On Chaotic Dynamics in Rational Polygonal Billiards

TL;DR

This paper investigates the complex interplay of regular and chaotic behaviors in rational polygonal billiards, analyzing how boundary effects influence particle trajectories and survival probabilities through deterministic and stochastic models.

Contribution

It introduces a novel analysis of regular and irregular motion in polygons using collision distribution and survival probability, highlighting the role of sliding orbits and vortices.

Findings

Late-time wall-collision events lead to circular-like regular trajectories.

Sliding orbits and vortices are distinguished by their relaxation dynamics.

Open polygons exhibit collective excitations related to boundary effects.

Abstract

We discuss the interplay between the piece-line regular and vertex-angle singular boundary effects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular motion in polygons is taken within the alternative deterministic and stochastic frameworks. The analysis is developed in terms of the billiard-wall collision distribution and the particle survival probability, simulated in closed and weakly open polygons, respectively. In the multi-vertex polygons, the late-time wall-collision events result in the circular-like regular periodic trajectories (sliding orbits), which, in the open billiard case are likely transformed into the surviving collective excitations (vortices). Having no topological analogy with the regular orbits in the geometrically corresponding circular billiard, sliding orbits and vortices are well distinguished in the weakly open polygons via the universal and non-universal relaxation dynamics.