Gauss map and Lyapunov exponents of interacting particles in a billiard

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.

arXiv comment13 pages, 2 figures

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Cited by in corpus (4)

• Intrinsic stickiness in open integrable billiards: tiny border effects
• Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems
• Three unequal masses on a ring and soft triangular billiards
• Chaotic properties of Coulomb-interacting circular billiards