倾斜与压缩：哈密顿鞍结分岔的相空间几何及其对化学反应动力学的影响
Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics
摘要：
在这篇文章中，我们提出了一个哈密顿鞍结分岔的高维相空间结构，调解反应动力学的影响。为了实现这一目标，我们确定的相空间不变流形使用拉格朗日描述符，这是一个基于概率的诊断适合于建设一个完整的“相空间断层扫描”通过分析低维切片上的动态。首先，我们建立了一个哈密顿系统与一个自由度（DoF）的模型反应，并研究增加一个参数的势能函数，控制井的深度的影响。然后，我们将这个框架扩展到两个自由度哈密顿量的鞍结分岔，该哈密顿量是通过将谐振子（即浴模式）耦合到系统中的另一个无功自由度来构造的。对于这个问题，我们描述的相空间结构的秩-1鞍平衡点的瓶颈区域，这是一个正常的双曲不变流形（NHIM）及其稳定和不稳定的流形。最后，我们解决了由于势能面的阱深的变化引起的鞍结分岔的反应动力学的哈密顿系统的定性变化。
英文摘要：
In this article we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete ``phase space tomography'' by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.
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