Stability And Bifurcation Of Stochastic Differential Equations

With the progress of science and technology, people realize the nature of the real world more and more, therefore, many scholars concerned the inevitable ran-dom and time delay factors of the reality system. In particular, a large number of random models in physics, engineering, biological engineering and many other areas of economic and financial evolution are derived in recent years, which encour-age many people to go to further investigation for stochastic system. Compared with the theoretical research of deterministic equations, the study of stochastic differential system is still in its infancy. Especially, there are many problems on stochastic stability and bifurcation which are worth thinking, and theoretical sys-tem still needs to be further improved. So the study on the dynamics properties of stochastic differential system has important theoretical value and practical sig-nificance. This dissertation aims at the well-posedness, stability and bifurcation analysis for the solutions of stochastic differential equations and stochastic par-tial differential equations. The dissertation consists of five chapters. The main contents are as follows:In chapter 1, the historical background, status and the up-to-date progress for all the investigated problems are introduced, the main contents of the dissertation are outlined, and some preliminaries are also presented.In chapter 2, the stability and bifurcation of a class of two-dimensional s-tochastic differential equations are investigated. Firstly, we employ Taylor expan-sions, polar coordinate transformation and stochastic averaging method to trans-form the original system into a stochastic averaging equation. Secondly, we provide a general analysis framework for the local stability, global stability and bifurca-tion of stochastic averaging equation and the original system. Finally, a class of stochastic closed orbit equations with multiple parameters and higher order terms are investigated, which is the important supplement of the previous analysis.In chapter 3, we study a class of two-dimensional stochastic differential equa-tions with small time delays. We first translate the system into a stochastic averag-ing equation by applying stochastic Taylor expansion, small time delay expansion, polar coordinate transformation, and stochastic averaging procedure. Then the lo-cal stability, global stability and bifurcation of stochastic averaging equation and the original system are investigated. Finally, a modified stochastic predator-prey model is provided to illustrate the effectiveness of our analytical procedure.In chapter 4, we consider the well-posedness and stability of mild solution-s to stochastic parabolic partial functional differential equations with space-time white noise. Firstly, we establish an existence-uniqueness theorem under the glob-al Lipschitz condition and the linear growth condition. Secondly, we show the existence-uniqueness property under the global/local Lipschitz condition but with-out assuming the linear growth condition. Then we consider the existence and uniqueness under weaker conditions than the Lipschitz condition. In particular, we obtain the non-negativity and comparison theorem of solutions, and utilize them to investigate the existence of non-negative mild solutions under the linear growth condition without assuming the Lipschitz condition. Finally, we provide the sufficient conditions for the stability of mild solutions to the equations.In chapter 5, we discuss the well-posedness and stability of parabolic stochas-tic partial differential equations. Firstly, we study the existence-uniqueness and stability of mild solutions for a class of parabolic stochastic partial differential e-quations driven by Levy space-time noise under the local/non-Lipschitz condition. Secondly, we analyze the existence-uniqueness and stability of mild solutions for a class of parabolic stochastic partial functional differential equations driven by Levy space-time noise under the local/non-Lipschitz condition. Finally, two examples are given to illustrate the effectiveness of our main results.