Bunimovich Stadium (Chaotic)

A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \(\mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}\), whenever it strikes the boundary. The Bunimovich stadium’s boundary is two straight edges, \(y = \pm r\) for \(|x| \le a\), joined by two semicircular caps of radius \(r\) centered at \((\pm a, 0)\), where \(2a\) is the straight-edge length. It is chaotic despite having no concave scatterer: the focusing semicircular caps defocus nearby trajectories after they refocus and diverge past the caps’ centers of curvature (unlike a full circle, whose focusing never gets interrupted by a flat stretch). This example plots a single trajectory and the resulting Poincare section.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.chaos.systems.billiards import BunimovichStadium
from physicskit.chaos.visualizers.phase_space import plot_billiard_trajectory, plot_poincare_section

billiard = BunimovichStadium(radius=1.0, straight_length=2.0)

Trajectory

fig, ax = plot_billiard_trajectory(billiard, pos=billiard.sample_interior_point(), vel=np.array([0.5, 0.9]), n_bounces=100)

Poincare section

As with the Sinai billiard, the section fills in densely rather than tracing out smooth invariant curves.

fig, ax = plot_poincare_section(billiard, n_rays=40, n_bounces=200)

plt.show()