chaotic phase space as in Fig. \[fig:Poincare-surface-of-section\], and no stable ray orbits around which high-Q modes can be form by the mechanism of paraxial optics [@BabicBook1972].

However, as suggested by Fig. \[fig:Two-self-intersecting-periodicOrbits\] (e), even in a chaotic region of phase space, the motion is nevertheless organized by the periodic orbits: Each unstable periodic orbit is a periodic point of the Poincar[é]{} map which has stable and one unstable manifolds. In the linear regime near a periodic point, they correspond to trajectories that either converge on, or recede from, that point. By launching a large number of rays from a small neighborhood of the periodic point and iterating the billiard map forward and backward in time, these manifolds are traced out, revealing a characteristic web of intersections [@LichtenbergBook1992] – the homoclinic tangle. In the main panel of Fig. \[fig:HusimiPhaseSpace\] (a), this has been done for the six-bounce orbit corresponding to the thick solid dots in Fig. \[fig:Poincare-surface-of-section\]. Embedded in the intersections between the manifolds are higher-order periodic points corresponding to orbits such as the one shown in Fig. \[fig:Two-self-intersecting-periodicOrbits\] (c). Guided by the homoclinic tangle, the phase space flow is far from random on intermediate time scales and in fact helps 