微分系统稳定性相关问题的研究

The study of stability of differential systems is one of very important topics in the theory of differential equations and dynamical systems. We will study in a systematic and deep way, the Lyapunov stability of periodic solutions of conservative systems with lower degrees of freedom, stability of non-autonomous differential systems in the setting of nonuniform exponential dichotomies, dynamical spectrum and the relationship with stability. Some important theories related to dynamical systems and differential equations will be used, for example, stability theory, Moser twist theory, Birkhoff normal form, nonuniform hyperbolicity, exponential dichotomies. We are mainly concerned with the main roles of the basic theory of linear systems (including the theories of spectrum, rotation number, Lyapunov exponents and nonuniform dynamical spectrum) and try to understand the main roles of linear systems playing in how to explain the nonlinear systems. We will try to approach the whole aim by studying some typical problems. Besides the dynamical systems defined by ordinary differential equations, we also consider the nonuniform hyperbolicity of random dynamcial systems in the setting of nonuniform behaviors, the existence and stability of periodic orbits of singular radially symmetric Keplerian-like systems. From these works we are devoted to develop new results on the dynamics of linear and nonlinear dynamical systems in a systematic way.

微分系统的稳定性研究是微分方程和动力系统领域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，包括稳定性理论、Moser扭转理论、Birkhoff标准型理论、非一致双曲理论、指数二分性理论等来较为系统而又深入地研究包括低自由度保守系统周期解的李雅普诺夫稳定性、基于非一致指数二分性条件下的非自治微分系统的稳定性、动力谱及其和稳定性的关系等相关课题。重点是研究线性系统的基本理论，如谱理论、旋转数理论、Lyapunov指数理论、非一致动力谱理论等，考虑它们在研究非线性系统动力学特征过程中所起到的重要作用。我们将通过一些典型问题的研究来达到总体研究目标。除由常微分方程所定义的系统外，我们还将对随机动力系统的非一致双曲理论和动力谱、非线性奇异开普勒型径向对称系统周期轨道的存在性和稳定性开展研究。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。

微分系统的稳定性研究是微分方程和动力系统领域非常重要的研究课题，本项目针对线性和非线性微分系统定性和稳定性中的几个重要问题做了系统研究。（1）低自由度保守系统周期轨的运动稳定性：证明了强迫相对钟摆方程稳定周期解的通有性质，二阶相对奇异方程周期解的稳定性，哑铃型卫星周期运动的稳定性。（2）指数二分性：证明了线性和非线性随机微分方程在双曲条件下的均方几乎自守解的存在性和唯一性，研究了线性随机系统的非一致均方指数二分性和指数稳定性之间的关系。（3）海洋流体动力学研究：建立北极地区环流运动模型，并证明了有界解存在性的系列结果，研究了任意维度的海洋水波欧拉方程精确解的存在性和不稳定性，赤道海洋稳态周期温跃层的存在性与稳定性。（4）线性系统的周期谱：用动力系统方法建立了Camassa-Holm方程和Modified Camassa-Holm方程周期特征值的分布，研究了特征值关于势能在弱拓扑意义下的强连续性，并得到了特征值的估计。这些结果发表在国际数学和应用数学的权威期刊Journal of Differential Equations、Discrete and Continuous Dynamical Systems等上。通过这些具体问题的研究我们实现了总体研究目标。