Study On Synchronization Of Complex Dynamical Networks With Coupling Delays

In the last decade, complex networks have attracted much attention fromdisciplines as diverse as physics, chemistry, biology, social science, telecommuni-cation and engineering. In this dissertation, stability theory of time delay system,control theory and matrix theory are introduced into the study of complex net-works. Modeling and synchronization analysis on complex networks have beenproposed, which are extremely significant in both theoretical development andpractical applications.Time delays are ubiquitous in almost every situation such as informationtransmission, chemical reactions and tra?c congestions. Hence it is inevitableto ignore the e?ect of delays in the analysis process. In the previous work oncomplex networks with coupling delay, dynamical node is modeled as a retardedfunctional di?erential equation interacting with other nodes, which is a simplifieddescription of real systems. In this dissertation, a generalized complex dynam-ical networks model with mixed coupling delay is proposed. It is an extensionof the model without time delay and that with the retarded delay. By studyingthe synchronization properties of complex networks, however, the emergence phe-nomena presented in real world networks are illustrated more clearly; and on theother hand, these theoretical results can be used, for example, to design of realnetworks to achieve excellent synchronous performance.The main contents and originalities of this dissertation are summarized asfollows:1. A generalized complex dynamical networks model with mixed coupling de-lays is presented. Compared with previous works, this model is more generalbecause it includes both retarded-type coupling delays and neutral-type ones.Namely, the dynamical complex networks are modeled into functional dif-ferential equations involving the past state variables and their change rateinformation. By some transformation, the synchronization problem of the complex networks is transferred equally into the asymptotical stability prob-lem of a group of uncorrelated neutral delay functional di?erential equations.Furthermore, local synchronization and global synchronization of complexnetworks are analyzed. Some less conservative su?cient conditions for bothdelay-independent and delay-dependent synchronization criteria are derivedin the form of linear matrix inequalities based on the free weighting matrixstrategy.2. Exponential estimates and su?cient conditions for the exponential synchro-nization of complex dynamical networks with bounded time-varying delaysare given in terms of Linear Matrix Inequalities (LMI). A generalized com-plex networks model involving both neutral delays and retarded ones is pre-sented. Exponential synchronization problem of the complex networks isconverted equally into the exponential stability problem of a group of un-correlated delay functional di?erential equations with mixed time-varyingdelays. By utilizing the free weighting matrix technique, a less conservativedelay-dependent synchronization criterion is derived.3. Synchronization of a class of complex dynamical networks with time-varyingcoupling delays is investigated. Our strategy is to apply impulsive controlto enhance the synchronizability of the complex networks. Su?cient con-ditions on global exponential synchronization are derived by introducing acomparison system and estimating the corresponding Cauchy matrix. Animpulsive controller is explicitly designed not only to achieve synchronizeddynamics for complex networks, but to ensure that the synchronous errorsconverge with a given decay rate. Compared with the other control meth-ods in the literature, our control scheme is e?cient and practical in dealingwith synchronization problems particularly in the mass data transmissioncircumstances.4. A scheme of secure communication based on synchronization in complexchaotic networks is proposed. In communication network, each endpoint isregarded as the node of complex networks. Transmitter end is composed ofa chaotic system and encryption rule, and receiver end contains the identical chaotic system and decryption rule. Synchronization of complex networks isviewed as a specified version of a state observer design problem. If complexchaotic networks achieve synchronization status, then the transmitting signaland key are recovered at the receiver ends. Thus synchronization acts as animportant role in the realization of practical secure communication.