Completely chaotic billiard, it is hyperbolic, ergodic, mixing, etc., but has a sticky bouncing ball family.
Toadstool mushroom with dispersing stem.
(Color) (a) Circular mushroom billiard with three members of the CFMUPO formed by the continuous family of period five orbits confined to the cap but situated on the phase space far from the KAM island. (b) Phase space showing the three periodic orbits in large colored dots.
(Color online) Phase space of a billiard in an ellipse. The points and correspond to the elliptic two-periodic orbit (small axis), while the points correspond to the hyperbolic two-periodic orbit (large axis). Closed curves around correspond to hyperbolic caustics while the invariant curves with projection covering entire segment , to elliptic caustics.
(Color) Cheburashka billiards. The blue trajectory shown in (a) is elliptic and is stuck in the head of Cheburashka (ellipse) for 38 bounces. The red trajectory shown in (a) is parabolic and is stuck for 100 bounces. Both trajectories are shown in the phase space in (b) with large colored dots. The lines labeled as correspond to the boundaries of the upper arc of the ellipse; and the lines are the boundaries of the lower arc of ellipse. (c) shows a Cheburashka billiard with a short head (small ellipse arcs) for which all internal stickiness is destroyed.
(Color) (a) Barbeque billiard with one sticky (elliptic) island and one non-sticky (parabolic) island. A chaotic trajectory is shown in green, and a portion of the trajectory is stuck the cap for 45 bounces before returning to the stems, while the trajectory in blue belongs to the elliptic KAM island, and the trajectory in red belongs to a the parabolic KAM island. These orbits can also be seen in the corresponding Poincaré map in (b) with large colored dots.
(Color) (a) Barbeque billiard with two non-sticky islands. The blue trajectory is regular while the red one is chaotic and it gets stuck near a CFMUPO of period 5 for 32 bounces. (b) shows the trajectories in phase space with large colored dots.
(Color) (a) Barbeque billiard with two non-sticky islands. The blue trajectory is in one KAM island while the red trajectory is chaotic, a portion of which is stuck in the elliptic cap for 43 bounces; the green trajectory is also chaotic and is stuck in the cap for 15 bounces. (b) shows the trajectories in phase space with large colored dots.
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