基于旋转数和盘旋曲线的时变哈密顿系统周期解研究

Hamiltonian system is an active research field in differential equations and dynamical systems. The existence and multiplicity of periodic solutions is one of the hot topic in differential equations. In the corresponding work, it always requires the global existence of solutions and sign condition. In order to solve the problem, we introduce the concepts of rotation numbers and spiral curves for time-dependent Hamiltonian systems. Based on rotation numbers and spiral curves, we consider the problem of periodic solutions for time-dependent and nonconservative perturbation Hamiltonian systems by using Poincare-Birkhoff twist theorem, topologcal degree and and qualitative analysis. We consider the following questions: (1) Reveal the relationship between the rotation numbers and the number of revolutions, and estimate the twist of solution via rotation numbers; (2) Investigate the spiral property by using spiral curves, explore the boundary of transforming original equations; (3) Establish a new twist theorem for non area-preserving continuous mappings with angle description，and explore the existence and multiplicity of periodic solutions for time-dependent and nonconservative perturbation Hamiltonian systems. Through the researches of the selected problems, we will establish a unified approach for time-dependent and nonconservative perturbation Hamiltonian systems without the global existence of the solutions and sign condition. The project will not only contribute to understand nonlinear dynamics of time-dependent Hamiltonian systems, but also provide new qualitative methods for the research of related models.

哈密顿系统是微分方程与动力系统中十分活跃的研究领域，周期解的存在性和多重性是哈密顿系统的热点研究课题之一。这方面相关的研究成果大多需要系统具有解的全局存在性和符号条件。为解决此局限，本项目将建立时变哈密顿系统的旋转数和盘旋曲线这两个新工具，并将其与Poincaré–Birkhoff扭转定理，拓扑度和定性分析等方法结合，探究时变哈密顿及其非保守扰动系统周期解的存在性和多重性。研究内容具体包括：（1）揭示旋转数与旋转圈数的关系进而利用旋转数刻画方程解的扭转；（2）基于盘旋曲线研究解的盘旋性质进而探究改造原方程的界；（3）构建非保面积连续映射具有角度描述的不动点定理，探索时变哈密顿及其非保守扰动系统周期解的存在性和多重性。有望在解的全局存在性和符号条件均缺失时，构建时变哈密顿及其非保守扰动系统周期解研究的一般方法，阐明时变哈密顿系统的非线性动力学机制，丰富相关的定性方法。

哈密顿系统是微分方程与动力系统中十分活跃的研究领域，周期解的存在性和多重性是哈密顿系统的热点研究课题之一。本项目通过旋转数和盘旋函数两个新的工具建立了哈密顿系统在解的唯一性、全局存在性及符号条件缺失时应用不动点定理研究周期解存在性和多重性的一般框架。主要的研究成果如下：平面哈密顿系统周期解的存在性和多重性；哈密顿弱耦合系统周期解的多重性；时变p-Laplacian方程周期解的多重性；碰撞振子碰撞周期解的存在性；非保守的哈密顿扰动系统的周期解的存在性；带参数的弱耦合奇异系统周期解的多重性。通过这些研究成果，阐明了时变哈密顿系统的非线性动力学机制，丰富了相关的定性方法。在国家自然科学基金项目的资助下，三年中先后在知名学术期刊发表论文5篇，均为SCI收录。