- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Journal of Applied Mathematics

Volume 2013 (2013), Article ID 823863, 11 pages

http://dx.doi.org/10.1155/2013/823863

## The Complex Network Synchronization via Chaos Control Nodes

^{1}School of Mathematics and Information Science and School of Physics and Electromechanical Engineering, Shaoguan University, Shaoguan, Guangdong 512005, China^{2}Department of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

Received 21 July 2012; Accepted 4 February 2013

Academic Editor: Xiaojun Wang

Copyright © 2013 Yin Li and Chun-long Zheng. 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

We investigate chaos control nodes of the complex network synchronization. The structure of the coupling functions between the connected nodes is obtained based on the chaos control method and Lyapunov stability theory. Moreover a complex network with nodes of the new unified Loren-Chen-Lü system, Coullet system, Chee-Lee system, and the New system is taken as an example; numerical simulations are used to verify the effectiveness of the method.

#### 1. Introduction

Since the most famous random graph model was proposed by Erdös and Rényi [1], the complex network has attracted much attention in many fields of research, such as biology, physics, computer networks, the World Wide Web (WWW) [2], and so on. Network synchronization has obvious advantages, it has great application value in practice. Therefore, Atay et al. [3] studied synchronization of complex network when delays exist among the nodes; Motter et al. [4] studied the influence of coupling strength on the synchronizing ability of a complex network; Timme et al. [5] studied the web synchronization law of pulse-coupled dynamical systems; Checco et al. [6] studied the synchronization of random web. Lü et al. [7] constructed general complex dynamical networks and studied the synchronization; Lu and Chen [8] studied synchronization analysis of linearly coupled networks of discrete time systems; Han and Lu [9] studied the changes of synchronization ability of coupled networks from ring networks to chain networks; He and Yang [10] studied adaptive synchronization in nonlinearly coupled dynamical networks; Hung et al. [11] studied globally generalized synchronization in scale-free networks.

Recently, network synchronization has been an important part of the dynamic study of complex network. It has aroused great interest of scholars both domestically and abroad to build weighted network models and study the characteristics of them. Lü et al. [7] studied chaos synchronization of general complex dynamic networks. Gao et al. [12] realized the adaptive synchronization of complex network. Pei et al. [13] made statistical analysis of a class of real networks; Barrat et al. [14] presented a weighted network model, and studied its dynamic character; Atay et al. [15] studied the synchronization of a complex network with time delay. Hung et al. [11] realized the generalized synchronization of a scale-free network. Qin and Yu [16] achieved the synchronization of the star-network of hyperchaotic Rossler systems. In addition to the above researches, much other work [17, 18] in the field have been done, and complex network synchronization become a focus of attention, such as a random network synchronization, small-world network synchronization, scale-free network synchronization, and so on. However, universal synchronization methods of weighted network still need further exploration and study.

The motivation in this paper lies in the complex network synchronization and chaos control importance. Network synchronization is one of the most practical and valuable issues. A synchronization of network means the situation in which the output of all nodes in the study of the complex network is consistent with any given external input signal under a certain condition. The working mechanism of a single chaotic system to track any given external input signal is relatively interesting and significance. Numerical simulations are used to verify the effectiveness of the proposed techniques.

It is organized as follows. Firstly, the theory and the method are presented in Section 2. Then, different order chaotic systems are adopted as the nodes of this complex network; the structure of the coupling functions among the connected nodes is obtained based on Lyapunov stability theory. With the help of symbolic computation, the temporal evolution of variables and node interaction of the dynamic equation are discussed and simulated by computer in Section 3. The theoretical analysis and the numerical simulations show that this is a universal method and the number of the nodes does not affect the stability of the whole network. Finally, the conclusions are given in Section 4.

#### 2. Theory and Method

We summarized the main steps for the complex network synchronization based on Lü et al. [19–23] and Chen et al. [24–27], as follows.

*Step 1. *Assume the state of node is , where and , then the dynamic function for node without coupling can be described as
where , is the linear coefficient matrix of the system, and is the coefficient matrix of control gain.

*Step 2. *Considering the coupling of network, the dynamic function for node can be described as
where denotes the coupling function. The proportional scale for node is , then we will have .

*Step 3. *The errors between the state variables of the network are defined as
We obtain from (2) and (3)
where , .

*Step 4. *Choose , then we will have
If we choose node 1 as target node, then the coupling function of node can be described as
In order to realize the synchronization of the complex network, if we choose other node as target node, the coupling functions of all nodes can also be obtained.

*Step 5. *Constructing the Lyapunov function according to the weighted complex network with different nodes,
and considering (3) and (4), we can obtain the derivative form of as
From (8), we can easily see that if
Then
According to Lyapunov stability theory [20] and the weighted complex network [19, 20, 28], the synchronization of the complex network can be realized.

