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.