Generalized Outer Synchronization Of Two-layer Complex Dynamical Networks

A large number of complex systems in the real world can be described by the net-work, such as World Wide Web, power grid networks, transport networks, neural net-works, communication networks, and so on. Even though these networks look different, more and more researches show that they have striking similarities. After the small-world and scale-free characteristics were found in lots of real networks, the researchers jointly consider topological structures of complex networks and their dynamical behav-iors, then a new research discipline comes out-complex dynamical networks, which is an efficient tool to study different network topologies and their dynamics characteristic.In recent years, the group behavior of complex networks have been paid much attention to, with special emphasis on the synchronization of one or two complex net-works. Synchronization is a widespread natural phenomenon in real world, such as birds flocking, fish schooling and fireflies flashing, and so on, and its research has attracted wide interest. Therefore, these become the research themes of this thesis.The thesis is consisted of five chapters. In chapter 1, we will briefly introduce some fundamental knowledge of complex networks, and we also will summarize some present works on network synchronization related 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 listed as follows:(1) In the real world, complex network often has some uncertain information, such as unknown parameters in the node dynamics. Undoubtedly, the case that the node dynamics parameters are unknown is challenging question in complex networks. In Chapter 2, the generalized outer synchronization (GOS) between two different complex dynamical networks with unknown parameters is discussed. That is to say, when the exact functional relations between the two complex networks are previously known, a sufficient criterion for GOS is derived based on Barbalat's lemma. But, the functional relations between two complex networks are often unknown. Therefore, In Chapter 3, the generalized outer synchronization between two complex networks with unknown functional relations is obtained, the auxiliary system method is employed and a sufficient criterion for GOS is derived.(2) Beyond the previous result of pinning control of one network, the general-ized synchronization (GS) in two-layer complex networks with unidirectional inter-layer coupling via pinning control is investigated. In Chapter 4, a sufficient condition for achieving GOS via pinning control according to the auxiliary-system approach is pre-sented. Numerical simulations are further provided to illustrate the correctness of the theoretical results. It is also revealed that the least number of pinned nodes needed for achieving GS decreases with the increasing density of the response layer.(3) For two-layer complex networks with identical number of nodes, each node in one layer is connected to a counterpart in the other layer. In Chapter 4, we assume that nodes in the drive layer are sorted from large to small degrees and consider three different pinning strategies for the response layer. It is found that the least number of pinned nodes needed for achieving GS is relevant to the intra-layer coupling strength of the response layer. That is to say, when the intra-layer coupling strength of the response network is large, nodes with larger degrees should be selected to pin first for the purpose of achieving GS. However, when the coupling strength is small, it is more preferable to pin nodes with smaller degrees. This work provides engineers with a convenient approach to realize harmonious coexistence of various complex systems, which can further facilitate the selection of pinned systems and reduce control cost.(4)As is well known, complex networks in reality are usually interconnected. To describe interconnected networks, a new kind of network named multiplex networks is proposed, and multi-layer network has become one of the frontier problems of complex networks. In Chapter 5, based on a class of continuously variable network models which are heterogeneously distributed in between scale-free networks and random networks. We have studied the synchronizability of two-layer networks with heterogeneous degree distribution. If the synchronous region is unbounded, the synchronizability of the two-layer network keeps enhancing, but it has a threshold. If the synchronous region is bounded, the synchronizability of the two-layer network depends on the heterogeneity of-?- the degree distribution, inter-layer coupling strength, and intra-layer coupling strength.