11, 17–20], the authors studied synchronization of complex network with time-varying coupling delay by designing a controller and adaptive updated laws. However, the controller designed for such an adaptive synchronization is usually quite complex, it will be useful to find a simple structure to solve this problem, the impulsive controller seems to have a simple structure, and impulsive control is an artificial control strategy which is cheaper to operate compared with other control strategy. Motivated by the above discussions, we investigate impulsive synchronization for such a complex networks model in this paper, and the novel synchronization criterion is derived.

The rest of this work is organized as follows. Section 2 gives the problem formulation. Section 3 gives synchronization scheme. Section 4 gives illustrative example. Section 5 gives the conclusion of the paper.

2. Problem Formulation

In [17], the authors achieve synchronization between two complex networks with multilinks by designing effective controller. For simplicity, the complex network model is written in the following form:

where , standing for the activity of an individual subsystem is a vector value function. is the subnetwork’s topological structure. The definition of is that in the sub-network, if there exists a link from node to , then . Otherwise, is time-delay of the subnetwork compared to the zero subnetwork () which is without time delayed.

Remark 2.1. In [17], is defined, we are not concerned whether the coupling matrix satisfies in this paper.

In the paper, we have the following mathematical preliminaries.

Assumption 2.2. We assume that is Lipschitz continuous on , that is, there exists a positive constant such that

Assumption 2.3. We also assume that is Lipschitz continuous on , and one can consider

where .

3. Synchronization Scheme

In this section, we will investigate impulsive synchronization of the complex networks with perturb functions. The multidelayed coupled complex network with perturb functions can be described by

We take the network given by (3.1) as the driving network and a response network with impulsive control scheme which is given by

where .

Let , then we have the following error system:

where , .

Theorem 3.1. Let Assumptions 2.2–2.3 hold, , . If there exists a constant such that

then the driving network (3.1) and the response network (3.2) can realize impulsive synchronization.

Proof. We choose a nonnegative function as

Then the differentiation of along the trajectories of (3.4) is

This implies that

On the other hand, when , we have

where .

When , , then

So,

In the same way, for , we have

In general for any , one finds that

Thus for all , , we have

From the assumptions given in the theorem

we have . That is .

When , from [21], this implies that the origin in system (3.4) is globally asymptotically stable or the driving network is synchronized with the response network asymptotically for any initial conditions. This completes the proof.

Remark 3.2. Systems (3.1)-(3.2) are the time-invariant complex networks. As discussed in [22–24], systems (3.1)-(3.2) are the time-varying complex networks, which is a more complicated research issue.

Remark 3.3. Normally, it is difficult to control a complex networks by adding the controllers to all nodes, so it would be much better to use the pinning control method since the most complex networks have large number of nodes [25]. Regarding for the pinning control of the network systems (3.1)-(3.2), are the next research topic for us.

Remark 3.4. For the transportation network, we all know that the transmission speed is different among highway network, railway network and airline network. So we can use multilinks delayed to describe these networks [17]. Also impulsive control is an artificial control strategy which is cheaper to operate compared with other control strategy, so impulsive control method of the network systems (3.1)-(3.2) should have potential applications.

4. Illustrative Example

It is well known that the Lorenz system families are typical chaotic systems and the Lü chaotic system is a member of the families which is known as [26]

when .

It is well known that the Lü attractor is bounded. Here we suppose that all nodes are running in the given bounded region. Our numerical analyses show that there exist constants .

Satisfying for . Therefore, one has

Obviously, . Thus the Lü system satisfies Assumption 2.2. In the same way, it can be seen that the Chen system, the Lorenz system, the unified chaotic system and the Lorenz system families also satisfy Assumption 2.2. So, in the simulations, we select the Lü chaotic system as an example to show the effectiveness of the proposed method [27, 28].

According to Section 3, we show that the network with 4 nodes described by

In numerical simulation, let

we choose , , and the gain matrixes as a constant matrix, , then . Let , from , then , we let . All initial values are , , , , , . Figure 1 shows the variance of the synchronization errors. We introduce the quantity [29] which is used to measure the quality of the control process. It is obvious that when no longer increases, two networks achieve synchronization.

(a)

(b)

(c)

(d)

Figure 1: Synchronization errors with time .

5. Conclusion

This paper deals with the problem of impulsive synchronization of multilinks delayed coupled complex networks with perturb effects. On the basis of the comparison theory of impulsive differential system, the novel synchronization criteriion is derived and an impulsive controller is designed simultaneously. Finally, numerical simulations are presented to verify the effectiveness of the proposed synchronization criteria.

Acknowledgments

The authors would like to thank the referees and the editor for their valuable comments and suggestions. This research is supported by the National Natural Science Foundation of China (61075060), the Science and Technology Research Key Program for the Education Department of Hubei Province of China (D20105001 and D20126002), the Excellent Young Key Talents Cultivation Plan of Hubei Province, China Postdoctoral Science Foundation (2012M511663), the Research Fund for the Technology Development Program of Higher School of Tianjin (20111004), the Social Science Planning Project of Tianjin (TJLJ11-009), and the young talents Fund of Anhui Provinc of China (2011SQRL164).

