Study On Synchronization Of Complex Networks

In recent years, the synchronization of the chaotic systems has been investigated extensively, many controlling and synchronization methods, which have become the highlight of nonlinear dynamical domain, have been proposed. Currently, many people have studied the complex networks synchronization from physics to biology even social science on the basis of chaotic theory. These investigations have theoretical and practical significance in real complex networks.In this paper, we focus on synchronization of the nearest-neighboring coupled dynamical networks, synchronization of the globally coupled networks, and synchronization of the star-coupled networks, and analysis of the coupled dynamical networks synchronization. The first two chapters review chaotic controlling methods and development of complex networks in detail. And the two chapters are the foundation of our works as following chapters (form the third chapter to the sixth chapter).We investigate synchronization of regular coupled networks in the third, forth and fifth chapters. The third chapter mainly studies the synchronization of the nearest neighboring coupled dynamical networks and obtains the conditions that achieve synchronization. Finally the curves of relationship between the largest nonzero eigenvalue A, and the number of nodes are plotted. We find that 1 will increase continuously with the development of the network, till it approaches to zero in the end, that is the network will lose synchronization. We also find that coupled strength has a great effect on synchronization. At the same time, seen from the graph, we find that it is difficult to achieve synchronization with the development of the networks.The forth chapter emphasizes synchronization of globally coupled networks. We take Lorenz oscillator as an example and regard its chaotic attractor as a node of the network. Firstly, through networks (with few nodes) study, we apply it to the networks with many nodes and get that networks, coupled with two random nodes, will synchronize on condition that the coupled strength is fixed.The fifth chapter analyses synchronization of the star-coupled networks. We also take Lorenz oscillator as an example and discuss its synchronization in the star coupled networks. We do theoretical analysis and achieve conditions that reach synchronization in the star-coupled networks. Meanwhile, we also find the star coupled networks can synchronize as long as coupled strength, which is not concerned with the scale of the networks, is slightly greater than the critical coupled strength.The sixth chapter analyses the dynamics of the coupled networks synchronization, still taking the Lorenz oscillator as an example and considering its chaotic attractor as networks nodes. Through the studies of coupled networks synchronization of single and multiple variables with two or three nodes, as well as the simulations of their Lyapunov exponents, we found that multi-variable coupled networks is easy to synchronize than single-variable coupled networks, because the critical coupled strength of multiple variables coupled networks is smaller by far than that of single variable coupled networks.