Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 142061, 8 pages

http://dx.doi.org/10.1155/2014/142061

Research Article

Group Synchronization of Nonlinear Complex Dynamics Networks with Sampled Data

Man Li,¹ Bo Liu,² Yuqing Zhu,¹ Lijun Wang,³ and Mei Zhou²

¹Mathematics and Physics, Nanyang Institute of Technology, Nanyang, Henan 473004, China

²College of Science, North China University of Technology, Beijing 100144, China

³School of Economics and Management, Beijing University of Technology, Beijing 100022, China

Received 6 December 2013; Accepted 5 February 2014; Published 24 March 2014

Academic Editor: Huaicheng Yan

Copyright © 2014 Man 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.

Based on a nonlinear consensus protocol, this paper considers the group synchronization of complex dynamical networks with sampled data. Using the Lyapunov method, the group synchronization of the nonlinear complex networks is analyzed. All the nodes in each group can converge to their own synchronous state asymptotically, if the sampled period satisfies some matrix inequality conditions. Furthermore, the theoretical results are verified by some simulations.

1. Introduction

Because of the wide application of the complex dynamical networks [1–26], the synchronization problem of the complex networks has become a hot topic recently. A lot of researches concentrate on the continuous information transmission; that is, every node can receive or detect neighbor information of the network all the time [1–3, 5–7, 10]. In [1], the authors considered the group synchronization of the nonlinear complex dynamical network via pinning control. A new criterion for the cluster synchronization of coupling networks with time-varying delays was established in continuous time in [2]. Moreover, Liu et al. [3] considered the adaptive synchronization of complex dynamical networks with switching topology by local Lipschitz nonlinearity. In [5], group consensus of multiagent systems with switching topologies and communication delays was studied in the continuous-time case.

However, in many engineering practices, the information communication between nodes could be interrupted at any time due to some factors, such as the unreliability of communication channels and the limitations of the node detection ability. Hence, it is necessary to consider the information transmission under the discrete state [9, 13–16, 18–21, 23, 26]. Information data on the discrete time can also be regarded as sampled data [11, 25]. In [11], the group consensus of linear multiagent systems with sampled-data was considered. Xiao and Wang further considered a discrete-time model with time delays in [13] and analyzed a consensus problem in the existence of the time delay when agents exchanged information between each other. Gao and Wang [16] studied the continuous-time consensus of multiagent systems with sampled data by time-varying topology. In [18], consensus of multiple dynamic agents with sampled information was considered. In [20], average consensus control of networks with sampled data and measurement noises was studied on continuous time. In this paper, we consider the group synchronization of nonlinear complex dynamical network with sampled data.

The rest of this paper is organized as follows. In Section 2, we consider the group synchronization of a complex dynamical network with sampled data and give some assumptions and lemmas. In Section 3, we analyze group synchronization of the proposed network and give the synchronization condition based on linear matrix inequality (LMI). In Section 4, we give the synchronization condition for a special case. In Section 5, the simulations verify the theoretical results. Conclusion is finally summarized in Section 6.

2. Preliminaries and Problem Statement

Consider a complex dynamical network of nodes with sampled data as follows:

with

where represents the state vectors of the node at time , is continuously differentiable, is coupling strength, and is an inner-coupling matrix, if the node links through its th state with its neighbors ; otherwise . In this protocol, let , and let , where . ,  ,  , with . is the neighbors of the node , , where , .

In protocol (2), is the control input:

where is an on-off control; if the system is sampling the data, then ; otherwise . If node can get information from node in the same group, then ; otherwise ; if node can get information from node between different groups, then ; otherwise . Thus the coupling configuration matrix can be written as

where

and represent the coupling configuration of the subgroups, respectively.

Network (1) is said to group synchronization if

where the nonlinear function satisfies

and , are the synchronous states.

Given a positive real number and a sampled period , we suppose that

where shows the discrete time that the node can obtain information from its neighbors and positive integer is a sampled time about the time (), and it satisfies . Under this condition, a linear consensus protocol based on a linear estimation-based sampling period is designed as follows [11]:

Thus, we have

Then the complex dynamical network (1) in every sampled period becomes

with

Some assumptions and lemmas are needed.

