Random Disturbances In The Power System

This thesis is mainly concerned with the asymptotic behavior of randomly perturbed dynamical systems,which is composed of two parts.In the first part,we present the general framework to study the asymptotic behavior of ?~?,which is a stationary(invariant)measure for the Markov process X~?= {X_?~t}t?0arising as a result of random perturbations of a dynamical system ? when the small noise intensity ? goes to 0.We prove that any weak limits of ?~?as ? ? 0 are all ? invariant,and their supports are located in the Birkhoff center of ?.Furthermore,theses results are applied to various stochastic evolution systems,including stochastic ordinary differential equations,a class of stochastic partial differential equations(such as stochastic reaction-diffusion equations,stochastic 2D Navier-Stokes equations and stochastic Burgers type equations,etc),stochastic functional differential equations driven by Wiener process or Lévy process and stochastic approximation with constant step.In the second part,we first prove the decomposition formula for the solution of stochastic Lotka-Volterra(sLV)system with the identical growth rate perturbed by white noise,which is expressed in terms of the solution of the corresponding deterministic LV system multiplied by the solution of stochastic 1D logistic system.By virtue of this formula,we show that the sLV system generates a random dynamical system.In the framework of random dynamical system,we study the asymptotic behavior of the pull back trajectories and their attracting domain.From the probabilistic point of view,we investigate the existence and uniqueness of the stationary measure for the solution of sLV system in the forward invariant set.Moreover,based on the nullcline equivalent classification for 3D competitive LV system as obtained a total 37 topological classes,we also present the complete classification for the 3D competitive sLV system in trajectory and distribution way,respectively.At last,the asymptotic behavior of stationary measure for the 3D competitive sLV system is studied.