Gauss map and Lyapunov exponents of interacting particles in a billiard

C. Manchein and M.W. Beims

Chaos, Solitons & Fractals, 2009, vol. 39, issue 5, 2041-2047

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 (γ=m2/m1) between particles. Besides the main qualitative behavior, some unexpected peaks in the γ 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.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:39:y:2009:i:5:p:2041-2047

DOI: 10.1016/j.chaos.2007.06.112

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