Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables

Abstract

We present a numerical study of some billiard tables depending on a perturbative parameter ɛ ≥ 0 and a hyperbolicity parameter h > 0 . These tables are ellipses for ɛ = 0 and circumferences in the limit h → 0 + . Elliptic billiard tables are integrable and have four separatrices, which break up when ɛ > 0 . We conjecture, based on numerical experiments, that as h → 0 + the area of the main lobes of the resulting turnstile (which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories) behaves like an exponential term ɛ e − π 2 / h times an asymptotic series ∑ j ≥ 0 α j ɛ h 2 j such that α 0 ɛ ≠ 0 . This series is Gevrey-1 of type ρ = 1 / 2 π 2 , so that its Borel transform is convergent on a disk of radius 2 π 2 . In the limit ɛ → 0 , the series ∑ j ≥ 0 α j 0 h 2 j is an analytic function which can be explicitly computed with a discrete Melnikov method. The asymptotic series ∑ j ≥ 0 ω j ɛ h 2 j associated to the second exponential term ɛ e − 2 π 2 / h has the same properties. Finally, we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space. Our computations have been performed in multiple-precision arithmetic (namely, with several thousands decimal digits) and rely strongly on the expansion of the local invariant curves up to very high orders (namely, with several hundreds Taylor coefficients). Our programs have been written using the PARI system.