3D billiards: visualization of regular structures and trapping of chaotic trajectories

TL;DR

This paper introduces a visualization method for 3D billiard dynamics, revealing how regular structures and resonance channels influence chaotic trapping and classical transport in phase space.

Contribution

It presents a novel 3D phase-space slicing technique to visualize mixed dynamics and analyzes trapping mechanisms and transport barriers in 3D billiards.

Findings

Long-trapped trajectories cause power-law decay in recurrence times.

Trapping occurs near regular structures outside the Arnold web.

Resonance channels govern sticky orbit dynamics and partial transport barriers.

Abstract

The dynamics in three-dimensional billiards leads, using a Poincar\'e section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincar\'e recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.