Analysis Of Synchronization Stability And Spreading Behavior In Complex Dynamical Networks

In this dissertation, we first introduce a class of double-layer star-shaped complex networks with relatively regular structure and analyze its synchronization stability. By using the linear stability analysis, we obtain the coupling strength regions for complete synchronization of these networks. These regions are shown to be dependent not only on the characteristics of local dynamics of individual nodes, but also generally on the topological structure of networks. At the same time, we find that deleting some links in a double-layer star-shaped complex network can enlarge its synchronization region to make the network synchronize more easily.Next, based on a discrete dynamical network, we present an analytical discussion to the relation between transverse stability and contraction stability of its synchronization, and prove that the contraction stability is stronger than the transverse stability. Moreover, by applying the partial contraction principle to a discrete dynamical network, a sufficient condition for achieving its synchronization is obtained.Then, by constructing and applying certain pinning control schemes and adaptive laws, we study the cluster synchronization of complex dynamical networks. On one hand, we consider the cluster synchronization in community networks by using the pinning control which is designed according to the topological structure of networks. Some conditions for local and global stability of cluster synchronization are obtained. On the other hand, we address the stability of adaptive cluster synchronization of a generalized linearly coupled network with time delay. Both the pinning control scheme and adaptive technique are adopted to realize the cluster synchronization of this network. Some sufficient conditions to ensure the global and asymptotical stability of cluster synchronization are obtained by using the invariance principle of functional differential equations and linear matrix inequality.And then, we investigate the quantitative and statistical relation between synchro-nizability and structural properties of complex networks. Based on the BA scale-free and WS small-world networks, this relation can be exposed by performing the linear regression analysis. It is shown that the synchronizability in these networks can be quantified by the eccentricity, variance of the degree distribution and clustering coefficient. The results indicate that both the eccentricity and clustering coefficient are positively linearly correlated with synchronizability, while the variance is negatively linearly correlated. Re- lying on these results, we further design some structure-evolving strategies to change the synchronizability of complex networks.Finally, we study the dynamical behavior of a new susceptible/infective/removed (SIR) model with effective contacts in complex networks, and also give a novel research to the relation between synchronization and epidemic spreading in complex networks by modeling and theoretical investigation, which can be considered as the integrated work of synchronization phenomenon and spreading behavior. By constructing some hybrid models between complex dynamical networks and epidemic networks and theoretical investigation, we obtain some results to combine successfully these two different dynamical behaviors in complex networks. With this work, it can also provide a new insight and method to explore the synchronization and epidemic behaviors in complex networks.In conclusion, these works will not only deepen and expand theoretically our understanding in the synchronization and epidemics in complex networks, but also provide theoretical and numerical criteria for analyzing and manipulating the real-world complex networks.Chapter 8 concludes this dissertation, and lists some further research problems.