Unstable periodic orbit and its stable manifold in a 2 degrees-of-freedom Hamiltonian system

(a) For a fixed excess energy, ∆E, above the critical value Ee, the permissible regions (in white) are connected by a bottleneck around the saddle equilibria. All motion from the well in quadrant 1 to quadrant 2 must occur through the interior of a stable manifold associated with an unstable periodic orbit in the bottleneck between the quadrants; seen as a 2D configuration space projection of the 3D energy manifold. We show the stable manifold (cyan) and the periodic orbit (black) for an excess energy of ∆E = 100 (cm/s)². A trajectory crossing the U1-section inside the stable manifold will transition (red) into the quadrant 2 well, while one that is outside stays (blue) inside quadrant 1. The zoomed-in inset in the figure shows the structure of the manifold and how precisely the separatrix divides transition and non-transition trajectories. (b) In the (x,y,v_y) projection, the phase space conduit for imminent transition from quadrant 1 to 2 is the stable manifold (cyan) of geometry R¹ × S¹ (i.e., a cylinder). The same example trajectories (red and blue) as in (a) that exhibit transition and non-transition behavior starting inside and outside the stable mani- fold, respectively, are shown in the 3D projection and projected on the (x, y) configuration space. A movie of a nested sequence of these manifolds can be found https://youtu.be/gMqrFX2JkLU.

(a) 对于高于临界值$E_e$的固定过剩能量$Delta E$，允许区域（白色区域）通过鞍平衡点（saddle equilibria）周围的瓶颈相互连通。从象限1势阱到象限2势阱的所有运动，都必须经由象限间瓶颈处与不稳定周期轨道（unstable periodic orbit）相关联的稳定流形（stable manifold）内部完成——这一过程对应三维能量流形（energy manifold）在二维位形空间（configuration space）上的投影。我们针对$Delta E=100 (mathrm{cm/s})^2$的过剩能量，给出了稳定流形（青色）与周期轨道（黑色）的可视化结果。穿过稳定流形内部U1截面的轨迹将发生跃迁（以红色标记）并进入象限2势阱，而位于稳定流形外部的轨迹则保持（以蓝色标记）在象限1势阱内。本图的放大插图展示了该流形的结构，以及分隔线（separatrix）如何精准区分跃迁轨迹与非跃迁轨迹。
(b) 在$(x,y,v_y)$投影中，用于描述从象限1到象限2即时跃迁的相空间通道（phase space conduit），是几何结构为$mathbb{R}^1 imes S^1$（即圆柱面）的稳定流形（青色）。与(a)中一致的两组示例轨迹——分别从稳定流形内部出发发生跃迁（红色）、从外部出发保持不跃迁（蓝色）——在三维投影与$(x,y)$位形空间投影中均有展示。可通过链接https://youtu.be/gMqrFX2JkLU查看这些流形嵌套序列的动画视频。