Brief paperOn pinning synchronization of complex dynamical networks☆
Introduction
Many large-scale systems in nature and human societies, such as biological neural networks, ecosystems, metabolic pathways, the Internet, the WWW, electrical power grids, etc., can be described by networks with the nodes representing individuals in the system and the edges representing the connections among them. Recently, the study of various complex networks has attracted increasing attention from researchers in various fields of physics, mathematics, engineering, biology, and sociology.
In the early 1960s, Erdös and Rényi, 1959, Erdös and Rényi, 1960 proposed a random-graph model, which had laid a solid foundation of modern network theory. In a random network, each pair of nodes is connected with a certain probability. To describe a transition from a regular network to a random network, Watts and Strogatz (1998) proposed an interesting small-world network model. Then, Newman and Watts (1999) modified it to generate another variant of the small-world model. Later, Barabási and Albert (1999) proposed a scale-free network model, in which the degree distribution of the nodes follows a power-law form. Thereafter, small-world and scale-free networks have been extensively investigated.
Synchronization, on the other hand, is a typical collective behavior in nature. Since the pioneering work of Pecora and Carroll (1990), chaos control and synchronization have received a great deal of attention (Yu and Cao, 2007, Yu, Cao, Wong et al., 2007, Yu, Chen et al., 2007) due to their potential applications in secure communications, chemical reactions, biological systems, and so on. Typically, there are large numbers of nodes in real-world complex networks. Therefore, a large amount of work has been devoted to the study of synchronization in various large-scale complex networks (Cao et al., 2006, W. Lu and T. Chen, 2004, Wang and Cao, 2006, Wang and Chen, 2002a, Wang and Chen, 2002b, Yu et al., 2008). In Wang and Chen, 2002a, Wang and Chen, 2002b, local synchronization was investigated by the transverse stability to the synchronization manifold, where synchronization was discussed on small-world and scale-free networks. In Wu (2005) and Wu and Chua (1995), a distance from the collective states to the synchronization manifold was defined, based on which some results were obtained for global synchronization of coupled systems (Cao et al., 2006, W. Lu and T. Chen, 2004, Wang and Cao, 2006). A general criterion was given in Yu et al. (2008), where the network sizes can be extended to be much larger compared to those very small ones studied in Cao et al. (2006), W. Lu and T. Chen (2004) and Wang and Cao (2006). However, it is still very difficult to ensure global synchronization in very large-scale networks due to the computational complexity.
In the case where the whole network cannot synchronize by itself, some controllers may be designed and applied to force the network to synchronize. However, it is literally impossible to add controllers to all nodes. To reduce the number of controlled nodes, some local feedback injections may be applied to a fraction of network nodes, which is known as pinning control. In Grigoriev, Cross, and Schuster (1997), pinning control of spatiotemporal chaos, and later in Parekh, Parthasarathy, and Sinha (1998) global and local control of spatiotemporal chaos in coupled map lattices, were discussed. Very recently, in Wang and Chen (2002c), both specific and random pinning schemes were studied, where specific pinning of the nodes with large degrees is shown to require a smaller number of controlled nodes than the random pinning scheme. However, there exist some fundamental and challenging problems which have not been solved to date. A key problem is how the local controllers on the pinned nodes affect the global network synchronization. This paper aims to address the following questions: (1) What kinds of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What types of controllers may be designed to ensure the synchronization? (3) How large should the coupling strength be used for a network with a fixed topological structure to effectively achieve global network synchronization?
The rest of the paper is organized as follows. In Section 2, some preliminaries are briefly outlined. The main theorems and corollaries for pinning synchronization on complex networks are given in Section 3. In Section 4, the pinning schemes on some scale-free networks are simulated to verify the theoretical analysis. Conclusions are finally drawn in Section 5.
Section snippets
Preliminaries
Consider a complex dynamical network consisting of identical nodes with linearly diffusive coupling (Cao et al., 2006, W. Lu and T. Chen, 2004, J. Lü and G. Chen, 2005, Wang and Chen, 2002a, Wang and Chen, 2002b, Yu et al., 2008), described by where is the state vector of the th node, is a continuously differentiable vector function, is the coupling strength, is the inner coupling
Pinning synchronization criteria for complex networks
In this section, some pinning criteria are established to ensure the global synchronization of complex dynamical networks.