#### 3. Application of Chaos Control Method to the Network Synchronization

The unified chaotic system, the New system, Coullet system, and Chee-Lee system are taken as nodes of the network to show the synchronization mechanism mentioned above. Simulation is made as the number of the nodes is .

The dynamic equation of the unified chaotic system [21, 29] is presented as follows: where . When , LCL system (11) belongs to the generalized Lorenz system [30]; when , LCL system (11) belongs to the generalized Lü system [23]; when LCL system (11) belongs to the generalized Chen system [31]. Figures 1(a)–1(c) display these chaotic attractors of the generalized Lorenz system (), the generalized Lü system (), and the generalized Chen system (), respectively.

The new chaotic system of three-dimensional quadratic autonomous ordinary differential equations [22], which can display the chaotic attractors: where , , and are real constants; and , and are status variables. This system is found to be chaotic in a wide parameter range and has many interesting complex dynamical characteristics. The system is chaotic for the parameters , and ; it displays the chaotic attractor as shown in Figure 2(a). Detailed dynamic properties of this system can be found in [22].

The dynamic equation of Coullet system [19] is described as follows: where , , and are parameters of the system. When the parameters are given as , , and , the chaos attractor of (13) is shown in Figure 2(b).

The dynamic equation of new system [23] is described as follows: where , , and are parameters of the system. When the parameters are given as , , and , the chaos attractor of (14) is shown in Figure 3(a).

The dynamic equation of the Chen-Lee system [32] is described as follows: where , , and , (15) is a chaotic system. Figure 3(b) displays the chaotic attractors in -space. It is easy to see that it admits a symmetry . That is, the system (15) is symmetrical about the three coordinate axes , and , respectively.

Choose the unified chaotic system (16) (node 1) as a target system, that is, coupling function , then the dynamic functions of node 2, node 3, node 4, and node 5 with coupling can be described, respectively, as where fulfills (6): The relative weight of the coupling strength between the nodes of the network is arbitrarily taken as , , then the network synchronization error between system (16) and system (17) will be , , and and the network synchronization error between system (17) and system (18) will be , , and , and the network synchronization error between system (18) and system (19) will be , , and , and the network synchronization error between system (19) and system (20) will be , , and . The simulations are shown in Figures 4, 5, 6, and 7.

For further details, the dynamic attractors for each node with coupling network synchronization are shown in Figures 8 and 9. The network synchronization temporal evolution of variables is shown in Figures 10, 11, 12, and 13. From Figures 8 and 9, we can see that the network synchronization has been realized on the required proportional scale.

#### 4. Summary and Discussion

In this paper, the complex network synchronization is investigated. With the help of symbolic computation, different order chaotic systems are adopted as the nodes of this complex network; the structure of the coupling functions among the connected nodes is obtained based on Lyapunov stability theory. Being of network and physical interests, the temporal evolution of variables and node interaction of the dynamic equation are discussed and simulated by computer. This method has universal significance for network synchronization, and the weight value of the coupling strength between the nodes and the number of the nodes does not affect synchronization of the whole network.

#### Acknowledgments

This work is supported by the NSF of China under Grant nos. 11172181, Guangdong Provincial NSF of China under Grant no. 10151200501000008 and 94512001002983, Guangdong Provincial UNYIF of China under Grant, and Science Foundation of Shaoguan University.

