Studies On The Periodic Solutions Of The Complex Networks

With the progress of times and the development of science, network enter and enrich human life gradually, network has involved in different fields of the real world, such as social networks, transportation networks, biological networks, electric power networks, ect. However, most of the networks in real life are not so regular, They have more complex features, people have had understanding of networks through the process from regular network and random network to complex network. Where the small-world networks and the scale-free networks is typical. In recent years, the research of complex networks has attracted a much attention at home and abroad, people reveal the regulation of the realistic network system and guide the actual engineering practice problems via the research of the complex network model.This paper firstly reviewed the development process of network science. The complex networks in real life and some basic parameters of network topology, such as the degree distribution, average path length and clustering coefficient. Then I introduce some models of complex network:the rules of the network model, random network model, small world network model, scale-free network model, the extension model of BA scale-free network model, Also introduced the classical model of small-world networks. For WS network model and NW network model, matlab simulation analysis of their network statistics, carries on the detailed comparison and research.The main work of this paper are:to explore periodic solutions and rotating waves of the two coupled ring networks, the mathematical description of the network is: Where x=(x1,x2,...,x6)T is the state of the node, k>0, a>0are the connection strength between nodes. H is a connection matrix between nodes, defined as: In the model, the ring network is formed by connecting the nods1,2,3and the nods4,5,6. The two rings are connected by the node3and4, where each node has an inhibition connection of self-feedback. By calculation, we know a rotating wave exists in the first ring, while the nodes vibration phase of the second ring differs by a constant. However, each node vibration phase differs by a equivalent constant. As the nodes vibration amplitude is not same in the second ring, we called it "class rotating waves".Then we also study a coupled network model with two nonlinear oscillators, the motion equation is as follows:Where k,k arc constant, a s mall amount. We knew the approximate solution with perturbation method, we found that the periodic solution of the above system is consistent with the numerical solution.