Numerical non-integrability of Hexagonal string billiard

TL;DR

This paper investigates a smooth hexagonal billiard table constructed via string method, revealing chaotic behavior near hyperbolic orbits, suggesting non-integrability and challenging the Birkhoff-Poritsky conjecture.

Contribution

It provides the first numerical analysis of the billiard's dynamics, demonstrating chaotic regions and supporting its non-integrability.

Findings

Chaotic regions are present near hyperbolic periodic orbits.

The billiard table exhibits non-integrable behavior.

Chaotic zones are very small due to near-circular shape.

Abstract

We consider a remarkable $C^2$-smooth billiard table introduced by Hans L.Fetter. It is obtained by the string construction from a regular hexagon for a special value of the length of the string. It was suggested as a possible counter-example to the Birkhoff-Poritsky conjecture. In this paper, we investigate numerically the behavior of this billiard and find chaotic regions near hyperbolic periodic orbits. They are very small since the billiard table is nearly circular.