Random periodic paths through random attractors, synchronizations and Lyapunov exponents

In this thesis, we discuss the existence of random periodic paths through studying random periodic attractors and synchronization of stochastic periodic semi-flows. We define a random periodic attractor as an invariant set attracting all trajectories when time goes to infinity. We prove the existence of random periodic attractors for stochastic periodic semi-flows and give conditions to enable a random periodic attractor to be a singleton which means that stochastic periodic semi-flows become synchronized. Then synchronization of stochastic periodic semi-flows can imply the existence of random periodic path. We also apply our results in Benzi-Parisi-Sutera-Vulpiani's stochastic differential equation and use some numerical methods to compute Lyapunov exponents. Finally with the help of the idea of random attractors and Lyapunov exponents, we find the existence of the random periodic path of this equation.