Duffing方程与开普勒系统运动稳定性及相关问题研究

The research of existence and dynamical behavior of periodic orbits for differential systems is one of the important topics. In the fields of differential equations and dynamical systems, we will study the analysis and dynamical problems, especially the nonlinear stability, the prevalence of stable periodic solutions and related topics, of Duffing equations and Keple systems, some important theories related to differential equations and dynamical systems will be used, for example, qualitative theory, stability theory, topological degree theory, average methods, Moser twist theory. We are mainly concerned with the linear stability will play an important role in studing the nonlinear stability. Some connections among stability of Duffing equation, their topological degree, and eigenvalues will be established. Meanwhile, we wish extend the third-order approximation to study the nonlinear stability of Keplerian systems and shall continue the development of analytical method to study a general differential systems.The results obtained here complement the work due to Ortega. Moreover, the existence of periodic solutions of singular differential equations plays important roles in stability theory, thus we will establish some methods of computing of topological degree. The abstract results are applied to the existence and uniqueness of periodic solutions of singular Duffing-like equations and singular Keplerian-like systems. The target of this project is to initially form a research system with some characteristics.

微分系统周期轨道的存在性及其动力学行为是微分方程与动力系统领域的一个重要研究课题。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，如定性理论、稳定性理论、拓扑度理论、平均方法、Moser扭转定理等来研究Duffing方程和开普勒系统等的分析学问题和动力学行为，尤其是它们的稳定性、稳定周期解的通有性及相关问题。重点是将Duffing方程的运动稳定性与拓扑度、特征值建立某种等价联系，探讨其线性系统稳定性在非线性系统稳定性中所起的重要作用；同时，考虑开普勒系统的三阶近似方法，将这种解析方法发展到更一般的的非线性系统，这是对西班牙数学家Ortega工作的重要补充。此外，由于奇异方程解的存在性在稳定性研究中发挥重要作用，我们将考虑拓扑度的计算方法，来研究奇异Duffing型方程和奇异开普勒型系统解的存在性、唯一性。我们的目标是通过努力，初步建立有一定特色的思路和体系。

微分系统周期轨道的存在性和已知轨道的动力学行为是微分方程和动力系统领域的一个重要研究课题。利用定性理论、稳定性理论、拓扑度理论等，我们研究了包括Duffing方程和Kepler系统的分析学问题和动力学行为，尤其是它们的稳定性、稳定周期解的通有性、动力谱及相关问题。重点是研究了在周期及反周期特征值条件下Duffing方程的渐近稳定性。首先，借助于Leray-Schauder型存在性定理、Floquet理论和Lyapunov稳定性理论等研究工具，分别在第一稳定区间和高阶稳定区间获得了耗散型Duffing方程渐近稳定性结果。 其次，为了满足非线性稳定性研究的需要，我们还着重讨论了保守型Duffing方程的线性稳定性，改进了相关文献中的结果。同时，考虑Kepler系统的三阶近似方法，将这种解析方法发展到更一般的的非线性系统，这是对西班牙数学家Ortega工作的重要发展。此外，我们试图将西班牙数学家Ortega关于保守型钟摆方程稳定周期解通有性方面的最新思想和结果进行推广，分别在Landesman-Lazer型和周期条件下考虑了耗散型和保守型二阶微分方程稳定周期解的通有性。