#### References

- P. Erdös and A. Rényi, “On the evolution of random graphs,”
*Publication of the Mathematical Institute of the Hungarian Academy of Sciences*, vol. 5, p. 17, 1960. View at Google Scholar - M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distributions and their applications,”
*Physical Review E*, vol. 64, Article ID 26118, 17 pages, 2001. View at Publisher · View at Google Scholar - F. M. Atay, J. Jost, and A. Wende, “Delays, connection topology, and synchronization of coupled chaotic maps,”
*Physical Review Letters*, vol. 92, no. 14, pp. 144101–1, 2004. View at Publisher · View at Google Scholar · View at Scopus - A. E. Motter, C. Zhou, and J. Kurths, “Network synchronization, diffusion, and the paradox of heterogeneity,”
*Physical Review E*, vol. 71, Article ID 016116, 9 pages, 2005. View at Publisher · View at Google Scholar - M. Timme, F. Wolf, and T. Geisel, “Topological speed limits to network synchronization,”
*Physical Review Letters*, vol. 92, Article ID 074101, 4 pages, 2004. View at Publisher · View at Google Scholar - P. Checco, M. Biey, and L. Kocarev, “Synchronization in random networks with given expected degree sequences,”
*Chaos, Solitons & Fractals*, vol. 35, no. 3, pp. 562–577, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - W. Lu and T. Chen, “Synchronization analysis of linearly coupled networks of discrete time systems,”
*Physica D*, vol. 198, no. 1-2, pp. 148–168, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Han and J. Lu, “The changes on synchronizing ability of coupled networks from ring networks to chain networks,”
*Science in China F*, vol. 50, no. 4, pp. 615–624, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. He and J. Yang, “Adaptive synchronization in nonlinearly coupled dynamical networks,”
*Chaos, Solitons and Fractals*, vol. 38, no. 5, pp. 1254–1259, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. C. Hung, Y. T. Huang, M. C. Ho, and C. K. Hu, “Paths to globally generalized synchronization in scale-free networks,”
*Physical Review E*, vol. 77, Article ID 16202, 8 pages, 2008. View at Google Scholar - Y. Gao, L. X. Li, H. P. Peng, Y. X. Yang, and X. H. Zhang, “Adaptive synchronization in a united complex dynamical network with multi-links,”
*Acta Physica Sinica*, vol. 57, no. 4, pp. 2081–2091, 2008. View at Google Scholar · View at MathSciNet - W. D. Pei, Z. Q. Chen, and Z. Z. Yuan, “A dynamic epidemic control model on uncorrelated complex networks,”
*Chinese Physics B*, vol. 17, p. 373, 2008. View at Publisher · View at Google Scholar - A. Barrat, M. Barthelemy, and A. Vespignani, “Weighted evolving networks: coupling topology and weight dynamics,”
*Physical Review Letters*, vol. 92, Article ID 228701, 4 pages, 2004. View at Publisher · View at Google Scholar - Y. Li, Y. Chen, and B. Li, “Anticipated function synchronization with unknown parameters in discrete-time chaotic systems,”
*International Journal of Modern Physics C*, vol. 20, no. 4, pp. 597–608, 2009. View at Publisher · View at Google Scholar - J. Qin and H. J. Yu, “Synchronization of star-network of hyperchaotic Rossler systems,”
*Acta Physica Sinica*, vol. 56, pp. 6828–6835, 2007. View at Google Scholar - Z. Zeng and J. Wang, “Complete stability of cellular neural networks with time-varying delays,”
*IEEE Transactions on Circuits and Systems*, vol. 53, no. 4, pp. 944–955, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cao, P. Li, and W. Wang, “Global synchronization in arrays of delayed neural networks with constant and delayed coupling,”
*Physics Letters A*, vol. 353, no. 4, pp. 318–325, 2006. View at Publisher · View at Google Scholar - L. Ling, L. Gang, G. Li, M. Le, J.-R. Zou, and Y. Ming, “Generalized chaos synchronization of a weighted complex network with different nodes,”
*Chinese Physics B*, vol. 19, Article ID 080507, 2010. View at Publisher · View at Google Scholar - L. Lü,
*Nonlinear Dynamics and Chaos*, Dalian Publishing House, Dalian, China, 2000. - J. Lü, G. Chen, D. Cheng, and S. Celikovsky, “Bridge the gap between the Lorenz system and the Chen system,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 12, no. 12, pp. 2917–2926, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lü, G. Chen, and D. Cheng, “A new chaotic system and beyond: the generalized Lorenz-like system,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 14, no. 5, pp. 1507–1537, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lü and G. Chen, “A new chaotic attractor coined,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 12, no. 3, pp. 659–661, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Chen, J. Zhou, and Z. Liu, “Global synchronization of coupled delayed neural networks and applications to chaotic CNN models,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 14, no. 7, pp. 2229–2240, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Chen and X. Li, “Function projective synchronization between two chaotic systems,”
*Zeitschrift für Naturforschung*, vol. 62, p. 176, 2007. View at Google Scholar - Y. Q. Yang and Y. Chen, “Chaos in the fractional order generalized Lorenz canonical form,”
*Chinese Physics Letters*, vol. 26, Article ID 100501, 2009. View at Publisher · View at Google Scholar - H. L. An and Y. Chen, “The function cascade synchronization method and applications,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 10, pp. 2246–2255, 2008. View at Publisher · View at Google Scholar - M. J. Wang, X. Y. Wang, and Y. J. Niu, “Projective synchronization of a complex network with different fractional order chaos nodes,”
*Chinese Physics B*, vol. 20, Article ID 010508, 2011. View at Publisher · View at Google Scholar - Y. Chen and Z. Yan, “New exact solutions of (2 + 1)-dimensional Gardner equation via the new sine-Gordon equation expansion method,”
*Chaos, Solitons and Fractals*, vol. 26, no. 2, pp. 399–406, 2005. View at Publisher · View at Google Scholar · View at Scopus - E. N. Lorenz, “Deterministic nonperiodic flow,”
*Journal of the Atmospheric Sciences*, vol. 20, pp. 130–141, 1963. View at Publisher · View at Google Scholar - G. Chen and T. Ueta, “Yet another chaotic attractor,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 9, no. 7, pp. 1465–1466, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. B. Sedra and A. El Boukili, “Some physical aspects of Moyal noncommutativity,”
*Chinese Journal of Physics*, vol. 47, no. 3, pp. 305–315, 2009. View at Google Scholar · View at Scopus