Gauss map and Lyapunov exponents of interacting particles in a billiard

TL;DR

This paper links the Lyapunov exponents of a billiard system with two interacting particles to the Gauss map of their mass ratio, revealing qualitative behavior and unexpected peaks in system stability.

Contribution

It introduces a novel approach using the Gauss map to analyze Lyapunov exponents in a non-KAM Hamiltonian system with interacting particles.

Findings

Lyapunov exponent behavior correlates with Gauss map properties.

Unexpected peaks in Lyapunov exponents depend on mass ratio.

Pseudo-integrable systems with complex invariant surfaces are more unstable.

Abstract

We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue measure zero from the Gauss map can be used to determine the main qualitative behavior of the LE of a Hamiltonian system. The Hamiltonian system is a one-dimensional box with two particles interacting via a Yukawa potential and does not possess Kolmogorov-Arnold-Moser (KAM) curves. In our case the Gauss map is applied to the mass ratio $\gamma = m_2/m_1$ between particles. Besides the main qualitative behavior, some unexpected peaks in the $\gamma$ dependence of the mean LE and the appearance of 'stickness' in phase space can also be understand via LE from the Gauss map. This shows a nice example of the relation between the "instability" of the continued fraction representation of a number with the stability of non-periodic curves (no KAM curves) from the physical model. Our results also confirm the intuition that pseudo-integrable systems with more complicated invariant surfaces of the flow (higher genus) should be more unstable under perturbation.