Transition Tube in Phase Space

Escape from a potential well is a paradigm to understand critical events in celestial mechanics, structural mechanics, chemical physics, and ship dynamics, to name a few examples. These occur via "transition tubes".
We show Poincaré surface-of-section as the magenta plane and the stable manifold of right saddle-center equilibrium point is shown as the cyan surface for a given energy E. The stable manifold is codimension-1 in the 3-dimensional energy surface shown (in variables (x,y,vy)) and has a cylindrical geometry; hence the name tube manifold. The stable manifold of a saddle up to its first intersection with the Poincare section is a boundary of the pathway that leads to imminent escape at this energy. An example 'transit trajectory' that leads to escape from the potential well is shown in red, and lies in the interior of the tube. Similarly, a 'non-transit trajectory' that stays inside the potential well is shown in blue and lies outside the tube.
These example trajectories are shown in the (x, y, vy) space for a given energy E and also as projection in the position space. The zero velocity curves are also shown in the position space projection as thin lines and bound where trajectories can go for this energy.
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