Research: Stability And Synchronization Of The Dynamic Behavior Of Complex Networks

In this dissertation,we mainly discuss two dynamical behaviors of complex networks: the global stability of the equilibrium and the synchronization phenomenon.The global stability and robustness of the equilibrium in complex networks is firstly investigated.We use the neural networks as the model,then give a rigorous mathematical analysis on the role of the unbounded time-varying delays,parameter uncertainties and stochastic disturbances,and some corresponding criteria based on linear matrix inequality(LMIs) approach are also presented.Secondly,the synchronization phenomenon of complex networks is the other issue of this thesis.The global stability of the equilibrium can be regarded as the convergence problem between the solution and the equilibrium,that is to say, the solution from any initial values will finally converge to the equilibrium;while the synchronization issue only concerns the identical or approach between any two nodes' final dynamical behavior,i.e.,there is no requirement on the final state of the solution,therefore,the synchronization issue is under a milder condition of complex networks.In the following discussion,we use the linearly coupled ordinary differential equations as the complex network model,and based on the geometrical analysis of synchronization manifold,we investigate the impact on the network synchronization of the dynamics of the uncoupled nodes,the network topology(the coupling matrix),the coupling strength,and the time delays,respectively.The control problem of network synchronization is also discussed.Moreover,we also investigate the synchronization problem in nonlinearly coupled complex networks,and analyze the role of the dynamics of the uncoupled nodes,the network topology(the coupling matrix),nonlinear coupling functions,etc.Finally,compared with previous results, a conjecture is also presented.