Assumption 1 (see [3]). If each of the nonlinear function in network (1) satisfies the local Lipschitz condition, for any compact set , there exists a positive constant matrix , such that

Assumption 2 (see [1]). Assume that protocol (1) satisfies the balance of effectiveness between the subgroups

Lemma 3 (see [1]). Define , and construct a closed space

where and is a constant. For , there exists a constant such that

Lemma 4 (see [1]). Suppose that and are vectors, and in matrix , the following inequality holds:

In this paper, means that is a positive (or semipositive, negative, or seminegative) definite matrix; , .

3. Group Synchronization Analysis in Complex Network with Sampled Data

In this section, we consider group synchronization problem of the complex networks with sampled data. We have the following theorem.

Theorem 5. For network (1) with protocol (12) of nodes, under Assumptions 1-2 and Lemmas 3-4, if , satisfy

then all the nodes in each group can converge to their own synchronous state asymptotically, where

Proof. Construct a Lyapunov function as follows:

where

Define

Then,

Under Assumption 2, we can know

Under Assumption 1 and Lemma 3, we can have

Then, we can get

Using Lemma 4, we can have

Thus, we can obtain

If

then

Therefore, decreases on an interval , , and . From the above discussion, we have

Then, network (1) with sampled data is group-synchronized under protocol (12).

4. A Special Case

In this section, we consider a special case about network (1). For convenience, we let and .

Lemma 6 (see [1]). If is a symmetric irreducible matrix with and , then for any matrix with all eigenvalues of the matrix are negative.

In networks, when the information transmission is the same between the nodes and , the coupling configuration matrix is symmetric. From Lemma 6, when is symmetric, and are negative. Hence, all of their eigenvalues are strictly negative; one denotes them as

Then, using Lemma 6, we can obtain the following result when the coupling configuration matrix is symmetric.

Theorem 7. For network (1) with protocol (12) of nodes, when the coupling configuration matrix is symmetric, under Assumptions 1-2 and Lemmas 3–6, if , satisfy

then all the nodes in each group can converge to their own synchronous state asymptotically, where and .

The proof of Theorem 7 is similar to that of Theorem 5 and here is omitted.

5. Simulations

In this section, we give some simulation results of the above discussions. For convenience, let , , , for all , , and , . We consider group synchronization of complex dynamical network with sampled data in the time interval . The curves in the graphs are the locations of nodes and the synchronous targets in the network.

From Figures 1, 2, and 3, the synchronous states are , . From these figures, we can see that all the nodes in each subgroup can converge to their own synchronous targets. But, in Figures 4, 5, and 6, all of the nodes can asymptotically converge to a synchronous state, when the synchronous state is the only one .

Figure 1: The positions of the nodes of network (1) on interval .

Figure 2: The positions of the nodes of network (1) on interval , sampled period .

Figure 3: The positions of the nodes of network (1) on interval , sampled period .

Figure 4: The nodes of network (1) on interval converge to the same target.

Figure 5: The nodes of network (1) on interval converge to the same target, sampled period .

Figure 6: The nodes of network (1) on interval converge to the same target, sampled period .

In Figures 1 and 4, we simulate synchronization of the network including two subgroups without sampled data. In Figures 2 and 5, we chose as the sampled period. That is, the complex network is sampling at the moment , for . In Figures 3 and 6, we chose as the sampled period; that is, the complex network is sampling at the moment , for .

In these simulation results, we can find that the nodes of the system keep their state value in the first time until before the moment , keeping their state value in time until the moment and so on. By comparing, when choosing the sampled data on every period, the rate of convergence of the nodes in the complex dynamics network is slow. Moreover, the bigger the sampled period is, the slower the rate of convergence of the nodes in the complex dynamics network is.

6. Conclusion

In this paper, we have investigated the group synchronization problem of a complex dynamical network with sampled data. We prove that the nodes of the network arrive at synchronization in two subgroups if the sampling period satisfies the condition based on the linear matrix inequality (LMI). In addition, we have given some simulation results about the proposed complex network.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61304049, Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201108055), and Science and Technology Development Plan Project of Beijing Education Commission (nos. KM201310009011 and KM201310009013).

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