We study three-degree-of-freedom Hamiltonian systems that are invariant under rotations about the $z$-axis and under reflection across the $xy$-plane. Fixing the angular momentum, such systems reduce to Hamiltonian systems with two degrees of freedom. We focus on the range of energy values for which the corresponding Hill regions are compact. First, under suitable assumptions on the topology of these compact Hill regions, we prove the existence of periodic solutions on each prescribed energy surface of the reduced system by means of a variational minimax method. These periodic solutions are obtained as saddle points of the Maupertuis functional. The resulting solutions are either nontrivial spatial periodic solutions or trivial planar brake solutions in the reduced system. Next, by computing the Morse index, we provide a sufficient condition ensuring that the periodic solutions obtained are nontrivial. Finally, we apply our results to the isosceles three-body problem and to the spatial anisotropic Kepler problem. In both cases, we verify the sufficient condition for nontriviality and thereby establish the existence of nontrivial periodic solutions.
Elementary exposition of realizing phase space structures relevant to chemical reaction dynamics
在这篇文章中，我们审查的分析和数值方法计算的相空间结构的两个自由度的哈密顿系统中出现的化学反应。特别地，这些相空间结构是与指数-1鞍相关联的不稳定周期轨道、周期轨道分割面以及不稳定周期轨道的稳定不变流形和不稳定不变流形。我们审查的背景下，一个两个自由度的哈密顿与一个四次潜在的二次潜在的方法。我们推导出了可积情形下相空间结构的解析形式，并将其在三维能量面上的几何形状可视化。然后，我们调查的分歧的分割面由于势能的参数的变化。我们还回顾了\emph{turning point}的数值方法，并提出了它的新的修改称为\emph{turning point based on configuration difference}计算两个自由度系统的不稳定周期轨道。这些方法在开源python包UPOsHam~ \cite{Lyu2020}中实现。
Bifurcation of dividing surfaces constructed from a pitchfork bifurcation of periodic orbits in a symmetric potential energy surface with a post-transition-state bifurcation
Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits
在作者最近的一篇论文中（K。Yagasaki，Nonintegrability of the限制三体问题，submitted for publication），开发了一种技术，用于确定几乎可积系统是否不是亚纯Bogoyavlenskij可积的，使得首次积分和交换向量场也亚纯依赖于小参数。在这里，我们将继续证明某些类别的动力系统的技术。特别是，我们考虑时间周期扰动的单自由度哈密顿系统和讨论的关系的技术与次谐波Melnikov方法，这使我们能够检测周期轨道的存在性和它们的稳定性。我们说明了理论的周期性强迫Duffing振荡器和两个额外的例子：二阶耦合振荡器和二维系统受到恒定的扭矩的双曲型。
