Chaotic Billiard — Bunimovich Stadium

Billiard Shape

Stadium length ratio a: 0.50 a=0: circle (integrable) | a>0: chaotic stadium

Trajectories

Trajectories: 3 Bounces: 300 Angle spread δ: 0.010

Statistics

Bounces0

Max Lyapunov λ—

Separation δ(t)—

Shape—

Theory

Bunimovich 1974: stadium = 2 semicircles + rectangle (a>0) is fully chaotic (ergodic, mixing).

Key: defocusing mechanism at curved walls — divergent nearby rays.

Lyapunov exponent λ > 0 ↔ chaos.
Poincaré section (s, sin θ): ergodic ↔ fills phase space uniformly.