| Complex network is a new subject on the research of complex systems, and has got widely concerns for researchers from different fields. Synchronization is an important content of the complex network dynamics, and has important implications to the real-world systems. Therefore, the research on synchronization of complex networks is quite promising. In this thesis, employing node dynamics as logistic mapping, we explore the synchronization process by studying regular networks with non-local links, including the realization and transition of partial phase synchronization, partial synchronization and complete synchronization. The studies are mainly based on numerical simulations, while compared with mathematical analysis.The MSF (Master Stability Function) is firstly reviewed in the research of complete synchronization. MSF determines the scope of coupling strength that ensures the stable synchronization of networks, by calculating the eigenvalues of network coupling Laplace matrix and analyzing threshold conditions of stability of networks. We demonstrate the correctness of MSF by quantitative analysis of small size networks.Partial synchronization (PaS) is a kind of synchronization phenomenon (or synchronous cluster) that some topology-symmetrical nodes in a network appear to be synchronous, when the coupling strength of a network is within a certain scope. The topology and connecting ways has direct impact on the appearance of PaS. We have reviewed judging criteria of PaS and used it in the theoretical analysis of small size networks. We have also verified it using quantitative simulations.In this thesis, we find an interesting phenomenon in analyzing the transition process of small size networks from desynchronization state to synchronization:the state of partial phase synchronization. Within a certain coupling strength, based on statistic analysis of long-time evolution, these exists static inverse-phase among symmetrical nodes. We try to explore the mechanism behind this phenomenon. We also analyze the quantitative relationship between this process and the coupling strength. Besides, we try to find the role of partial phase synchronization behavior in the transition to global network synchronization.It must be noticed that the research in this thesis is just a special example of the research on synchronization process of complex systems. Hence, much work should be done further. Although our research on partial phase synchronization is also realized in small size networks, it is our hope that these results could be helpful to the study of large networks with more complex topology structure. In addition, we still can not judge the occurrence partial phase synchronization through theoretical computation and we can not compute threshold conditions of partial phase synchronization. Finally, our quantitative research is kind of tough. We hope to address these in future studies. |
