Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 1 (2005), 014, 9 pages      math-ph/0511076      https://doi.org/10.3842/SIGMA.2005.014

On Chaotic Dynamics in Rational Polygonal Billiards

Valery Kokshenev
Departamento de Fisica, Universidade Federal de Minas Gerais, Instituto de Ciencias Exatas, Caixa Postal 702, CEP 30123-970, Belo Horizonte, MG, Brazil

Received June 23, 2005, in final form October 29, 2005; Published online November 13, 2005

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.

Key words: polygons; hyperbolic systems with singularities; stochastic system; chaotic dynamics; anomalous diffusion process.

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