  Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic
trajectories inside any smooth strictly convex billiard table, and all of them
have the same length when the table is an ellipse or a circle. We quantify the
chaotic dynamics of axisymmetric billiard tables close to their borders by
studying the asymptotic behavior of the differences of the lengths of their
axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. 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 $q^{-3} e^{-r q}$ 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 restricted to some perturbed ellipses and circles, which allows us to
compare the numerical results with some analytical predictions obtained by
Melnikov methods and also to 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.
