Research Of Synchronization Of Dynamical Systems On Complex Networks

Complex networks have been considered as an important approach for describing and understanding complex systems. Any complex system is composed of interacted individuals, which can be naturally represented by graphs with individuals denoted by nodes and interactions by links. From this point of view, complex networks are ubiquitous, ranging from nature and biological system to society. The well-known and extensively studied networks include Internet, World Wide Web, the power grid, airport networks, protein-protein interaction networks, collaboration networks, epidemic spreading networks, etc. In the past few years, we have witnessed a great devotion by scientists to understand the underlying mechanisms of complex networks together with dynamical process taking place on them.The ultimate goal of studying complex networks is to understand how topological properties affect the dynamical processes taking place on them. The synchronization of dynamical systems on networks is an important issue. The several stages it has gone through include the synchronization phenomena on complex networks and the synchronization stability analysis, the investigation of the relationship between network ingredients and network synchronizability, the method that improve the network synchronizability and the application of network synchronization. The works we do in this field are following.With the help of random interchanging edges method, we give the precise relationship between average distance, degree distribution and network synchronizability. We conclude that shorter average distance and homogeneity solely will lead to better synchronizability, but only their combination could make the network easy to synchronize.We bring forward a structural perturbation method to enhance network synchronization. We divide each node with heavy load into several nodes, thus make the several nodes take on the work that did by the old one. Although the operation on network stucture is easy and makes the network almost unchanged, the network synchronizabily is enhanced remarkably.By making use of the hub nodes, we bring forward another method to enhance network synchronization. During the evolution, the nodes with larger degree have much stronger influence on nodes with smaller degree, and make the latter synchronize to the former much quicker, thus induce the whole network synchronize much easier. A coupling method that can make the network have the best synchronizability is brought forward. Make each node only affected by the neighbor with the largest degree, and the node in the network with the largest degree is not coupled by any others, the network will synchronize with proper coupling strength, no matter what oscillators the nodes are.We also investigate the synchronization on networks with different sort of communities, and find that when the community structure is strong, only the oscillators on the same community synchronize, with the weakening of the community strength, oscillators on the same sort of communities will synchronize independently, only when the community structure is not evident that all the oscillators on the network can synchronize.