Transition to chaos with conical billiards

TL;DR

This paper investigates how particle trajectories on a conical surface transition to chaos, revealing conditions for ergodic behavior and using Poincaré maps to visualize the dynamics.

Contribution

It introduces a novel analysis of conical billiards, exploring the transition to chaos and ergodicity as functions of cone tilt and sharpness, with a new visualization method.

Findings

Identified regions with different trajectory sampling behaviors.

Observed transition from regular to chaotic dynamics with increasing cone sharpness and tilt.

Developed a Poincaré map approach to visualize the transition to chaos.

Abstract

We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane, with tilt angle $\gamma$. We explore the dynamics as a function of $\gamma$ and the cone deficit angle $\chi$ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the ($\gamma, \chi$) plane where, depending on the initial conditions, either (A) the trajectories sample the entire cone base and avoid the apex region; (B) sample only a portion of the base region while again avoiding the apex; or (C) sample the entire cone surface much more uniformly, suggestive of ergodicity. The special case of an untilted cone displays only type A trajectories which form a ring caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently large $\chi$ and $\gamma$. A Poincar\'e map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps.