Phase Space Structure and Transport in a Caldera Potential Energy Surface

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

我们使用非线性动力学技术研究二维破火山口势能面（PES）中的相空间输运。破火山口势能面的特征是其中心有一个平坦区域或浅的极小值，周围是势垒以及多个与对称性相关的一阶鞍点，这些鞍点允许进出这个中间区域。我们发现了四种在性质上截然不同的相空间结构情况，它们支配着相空间输运。这些情况是根据总能量以及与中心极小值族相关的周期轨道的稳定性、同一族的分岔以及一阶鞍点的能量可达性来分类的。在每种情况下，我们都计算了势能中心区域不稳定周期轨道的不变流形，以及与一阶鞍点相关的周期轨道族的不稳定周期轨道的不变流形。对于第一种情况，中心区域的周期轨道是周期为10的不稳定周期轨道，它们位于中心极小值族的稳定周期轨道的稳定区域之外。此外，对于第二和第三种情况，中心区域的周期轨道是中心极小值族的不稳定周期轨道，而对于第四种情况，是中心极小值族的倍周期分岔中周期为2的不稳定周期轨道。我们发现，由不稳定周期轨道的不变流形结构决定了三种不同的机制，它们支配着相空间输运。第一种机制解释了轨迹从低能鞍点区域进入破火山口的性质，以及它们如何可能被困在势能的中心区域。第二种机制描述了从破火山口中心区域出发的轨迹的捕获，它们向鞍点区域的输运，以及它们从破火山口离开的性质。第三种机制描述了负责轨迹动态匹配的相空间几何结构，这最初是由卡彭特提出的，并在[柯林斯等人，2014]中针对我们所考虑的二维破火山口势能面进行了描述。