随机动力系统的非一致指数二分性及其数值模拟

项目名称： 随机动力系统的非一致指数二分性及其数值模拟

项目编号： No.11301001

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 朱海龙

作者单位： 安徽财经大学

项目金额： 22万元

中文摘要： 研究随机动力系统的非一致二分性是微分方程和动力系统领域新颖且又非常重要的研究课题之一. 本项目旨在综合运用涉及微分方程和动力系统的多个分支, 来较为系统地研究包括无穷维空间在内的随机动力系统的非一致二分性, 同时考虑相应的数值计算方法和数值模拟. 本项目目标是建立随机动力系统的Lyapunov正则性理论; 研究随机动力系统在Lyapunov意义下的稳定性; 探究不变稳定流形和不变不稳定流形的存在性. 与此同时, 计算随机动力系统的Lyapunov指数, 对相应的非一致行为进行数值模拟. 与理论结果相比对, 探究其在研究过程中发挥的作用. 我们的目标是经过努力, 初步形成有一定特色的研究思路和体系.

中文关键词： 随机微分方程；依均方指数二分性；鲁棒性；依均方概自守；Lyapunov正则性

英文摘要： The research of nonuniform exponential dichotomy of random dynamical systems is one of the new and important topics in the theories of differential equations and dynamical systems. The aim of this project is to systematically study the nonuniform exponential dichotomy of random dynamical systems including infinite dimensional spaces by using all aspects of differential equations and dynamical systems, with a numerical method and numerical simulation on random dynamical systems considered simultaneously. The goal of this project is to establish the Lyapunov regularity theory of random dynamical systems; to study the stability of Lyapunov exponents of random dynamical systems; and to explore the existence of invariant stable and unstable manifolds. Meanwhile, we will compute the Lyapunov exponents of random dynamical systems and simulate the nonuniform behaviors of the systems. Compared the computed results with the theory analysis, we try to understand the essential effect of the nonuniform dichotomies in our research. The target of this project is to initially form a research system with some characteristics.

英文关键词： Stochastic differential equations；Mean-square exponential dichotomy；Roughness；Mean-square almost automorphic；Lyapunov regularity