On Chaotic Dynamics in Rational Polygonal Billiards

Abstract

We discuss the interplay between the piece-line regular and vertex-angle singular boundary eects, 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, sim- ulated 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 geomet- rically corresponding circular billiard, sliding orbits and vortices are well distinguished in the weakly open polygons via the universal and non-universal relaxation dynamics. process caused by the boundary, formed by the disk and the square, and the interplay between boundary segments, formed by the semi-circles and the square. Moreover, rational polygons of m equal sides and m equal vertices (hereafter, m-gons (1)) exhibit chaotic-like changes in the associated quantum-level spectra (4), but the fluctuations found (5) are very close to those known in the universal Gaussian statistics. These controversial evidences for the chaotic and regular behavior of polygons are a challenge to dynamic theory of billiards. In the present paper we develop a physical insight into the problem, based on analysis of the simulation data on the billiard collision statistics. Within a more general context, the delicate problem of the interplay between regular and irregular segments, which constitute the total billiard boundary, is ultimately related to apparent controversy between the causality and randomness. Our approach to the problem is developed through the alternative deterministic and stochastic frameworks, which elucidate the duality of chaotic and non-chaotic features in the intrinsic dynamics of the rational polygons. The singular eects,