奇异系统解的定性研究

Second order singular differential equations, as a basic model of the nonlinear vibration, come from a large number mathematical questions in physics, mechanics and engineering research, which is a very active area of research. The physical background of research for this project is based on the radial symmetric system and Bose-Einstein condensates system. We will understand a wide range of dynamical behavior of the solutions from the point of view of geometry, topology. Combining with the method of phase plane analysis, Hamiltonian perturbation theory and KAM method, we study the qualitative behavior of solutions for the disturbed singular Isochronous system, including the study of existence and multipicity of periodic solutions, the coexistence of periodic solutions and unbounded solutions, the boundedness and related issues. We will master a wide range of behavior of the dynamic mechanism and develop the qualitative methods for singular nonlinear systems via the study of these problems.

二阶奇异微分方程作为非线性振动的一类基本模型，存在大量来自于物理、力学和工程的数学研究问题，是十分活跃的研究领域。本项目的研究将以径向对称系统和玻色-爱因斯坦凝聚系统为物理背景，立足于从几何、拓扑的观点来理解解的大范围动力行为，结合微分方程相平面分析方法、哈密顿扰动理论以及KAM方法对扰动的奇异等时系统解的定性行为进行研究，包括周期解的存在性及多样性问题、周期解与无界解的共存问题以及解的有界性及相关问题。通过对这些问题的研究，掌握奇异非线性系统的大范围行为动力学机制，发展奇异非线性系统的定性方法。

本项目的研究成果主要包括三个方面的内容：.(1) 扰动的奇异等时系统的周期解、无界解以及解的有界性：我们证明了超线性扰动奇异等时系统解的有界性。我们的结果指出对于半线性奇异扰动的奇异等时系统，周期解与无界解的存在取决于某类泛函的振荡性质。.(2) 带有小参数的弱扰动周期与拟周期奇异等时系统：我们以玻色-爱因斯坦系统为研究对象，对系统的轨道进行了分类，并给出了调振幅波的恰当表达式或高阶近似。.(3) 非周期扭转映射完全轨道的研究：对于具有指数型生成函数的非周期扭转映射，我证明了映射的无穷多完全轨道的存在性，并且已非周期碰撞振子为模型，给出了应用实例。