【作者】 刘显明；

【导师】 段金桥； 刘继成；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2010， 博士

【摘要】 在本文中研究两个方面的内容.一是研究Levy过程驱动系统的同步化现象.二是讨论Levy过程驱动系统的随机稳定和不稳定流形.首先我们研究了加性及乘性Levy过程驱动的动力系统的同步化现象.具体的,我们考虑欧氏空间Rd上Levy过程驱动的耦合动力系统和其中a,b, c, d为实数或向量,Lti,i= 1,2,3,4是独立的Levy过程,连续函数f,g满足单边耗散Lipschitz条件,u*是两个Langevin方程随机稳态解之和.在讨论了两个系统的稳态解和随机吸引子的基础上,证明了同步化结果.这揭示了耦合系统的一种动力系统渐近现象.这两个结果是Caraballo和Kloeden05年和08年的结果的进一步发展.在随机动力系统研究中不变流形是个重要的几何研究对象,对不变流形研究有助于对随机影响下复杂动力系统性态的认识.我们考虑Marcus系统假定矩阵S,U和非线性项f,g满足适当的条件,我们考虑了上述系统生成随机动力系统的稳定和不稳定流形.目前已有的结果都是考虑连续噪声的情形.进一步,我们考虑了随机稳定流形和相应的确定性稳定流形之间的渐近关系.最后,我们讨论了下述系统的随机慢流形,当ε↓0+时得到了一个简单的渐近结果.更多还原

【Abstract】 In this thesis, we study synchronization phenomena and invariant stable and un-stable manifolds for dynamical systems driven by non-Gaussian Levy noises.First we investigate synchronization of systems under additive as well as multi-plicative Levy noises. In particular, we consider coupled SDE systems in Rd, driven by Levy motion and where a, b, c, d are constants, Lti,i= 1,2,3,4 are independent two-sided Levy mo-tions, continuous functions f, g satisfy the one-sided dissipative Lipschitz conditions, and u* is the sum of stationary solutions of two Langevin equations.After discussing the stationary orbits and random attractors, a synchronization phenomenon is shown to occur. The synchronization result implies that coupled dy-namical systems share a dynamical feature in some asymptotic sense. This further develops the results of Caraballo and Kloeden.Random invariant manifolds are geometric objects useful for understanding com-plex dynamics under stochastic influences. We consider a Marcus systemUnder suitable assumptions on matrixes S, U and nonlinear terms f, g, invari-ant stable and unstable manifolds for random dynamical systems generated by above system are considered. The existing work in this area is for stochastic differential equations driven by noises that are continuous in time. Moreover, when the noise in- tensity is small, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold.Finally, the slow manifold of the following system is considered and an asymptotic result forε↓0+ is shown.更多还原