Synchronization Control Of Some Classes Of Complex Dynamical Networks

Complex system is the research object of complexity science. As a new branch for complex systems, complex networks have been used widely in different scientific fields, such as sociology, biological sciences, computer sciences, physics, engineering and so on, which have become a brilliant research topic of complexity science field. More and more domestic and foreign scholars are attracted by it. Synchronization is widespread in the various kinds of complicated network systems, and which is a typical collective behavior in complex networks and one of the most important dynamic characteristics of complex networks as well. Synchronization control of complex dynamical networks is a key link of research and application of complex networks and has a very high theoretical value and practical application value, which is an important topic in the field of control and becomes one of frontier science and technology strategic issues in the 21st century. Synchronization control method of complex dynamical networks is mainly divided into two kinds: one is to improve the network synchronization capability by changing the properties of the network itself, such as topology structure, coupling strength etc; another is to use control method which is a representative of control theory, mainly including drive-response synchronization method, variable feedback control method, adaptive control method, impulse control method, pinning control method, slide control method.This paper studies the synchronization control of complex dynamical networks, including inner synchronization and outer synchronization. Some network models are mainly studied, such as weighted complex networks with multi-links and nonlinear coupling, nonlinearly coupled complex net-work with stochastic perturbations, bipartite dynamic networks with distri- buted delays and nonlinear derivative coupled, delayed complex networks with uncertain system parameters and unknown topological structure etc. Based on stability theory, stochastic differential equation theory, matrix theory, control theory and graph theory, we use variable feedback control, adaptive control, pinning control and impulsive control methods to study the synchronization control problems of these complex dynamics networks, and obtain some criteria to realize completely synchronization, projection synchronization, generalized linear synchronization and generalized syn-chronization. For uncertain network systems, when they meet some conditions, we can also identify system parameters and the topology structure of the networks. Numerical simulations are given to illustrate the effectiveness of the results.This paper is composed of eight chapters. In chapter 1, we introduced the background briefly and study progress of complex networks and synchronization control, the basic concept, symbols, and some lemmas were also presented. Main results and ideas were given from Chapter 2 to Chapter 7. Some existing problems as well as the future research were pointed out in Chapter 8. The main contents are summarized as follows.1. Put forward the weighted complex networks with multi-links and nonlinear coupling, which single node contains the delay. Based on an idea of network split, networks can be splited into several sub-networks accor-ding to delays, and this model is built, which is very consistent with the actual situation. Synchronization of this kind network is realized via using linear feedback control and adaptive control, respectively.2. Synchronization is investigated for a class of nonlinearly coupled complex dynamical network with stochastic perturbations, which can be certain or uncertain case containing uncertain the network system parame- ters and unknown network topology structure. Some synchronization criteria are obtained by delayed stochastic differential equation theory, pinning control method and variable feedback control method.3. Using the idea of bipartite graph in theory of graph, Hopfield neural network as the nodes’dynamics and considering distributed delays and nonlinear derivative coupling, a bipartite dynamic network model is built, and the weight matrix is not necessarily symmetric or irreducible. Through special nonlinear controllers and adaptive laws, we can get some new synchronization criteria, and prove them by Barbalat lemma.4. Projective synchronization in a time-delayed drive-response dyna-mical network is considered, where the nodes are not necessarily partially linear and the scale factors may be different from each other. Based on stability theorem and impulsive control method, we derive some synchroni-zation criteria for the projective synchronization, in which there need no extra controllers with limitation conditions and simple controllers can be added to achieved synchronization without limitations.5. Linear generalized synchronization is realized between two diffe-rent delayed complex networks with uncertain system parameters by adap-tive control method, and the parameters can also be identified. In addition, we can also construct a response network to realize linear generalized synchronization with the drive network and a given linear mapping, and the unknown parameters and topology structure of the network system can be identified, too.6. Generalized synchronization between two complex networks with nonlinear coupling and time-varying delay is investigated. The novel adaptive schemes of constructing controller response network are proposed to realize generalized synchronization with a given reversible mapping and a drive network with unknown topology structure. We can identify the to-pology structure, too.