Simulation examples
In this section, some simulation examples are given to verify the criteria established above.
In 1963, Lorenz found the first chaotic system (Lorenz, 1963). Then, in 1999, Chen and Ueta found the dual of the Lorenz system (Chen & Ueta, 1999). Later in 2002, Lü and Chen discovered another new chaotic system (Lü & Chen, 2002), which bridges the gap between the Lorenz system and the Chen system. Generally, chaotic systems are more difficult to synchronize than non-chaotic systems. Here, consider
Conclusions
In this paper, pinning synchronization of a class of complex dynamical networks has been investigated in detail. A general criterion for ensuring network synchronization has been derived. Some analytical and adaptive techniques have been proposed to obtain appropriate coupling strengths for achieving network synchronization. It is surprising to find that a network can realize synchronization under any linear feedback pinning scheme by adaptively adjusting the coupling strength. Furthermore,
Wenwu Yu received the B.S. degree in information and computing science and M.S. degree in applied mathematics from the Department of Mathematics, Southeast University, Nanjing, China, in 2004 and 2007, respectively. Currently, he is working towards the Ph.D. degree at the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. He held several visiting positions in Hong Kong, USA, and China. His research interests include multi-agent systems, nonlinear dynamics and
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Wenwu Yu received the B.S. degree in information and computing science and M.S. degree in applied mathematics from the Department of Mathematics, Southeast University, Nanjing, China, in 2004 and 2007, respectively. Currently, he is working towards the Ph.D. degree at the Department of Electronic Engineering, City University of Hong Kong, Hong Kong. He held several visiting positions in Hong Kong, USA, and China. His research interests include multi-agent systems, nonlinear dynamics and control, complex networks and systems, neural networks, cryptography, and communications.
Guanrong Chen received his M.Sc. degree in computer science from Zhongshan (Sun Yat-sen) University, China in Fall 1981 and Ph.D. degree in applied mathematics from Texas A & M University in Spring 1987. He is currently a chair professor and the founding director of the Centre for Chaos and Complex Networks at the City University of Hong Kong, prior to which he was a tenured Full Professor at the University of Houston, Texas, USA. He is a Fellow of the IEEE (Jan. 1997), with research interests in chaotic dynamics, complex networks and nonlinear controls.
Jinhu Lü received the Ph.D degree in applied mathematics from the Chinese Academy of Sciences, Beijing, China in 2002. Currently, he is an Associate Professor with the Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China. He held several visiting positions in Australia, Canada, France, Germany, Hong Kong and USA, and was a Visiting Fellow in Princeton University, USA from 2005 to 2006. He is the author of two research monographs and more than 70 international journal papers published in the fields of nonlinear circuits and systems, complex networks and complex systems. He served as a member in the Technical Committees of several international conferences and is now serving as a member in the Technical Committees of Nonlinear Circuits and Systems and of Neural Systems and Applications in the IEEE Circuits and Systems Society. He is also an Associate Editor of IEEE Transactions on Circuits and Systems II, Journal of Systems Science and Complexity and DCDIS-A. Dr. Lü received the prestigious Presidential Outstanding Research Award from the Chinese Academy of Sciences in 2002, the National Best Ph.D. Theses Award from the Office of Academic Degrees Committee of the State Council and the Ministry of Education of China in 2004, the First Prize of Natural Science Award from the Ministry of Education of China in 2007 and the Lu Jiaxi Youth Talent Award from the Chinese Academy of Sciences in 2008. He is the co-author of the Most Cited SCI Paper of Chinese Scholars in the field of mathematics during the periods of 2001–2005 and 2002–2006. He is also an IEEE Senior Member.
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This work was supported by the NSFC-HKRGC Joint Research Scheme under Grant N-CityU107/07, the Hong Kong Research Grants Council under the GRF Grant CityU 1117/08E, the National Natural Science Foundation of China under Grants, No. 60221301 and No. 60772158, the National Basic Research (973) Program of China under Grant 2007CB310805, the Important Direction Project of Knowledge Innovation Program of Chinese Academy of Sciences under Grant KJCX3-SYW-S01, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen.