Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

Let q ≥ 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisy...

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Bibliographic Details
Published in:Experimental mathematics Vol. 25; no. 4; pp. 416 - 440
Main Authors: Martín, P., Ramírez-Ros, R., Tamarit-Sariol, A.
Format: Journal Article Publication
Language:English
Published: Taylor & Francis 01.10.2016
Subjects:
35 Partial differential equations
35J Partial differential equations of elliptic type
37 Dynamical systems and ergodic theory
37E Low-dimensional dynamical systems
37E40, 37J50, 65P10, 52A10
52 Convex and discrete geometry
52A General convexity
65 Numerical analysis
65P Numerical problems in dynamical systems
Billiards
Classificació AMS
exponentially small phenomena
Hamiltonian systems
length spectrum
Matemàtiques i estadística
Melnikov method
numeric experiments
Sistemes hamiltonians
Àrees temàtiques de la UPC
ISSN:1058-6458, 1944-950X
Online Access:Get full text
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Summary:Let q ≥ 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1, q)-periodic trajectories as q → +∞. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor q −3 e −rq times either a constant or an oscillating function, and the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1, q)-periodic trajectories. Our experiments are focused on some perturbed ellipses and circles, so we can compare the numerical results with some analytical predictions obtained by Melnikov methods. We also detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.
ISSN:1058-6458
1944-950X
DOI:10.1080/10586458.2015.1076361