Stochastic Stability And Synchronization Of Complex Networks

The deterministic stability and synchronization analyses of complex networks have got remarkable achievements in the last decade. However, the stochastic perturbations in systems may significantly change the quantitative behaviors of complex networks. and even may completely change the qualitative behaviors. The present dissertation is concentrated on the stochastic stability and stochastic synchronization of complex networks. whose node is nonlinear dynamical systems subjected to stochastic excitation.In the first part, the stochastic stability of complex networks is studied. Firstly, by linearization and variable transformation, the problem of stochastic stability of complex networks can be reduced to the analysis of the largest Lyapunov exponent of each independent linear subsystem. The criterion for the local stochastic stability of complex networks is given by using the method of largest Lyapunov exponent. In some special cases that the system parameter satisfying certain conditions. the criterion can be simplified to approximate formulas. And it is found that the stochastic stability of complex networks is determined by the subsystem largest Lyapunov exponents corresponding to the maximum and minimum eigenvalues of complex network matrix. The numerical simulation results are given to demonstrate the validity and accuracy of the theoretical analysis. Secondly, the proposed procedure is extended to analyze the local stochastic stability of complex networks under pinning control. The stability criteria for two cases, i.e, positive damping coefficient and negative damping coefficient, are obtained, respectively. The effect of pinning control on the stochastic stability and synchronization is discussed as well. It is found that the pinning control generally destabilize the networks with positive damping coefficient, while it may stabilize the networks with negative coefficient in the proper cases. The results are verified by numerical simulation. In the second part, the stochastic synchronization of complex networks is studied by numerical simulation. Firstly. the synchronizations of van der Pol oscillators under barametric stochastic excitations in complex networks with random coupling strength are studied. It is found that the networks can achieve stochastic synchronization under suitable parameters. Moreover, the stochastic perturbation can obviously accelerate synchronization of complex networks in some cases such as certain initial condition and suitable intensity of stochastic perturbation. Then. the stochastic synchronization of scale-free networks in which nodes are with different nonlinear stochastic dynamical systems. is studied. Two cases are considered. One is a complex network with van der Pol oscillators and linear oscillators. The other is a complex network with two different kinds of van der Pol oscillators. It is found by numerical simulation that the complex networks can achieve stochastic synchronization if the stochastic perturbation is suitable. and the complex networks cannot achieve stochastic synchronization if the stochastic perturbation is too large. At last, the differences of amplitude and frequency between synchronous response and original response of subsystems are discussed.