Study On Synchronization And Control Of Fractional-order Complex Networks

With the rapid development of complexity science and Internet technology,complex networks can be found everywhere in daily life, such as the Internet,the World Wide Web (WWW), science citation networks, metabolic networks,biological networks, social networks, etc. As an interdisciplinary new feld, moreand more people are aware of the importance of the networks. The complexity ofthe formation mechanism and evolution of the networks has attracted widespreadinterest of researchers. In particular, there has been increasing interest in the studyof synchronization of complex networks.On the other hand, the study of the fractional calculus has been300yearsold. However, due to the limitations of the research tools, the study of fractionalcalculus developed slowly. The rapid development of computer technology andthe emergence of efcient numerical algorithms promote applications of fractionalcalculus in diferent felds greatly. A large amount of results about synchronizationof complex networks have been got, which the network’s nodes are mainly integer-order diferential systems. Fractional diferential system has more complex naturethen integer-order diferential system, so stability theory of integer-order diferen-tial system can not be applied directly in the fractional diferential system. Thestability of the system is the key to study synchronization of the system so thatthe research on synchronization of fractional-order complex network has developedslowly.The main contribution of this paper is as follows:Adding controllers to all nodes of the networks not only can lead to excessivecontrol cost, but also is impractical because a real-world complex network usuallyconsists of a large number of nodes. So pinning control method is employed tocontrol the networks. The pinned nodes are chosen usually based on the highdegree pinning scheme or randomly pinning scheme. Since the nodes with highdegree may not be the center of the networks, a new pinning scheme to selectpinned nodes is proposed based on the closeness centrality. Furthermore, it is foundthat for some networks, the networks synchronizing capacity is greatly improvedbased on the closeness centrality scheme.Adaptive control can change control parameters during the working system,so it is widely applied in the design of the controller. This paper has obtained fractional-order complex networks with the adaptive coupling law based on theadaptive control method, and investigated the adaptive synchronization and gen-eralized projective synchronization of fractional-order complex networks.The results on robust stability and stabilization of uncertain fractional-ordersystems are mainly limited to fractional linear systems. Few results on fractional-order nonlinear systems are reported in the literature. The robust stability andsynchronization is investigated based on the LMI method. Furthermore, the ro-bust synchronization of a class of uncertain fractional-order complex networks isinvestigated. In addition, synchronization of uncertain fractional-order complexnetworks is investigated based on the adaptive control method. The advantagesand disadvantages of two control methods are compared.Impulsive control has received much attention of many researchers as it hasadvantages of accomplish easily, lower cost. It is widely used to control chaot-ic systems and complex networks. The studies on impulsive synchronization offractional-order systems have been published, but the methods is mainly approxi-mation of integer-order impulsive control systems based on the comparison methodof impulsive systems. Few theoretical studies on impulsive synchronization offractional-order systems are reported in the literature. This paper studies thestability of impulsive fractional-order systems. A new synchronization criteria offractional-order chaotic systems is proposed based on the stability theory of im-pulsive fractional-order systems. Furthermore, this paper investigates comparisonmethod of fractional-order impulsive systems, and establish stability criteria ofsystem based on the Lyapunov functions. Impulsive synchronization of a class offractional-order systems and uncertain fractional-order systems is investigated.Real complex networks are more likely to be time-varying networks, and itis realistic to consider the sudden changing on the nodes’ states. The equivalentVolterra integral equation and generalized Gronwell-Bellman’s inequality are em-ployed to study impulsive synchronization of fractional-order complex networkswith time-varying coupling. Moreover, the synchronization a class of impulsivefractional-order complex networks is studied. The results show that the impul-sive fractional-order complex networks can achieve synchronization under a singlecontroller if the pulse information of network nodes can be obtained.The dissertations is supported by the National Natural Science Foundation ofChina (No.11071060,11201168).