非光滑和奇异哈密顿动力系统的共振和拉格朗日稳定性

Hamiltonian dynamical system is the one of the active research fields in dynamical systems. The global dynamics related the stability, such as periodic and quasi-periodic motions, resonance phenomena (boundedness and unboundedness of the solutions, chaos), invariant tori, are always the hot topics. ..This project takes some important non-smooth and singular Hamiltonian models, such as impact oscillators，Hamiltonian impulsive equations, second order singular equations from the researches of radially symmetric systems or Bose-Einstein condensates, considers their global dynamics related the resonance and stability. The project includs the researches of periodic and quasi-periodic motions for impact oscillators，resonance and chaos phenomena for Hamiltonian impulsive equations, resonance and Lagrange stability of radially symmetric systems, modulation amplitude waves in Bose-Einstein condensates, related problems for topology and mappings...The researches of the project consider the qualitative dynamics from the geometric point of view, eliminate non-smoothness by using averaging method and the transformation of time and spaces, reduce the problems of higher-dimensional or infinite dimensional system to the problems for planar mapping. The topologcal methods, nonlinear oscillations,variational and qualitative methods will be applied comprehensively. ..Through the researches of the selected problems, the project will not only contributes to understand nonlinear dynamics of non-smooth or singular Hamiltonian dynamical systems, but also provide new qualitative methods for the research of related models.

哈密顿动力系统是微分方程和动力系统十分活跃的研究领域。其中周期和拟周期运动、共振(解的有界、无界、混沌)，不变环面等与稳定性相关的大范围的动力行为，一直是属于研究的热点。. 本项目将选择一些重要的非光滑和奇异的哈密顿动力系统模型：碰撞振子、脉冲哈密顿方程，径对称系统和玻色-爱因斯坦凝聚系统中出现的二阶奇异方程，研究它们的解与共振及稳定性相关的大范围的动力行为，包括：碰撞振子的周期和拟周期运动；脉冲哈密顿方程的共振和混沌；径对称系统的共振和拉格朗日稳定性；玻色-爱因斯坦凝聚系统的调制振幅波以及相关的映射和拓扑问题。.本项目的研究通过平均和时空变换消除非光滑性，并把高维或无穷维系统的问题约化到平面映射上，用几何的观点来理解这些模型的解的定性行为，方法上综合运用拓扑、非线性振动、变分和定性分析等手段。通过所选问题的研究，理解非光滑和奇异哈密顿系统的非线性动力学机制，发展相关的定性方法。

本项目研究非光滑和奇异哈密顿动力系统的共振和Lagrange稳定性，在脉冲哈密顿方程的几何方法，碰撞振子的共振，混沌和Lagrange稳定性，径对称系统的共振现象，Bose-Einstein凝聚态研究和不动点定理及相关应用等方面得到了丰富的成果。.. 主要成果包括：用Poincare-Birkhoff扭转定理研究脉冲方程的无穷多周期解的存在性；变号位势的Hill型碰撞振子的共振和混沌的研究；相对场方程解的Largrange稳定性；等时位势扰动的径对称方程的周期解,拟周期解和无界解研究；用壳函数和坐标变换研究弹性拟周期碰撞振子的Lagrange稳定性，通过相平面上解的盘旋性质研究平面时变哈密顿系统的周期解；发展带参数的扭转型拓扑不动点分支定理研究超线性径对称方程的无穷多周期解以及用非线性极限平均法和高维不动点定理研究Bose-Einstein凝聚态问题。.. 这些成果揭示了非光滑和奇异哈密顿系统的非线性动态机制，丰富和发展了相关模型研究的理论和方法。.. 项目组成员在 J.Differential Equation, Disc. Cont. Dyn. Sys-A，Nonlinear Differ. Equ. Appl. 等重要的 SCI 期刊发表论文 11 篇，《中国科学》上发表论文 1 篇。