Synchronization Of Nonlinearly-coupled Networks With Markov Jump And Brownian Motion

Complex networks exist in nature and society widely. For example, the Internet, the electric power networks, the Transportation network,the World Wide Web, Human neural network, social networks, and so on. A typical complex networks is made up of a large set of nodes and the edges that are interconnected by a set of notes. The nodes represent individuals of the networks, and the edges represent the connection between individuals. In the neural network, we can treat the neural cells as nodes and the neural fibers as edges. In the field of computer network, the computer was treated as nodes, and the communication media as edges, such as cable, twisted pair, coaxial-cable.Therefore, the research on complex networks have very important significance. At present, the studies of complex networks have focused on the following directions: the topology of the network and topology properties,the communication mechanism of networks, synchronization and control. Synchronization has been an important research direction of the complex networks, including global synchronization and local synchronization. Synchronization is that some of similar subsystems will achieve the same state in the end by the influences between the subsystems, although their initial state are different. In this paper, we attempt to study the problem of synchronization of two classification of nonlinearly-coupled complex networks The first one is asymptotically synchronization of nonlinearly-coupled complex networks with markov jump topology and time-varying delay; and the second is exponential synchronization of nonlinearly-coupled complex networks with markov jump topology and Brownian motion. Main works of this paper are as follows:(1) Firstly, we made a simple introduction of complex networks and the synchronization of complex networks, and its research status;(2) Secondly, we investigated the asymptotically synchronization of nonlinearly-coupled complex networks with markov jump topology and time-varying delay. The coupling matrix can jump between finite state by a markovian. Meanwhile, the coupling strength is modeled as a random variable. Based on the matrix theory, the Lyapunov stability theory, The nonlinear theory, and LMI approach, some synchronization criterias are derived to ensure the globally asymptotically synchronization in mean square Numerical simulations by Matlab are also given to confirm the theoretical results;(3) we studied the exponential synchronization of nonlinearly-coupled complex networks with markov jump topology and Brownian motion on account of control policy. We assumes that the complex networks are suffering from stochastic perturbations, it realized the synchronization among the whole network by adding a pinning controller and controlling the the nodes.We use stochastic process theory, Lyapunov stability theory, The nonlinear theory,LMI approach and some other theory, analyze the synchronization conditions of reaching mean square exponentil, and offer the related Numerical simulations to verify our theoretical results.