奇异微分系统周期解的存在性与稳定性

The study of existence and stability of periodic solutions of singular differential systems is one of very important topics in the theory of differential equations. We will study in a systematic and deep way, the existence and Lyapunov stability of periodic solutions of singular differential systems. Some important theories related to ordinary differential equations and nonlinear functional analysis will be used, for example, stability theory, topological degree theory, Poincare-Birkhoff theorems and the method of third order approximations. Our main aim is to establish the existence results of periodic solutions of singular Keplerian-like systems, singular planar systems in a box, and singular quaternionic-valued differential equations using the topological degree theory and Poincare-Birkhoff theorems, and to apply the stability theory, especially the method of third order approximations, to study the Lyapunov stability of periodic solutions for singular Keplerian-like systems. From these works we are devoted to develop new results for singular differential systems in a systematic way.

奇异微分系统周期解的存在性和稳定性研究是微分方程领域中非常重要的研究课题之一。本项目旨在综合运用涉及常微分方程和非线性泛函分析的多个分支，包括稳定性理论、拓扑度理论、Poincare-Birkhoff定理、三阶近似方法等，系统深入地研究奇异微分系统周期解的存在性及其在李雅普诺夫意义下的运动稳定性。重点是运用拓扑度理论、Poincare-Birkhoff定理来建立奇异开普勒型系统、定义在矩形上的平面奇异系统、奇异四元数微分方程等周期解的存在性；运用稳定性理论，尤其是三阶近似方法建立奇异开普勒型系统周期解的运动稳定性。我们的目标是经过努力，针对奇异微分系统初步形成有一定特色的研究思路和体系。

奇异微分系统周期解的存在性和稳定性研究是微分方程领域中非常重要的研究课题。本项目应用稳定性理论、拓扑度理论、Poincare-Birkhoff定理、三阶近似方法等系统地研究了奇异开普勒系统周期轨的径向稳定性、哑铃卫星方程周期解的存在性、带有阻尼的强迫钟摆方程周期解的存在性和稳定性。这些结果发表在国际数学和应用数学期刊Nonlinear Analysis-Real World Applications、Nonlinear Dynamics等上。通过这些具体问题的研究我们实现了总体研究目标，并对继续要开展的研究奠定了坚实的基础。