Synchronization Analysis And Computation Of Several Fractional Complex Networks

“I think the next century will be the century of complexity,” famous physicist Stephen Hawking says.Complex networks,as a new branch of complexity science,exist widely in many areas including biology,physics,society,computer,engineering and so on.Synchronization is widespread in a variety of complex networks.It is not only a typical collective behavior,but also a most important dynamics character in complex networks.Synchronization and control of complex networks can explain and solve many problems in nature and society.In the recent decades,complex networks have been generalized to fractional case.On one hand,fractional complex networks can excellently describe the memory and hereditary properties of various model.On the other hand,fractional complex networks increase one degree of freedom by the order of fractional derivative,which displays the rich dynamic behaviors.So,synchronization and control of fractional complex networks have more wider applications.This dissertation aims at synchronization of fractional complex networks,pays attention to the models including fractional fuzzy neural networks,fractional complex networks with time delays,fractional Takagi-Sugeno(T-S)fuzzy complex networks and general fractional complex networks.Theoretical analysis is used to confirm the rationality of adaptive controller,pinning controller,impulsive controller,pinning impulsive controller,and so on.All involved numerical simulations verify the effectiveness of the proposed method.These are the topics we cover.The first chapter introduces the background and motivations of complex networks and fractional complex networks synchronization,and reviews the recent research advances.The main contributions and innovations of this thesis are given in the end of this chapter.In Chapter 2,to enhance the coupling strength dynamically,fuzzy theory and interactions are introduced,and the fractional fuzzy neural networks with interactions are obtained.The uniqueness of equilibrium point and boundedness of solution are proven by properties of fractional derivatives and contraction mapping principle.By the fuzzy theory and Lyapunov principle of fractional differential equations,some new stability criteria are obtained via adaptive control.Finally we give some numerical examples to show the effectiveness of the obtained results.Our findings can provide insights into the dynamics of fractional coupled networks with diverse connections.In Chapter 3,the pinning synchronization between two fractional complex dynamical networks with nonlinear coupling,time delays and external disturbances is investigated.A asymptotic stability theorem for the fractional system with time delays is obtained.A class of novel controllers are designed for the pinning synchronization of fractional complex networks with disturbances.By using this technique,fractional calculus theory and linear matrix inequalities,all nodes of the fractional complex networks reach complete synchronization.In the above framework,the coupling-configuration matrix and the inner-coupling matrix are not necessarily symmetric.All involved numerical simulations verify the effectiveness of the proposed scheme.Chapter 4 focuses on impulsive synchronization of fractional T-S fuzzy complex networks.A novel comparison principle is built for the fractional impulsive system which could be used to make comparison between solutions of general fractional impulsive systems.The T-S fuzzy system is adopted as dynamic system of complex networks,which has advantage of local linearity.Then a impulsive synchronization criterion is established for the fractional T-S fuzzy complex networks by utilizing the comparison principle and fractional Gronwall-Bellman inequality.Numerical examples are also presented to show the effectiveness of the criterion.In Chapter 5,a class of fractional complex dynamical networks are synchronized via pinning impulsive control.Combining both the advantages of pinning control and impulsive control,a targeted pinning impulsive control method for fractional networks is proposed which can solve the problem of the choice of pinning nodes to some extent.Then the synchronization criterion is obtained for general fractional complex networks by using the derived comparison principle and Volterra integral form of fractional system.Examples are given to illustrate the correctness of theoretical results and effectiveness of the method.Finally,a summary of the full dissertation and possible research directions are included in the last chapter.