Research On Synchronization Problems Of Complex Networks

Recent years, some important progresses have been made in the study of complex systems from the view point of complex networks. The system is regarded as a whole, and the effect of the interaction between individuals on the whole system behavior is considered. The research approach breaks the limitations of reductionism so the overall system behavior can be better predicted. Synchronization behavior of complex dynamical networks can well explain many common phenomena in the natural world and the field of engineering. However, in the earlier study of synchronization in complex networks, most work focused on analysis of the rules topology network synchronization features, such as lattices and fully-connected graphs. With the in-depth understanding of the structural properties of real complex networks, in particular, the finding of the small-world and scale-free properties, more and more attention has been focused on the effects of network structure on network dynamical behavior, including synchronization behavior. Research on synchronization phenomena in networks with complex topologies has become an important topic in the area of complex network research. This dissertation discusses complex dynamic network synchronization issues, concerning enhancing the ability of synchronization of complex network, synchronization criteria of complex network, as well as the control to synchronization of complex network.The main contents and achievements of this dissertation include:For the problem of enhancing synchronizability of complex network, firstly, we present a weighted method based upon the node betweenness, and discover that synchronizability of the complex networks can be enhanced by the weighted method. And we demonstrate the validity of this method by applying this weighted method to two classes of networks with high homogeneous and heterogeneous degree distribution. The optimal synchronable condition can be obtained by a tunable parameter. Secondly, by defining the characteristic path length of every node, we introduce another weighted method and analyse the synchronizability effect of small world network with different number K of coupled edge, different link probability p, and different network scales. The simulation results display this weighted model significantly improved synchronizability of the small-world network. Finally, a rewiring algorithm is presented to enhance synchronizability of the complex dynamical network. Utilizing the algorithm for an unweighted and undirected network, one can obtain an optimized network with the same number of nodes and edges but that has enhanced synhcronizability.The local and global synchronization problems of uncertain complex dynamical networks are studied. Firstly, novel synchronization critera are presented, and linear controllers are designed via the Lyapunov stability theory. Especially, it is shown that the controlled networks are exponentially synchronized with a given convergence rate. Then, several local and global synchronization controllers are designed using sliding mode variable structure method, and stability have been proved by using Lyapunov stability theory. Finally, the famous Lorenz system is stimulated as the dynamical nodes of complex networks, and the results show the effectiveness of our designed controllers.The global exponential synchronization problems of uncertain complex dynamical networks with delay coupling are considered. The network model considered is general dynamical delay networks with unknown network structure and unknown coupling functions but bounded. Novel adaptive controllers and linear controllers are designed via the Lyapunov stability theory. Especially, it is shown that the controlled networks are globally exponentially synchronized with a given convergence rate. An example of typical dynamical network, having the Lorenz system at each node, has been used to demonstrate and verify the novel design proposed. And, the numerical simulation results show the effectiveness of proposed synchronization approaches.For the characteristics of complex network synchronization criteria is the N-1 time-varying linear system, several new criteria for states synchronization stability of general complex dynamical network are proposed based on the recently developed parallel differential (PD) eigenevalue and series differential (SD) eigenevalue theory. These two criteria need not to verify N-1 the stability of time-varying system, and as long as the verification of the real part of the PD spectrum positive and bounded eigenvectors, or verify the real part of the SD spectrum and boundedness. Moreover, the effectiveness of these synchronization criteria has been demonstrated by theory proof and numerical simulations. States in complex networks systems are difficult to be observed or measured, even the states can not be observed or measured at all; and the synchronization of only a part of states can satisfy the practical requirements, namely it is unnecessary to achieve synchronization for all states. For these phenomena, the problem of output synchronization of complex network is discussed. First, the definition of output synchronization of complex network is proposed. Then, we consider two cases of the output matrix:singular and nonsingular matrix. And the stability of the output synchronization of complex network is analyzed. Some sufficient conditions for output synchronization are shown. Finally an output synchronization example is provided to understand the significant role of output synchronization, while the states synchronization can not be achieved.