Chaos and stability in a two-parameter family of convex billiard tables

Abstract

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin–Strelcyn (1978 Phys. Rev. A 17 773–85). We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.