<< ️(AA) 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 γ. >>
<< ️(They) explore the dynamics as a function of γ and the cone deficit angle χ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. >>
<< ️(AA) find regions of the (γ,χ) 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, (They) observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently large χ and γ. A Poincaré map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. (AA) then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps. >>
Lara Braverman, David R. Nelson. Transition to chaos with conical billiards. arXiv: 2508.02786v1 [nlin.CD]. Aug 4, 2025.
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Keywords: gst, billiard, particles, transitions, chaos.