References

J. Lü, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 4, pp. 787–796, 2004.

X. Li and G. Chen, “Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint,” IEEE Transactions on Circuits and Systems I, vol. 50, no. 11, pp. 1381–1390, 2003.

Y. Xu, W. Zhou, J. Fang, and W. Sun, “Adaptive lag synchronization and parameters adaptive lag identification of chaotic systems,” Physics Letters A, vol. 374, no. 34, pp. 3441–3446, 2010.

J. Zhou, L. Xiang, and Z. Liu, “Global synchronization in general complex delayed dynamical networks and its applications,” Physica A, vol. 385, no. 2, pp. 729–742, 2007.

H. Peng, L. Li, Y. Yang, and F. Sun, “Conditions of parameter identification from time series,” Physical Review E, vol. 83, no. 3, Article ID 036202, 8 pages, 2011.

B. Shen, Z. Wang, H. Shu, and G. Wei, “On nonlinear filtering for discrete-time stochastic systems with missing measurements,” IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2170–2180, 2008.

B. Shen, Z. Wang, and Y. S. Hung, “Distributed -consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case,” Automatica, vol. 46, no. 10, pp. 1682–1688, 2010.

B. Shen, Z. Wang, H. Shu, and G. Wei, “Robust finite-horizon filtering with randomly occurred nonlinearities and quantization effects,” Automatica, vol. 46, no. 11, pp. 1743–1751, 2010.

Y. Xu, W. Zhou, J. Fang, and H. Lu, “Structure identification and adaptive synchronization of uncertain general complex dynamical networks,” Physics Letters A, vol. 374, no. 2, pp. 272–278, 2009.

Y. Xu, W. Zhou, J. Fang, and W. Sun, “Adaptive synchronization of the complex dynamical network with non-derivative and derivative coupling,” Physics Letters A, vol. 374, no. 15-16, pp. 1673–1677, 2010.

Y. Tang, R. Qiu, J. Fang, Q. Miao, and M. Xia, “Adaptive lag synchronization in unknown stochastic chaotic neural networks with discrete and distributed time-varying delays,” Physics Letters A, vol. 372, no. 24, pp. 4425–4433, 2008.

J. Zhou, L. Xiang, and Z. Liu, “Synchronization in complex delayed dynamical networks with impulsive effects,” Physica A, vol. 384, no. 2, pp. 684–692, 2007.

X. Wu, “Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay,” Physica A, vol. 387, no. 4, pp. 997–1008, 2008.

Z.-Y. Wu, K.-Z. Li, and X.-C. Fu, “Parameter identification of dynamical networks with community structure and multiple coupling delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3587–3592, 2010.

Y. Tang, J. Fang, M. Xia, and X. Gu, “Synchronization of Takagi-Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays,” Applied Mathematical Modelling, vol. 34, no. 4, pp. 843–855, 2010.

Y. Xu, W. Zhou, and J. Fang, “Topology identification of the modified complex dynamical network with non-delayed and delayed coupling,” Nonlinear Dynamics, vol. 68, no. 1-2, pp. 195–205, 2012.

H. Peng, N. Wei, L. Li, W. Xie, and Y. Yang, “Models and synchronization of time-delayed complex dynamical networks with multi-links based on adaptive control,” Physics Letters A, vol. 374, no. 23, pp. 2335–2339, 2010.

J. Zhou, L. Xiang, and Z. Liu, “Synchronization in complex delayed dynamical networks with impulsive effects,” Physica A, vol. 384, no. 2, pp. 684–692, 2007.

K. Li and C. H. Lai, “Adaptive-impulsive synchronization of uncertain complex dynamical networks,” Physics Letters A, vol. 372, no. 10, pp. 1601–1606, 2008.

S. Cai, J. Zhou, L. Xiang, and Z. Liu, “Robust impulsive synchronization of complex delayed dynamical networks,” Physics Letters A, vol. 372, no. 30, pp. 4990–4995, 2008.

T. Yang, Impulsive Control Theory, Springer, Berlin, Germany, 2001.

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.

J. Zhou, J. Lü, and J. Lu, “Adaptive synchronization of an uncertain complex dynamical network,” IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 652–656, 2006.

J. Lü, X. Yu, and G. Chen, “Chaos synchronization of general complex dynamical networks,” Physica A, vol. 334, no. 1-2, pp. 281–302, 2004.

J. Zhou, J. Lü, and J. Lu, “Pinning adaptive synchronization of a general complex dynamical network,” Automatica, vol. 44, no. 4, pp. 996–1003, 2008.

J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos, vol. 12, no. 3, pp. 659–661, 2002.

Y. Xu, B. Li, Y. Wang, W. Zhou, and J. Fang, “A new four-scroll chaotic attractor consisted of two-scroll transient chaotic and two-scroll ultimate chaotic,” Mathematical Problems in Engineering, vol. 2012, Article ID 438328, 12 pages, 2012.

Y. Xu, W. Zhou, J. Fang, J. Ma, and Y. Wang, “Generating a new chaotic attractor by feedback controlling method,” Mathematical Methods in the Applied Sciences, vol. 34, no. 17, pp. 2159–2166, 2011.

W. Lu, T. Chen, and G. Chen, “Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay,” Physica D, vol. 221, no. 2, pp. 118–134, 2006.