Title: Navigating Phase Space Transport with the Origin-Fate Map Show Chinese title

Title: 利用起源-命运图导航相空间输运

Abstract: We introduce and demonstrate the usage of the origin-fate map (OFM) as a tool for the detailed investigation of phase space transport in reactant-product type systems. For these systems, which exhibit clearly defined start and end states, it is possible to build a comprehensive picture of the lobe dynamics by considering backward and forward integration of sets of initial conditions to index their origin and fate. We illustrate the method and its utility in the study of a two degrees of freedom caldera potential with four exits, demonstrating that the OFM not only recapitulates results from classical manifold theory, but even provides more detailed information about complex lobe structures. The OFM allows the detection of dynamically significant transitions caused by the creation of new lobes, and is also able to guide the prediction of the position of unstable periodic orbits (UPOs). Further, we compute the OFM on the periodic orbit dividing surface (PODS) associated with the transition state of a caldera entrance, which allows for a powerful analysis of reactive trajectories. The intersection of the manifolds corresponding to this UPO with other manifolds in the phase space results in the appearance of lobes on the PODS, which are directly classified by the OFM. This allows computations of branching ratios and the exploration of a fractal cascade of lobes as the caldera is stretched, which results in fluctuations in the branching ratio and chaotic selectivity. The OFM is found to be a simple and very useful tool with a vast range of descriptive and quantitative applications. Show Chinese abstract

Abstract: 我们引入并展示了起源-命运图（OFM）作为一种工具，用于详细研究反应物-产物类型系统中的相空间输运。 对于这些具有明确起始和结束状态的系统，通过考虑初始条件集的反向和正向积分以索引它们的起源和命运，可以构建出完整的涡通量动力学图景。 我们通过一个具有四个出口的两自由度火山口势能的研究来说明该方法及其效用，证明OFM不仅重现了经典流形理论的结果，还提供了关于复杂涡结构的更详细信息。 OFM能够检测由新涡的形成引起的动态显著转变，并且还能引导不稳定周期轨道（UPO）位置的预测。 此外，我们在与火山口入口过渡态相关的周期轨道分割面（PODS）上计算了OFM，这使得对反应轨迹的强力分析成为可能。 对应于这个UPO的流形与其他相空间中的流形相交，在PODS上形成了涡，这些涡直接由OFM分类。 这允许计算分支比并探索涡的分形级联，当火山口被拉伸时，这会导致分支比的波动和混沌选择性。 发现OFM是一种简单而非常有用的工具，具有广泛的描述性和定量应用范围。