Rafael Ram rez-Ros
Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables
(1089K, Postscript)
ABSTRACT. We present a numerical study of some billiard tables depending on a
perturbative parameter $\epsilon \ge 0$ and a hyperbolicity parameter
$h > 0$. These tables are ellipses for $\epsilon = 0$ and circumferences
in the limit $h \to 0^+$. Elliptic billiard tables are integrable and
have four separatrices connecting their hyperbolic two-periodic points.
These connections break up when $\epsilon > 0$. As $h\to 0^+$, 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
$\epsilon \e^{-\pi^2/h}$ times an asymptotic series
$\sum_{j\ge 0} \alpha^\epsilon_j h^{2j}$ such that
$\alpha^\epsilon_0 \neq 0$. This series is Gevrey-1 of type
$\rho=1/2\pi^2$, so that its Borel transform is convergent on a disk of
radius $2\pi^2$. In the limit $\epsilon \to 0$, the series
$\sum_{j\ge 0} \alpha^0_j h^{2j}$ is an analytic function which can be
explicitly computed with a discrete Melnikov method. The asymptotic
series $\sum_{j\ge 0} \omega^\epsilon_j h^{2j}$ associated to the second
exponential term $\epsilon\e^{-2\pi^2/h}$ 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.