Synchronization And Topological Structure Indentification Of Complex Dynamical Networks

Synchronization is related to various research topics in natural and man-made systems, such as animal groups, power grids, sensor networks, cellular neural networks, and so on. In recent years, more and more researchers have become to be interested in the synchronization behavior in complex dynamical networks, since the "scale-free" and "small-world" properties of real-world net-works were discovered in the end of 20th century. Scientists jointly consider topological structures of complex networks and their dynamical behaviors, then a new research discipline comes out-complex dynamical networks, which is an efficient tool to study different network topologies and their dynamics character-istic.Based on several synchronization criteria of four kinds of complex dynamical networks, the thesis has studied on the following topics:how to identify uncertain topological structures of complex dynamical networks via time series of nodes' dynamics; how topological structures of complex networks affect their general-ized synchronization behaviors; how the impulsive coupling strategy affects the synchronizability of discrete dynamical networks.The thesis is composed of six chapters. In Chapter 1, we will introduce some fundamental knowledge and key concepts of nonlinear dynamical systems and complex networks. And we will also summarize some present work on the synchronization of complex dynamical networks which are related to the topic of the thesis. Then, main results and ideas of our work will be given in Chapter 2-Chapter 5. In Chapter 6, some outlooks of our further research work are discussed. The main contents and innovation points are summarized as follows.1) Topology identification and parameter identification are challenging ques-tions in complex networks. By the synchronization-based estimation theory, iden-tification of the topological structure and unknown parameters of a complex dy-namical network with nonidentical nodes/identical nodes is carefully studied. Based on rigorously theoretical analysis, it points out that the so-called linear in-dependency of drive signals is essential for an effective and correct estimation of the topological structure and unknown parameters. Moreover, how the coupling strength affects the topological identification is analyzed through the method of Master Stability Function. And one key factor-nodes' dissimilarity-that de-termine the efficiency of the proposed adaptive control approach is then further investigated.2) Time delay often appears in the state variables or coupling coefficients of various practical complex networks. The paper initiates a novel approach for simultaneously identifying the topological structure and unknown parameters of uncertain general complex networks with time delay. In particular, this method is also effective for uncertain delayed complex dynamical networks with differ-ent node dynamics. Moreover, the proposed method can be easily extended to monitor the on-line evolution of network topological structure.3) Generalized synchronization, which is weaker than complete synchroniza-tion, plays an important role in many networked systems. The proposed general-ized synchronization strategy is to adjust adaptively a node's coupling strength based on the node's local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. We investigates generalized synchronization in three typical classes of complex dy-namical networks:scale-free networks, small-world networks, and interpolating networks. It is demonstrated that the coupling strengths are affected by topolo-gies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we dis-cuss in detail how a network's degree of heterogeneity affects its generalization synchronization behavior.4) Over the last decade, impulsive control and synchronization of continu-ous dynamical networks has been extensively investigated in various disciplines. However, impulsive control and synchronization of discrete dynamical networks has only lightly been covered. In this paper, a novel model is proposed for the synchronization of a class of discrete dynamical networks through impulsive cou-plings. Moreover, the global and local stability of synchronization manifold are then further studied. As a result, several effective synchronization criteria are attained, which describe conditions for the impulsively coupling matrix, coupling strengths, and the impulsive intervals, as the nodes' dynamics is given.