TITLE:
Bi-asymptotic billiard orbits inside perturbed ellipsoids

AUTHORS:
Sergey Bolotin (1), Amadeu Delshams (2), Yuri Fedorov (1) 
and Rafael Ramirez-Ros (2)

(1) Department of Mathematics and Mechanics,
    Moscow State University,
    Moscow 119899, Vorob'eby Gory, Russia

(2) Departament de Matematica Aplicada I
    Universitat Politecnica de Catalunya
    Diagonal 647, 08028 Barcelona, Spain

E-mails: bolotin@math.wisc.edu, Amadeu.Delshams@upc.edu, 
         fedorov@mech.math.msu.su, rafael@vilma.upc.edu

ABSTRACT:
The billiard motion inside an ellipsoid of the three-dimensional
euclidean space is completely integrable.

In the generic case (an ellipsoid with three different axis),
the periodic orbit associated to the diameter of the ellipsoid 
is hyperbolic and its unstable and stable invariant 
surfaces are doubled but not completely doubled.
In particular, they have a transverse intersection along eight 
one-dimensional curves, called loops.
Inside any small enough perturbation of a generic ellipsoid there exist at least 
sisteen (primary) heteroclinic orbits close to the unperturbed loops.

In the prolate case (an ellipsoid of revolution around its major axis),
the periodic orbit is still hyperbolic, but its invariant surfaces 
are completely doubled giving rise to a separatrix.
Inside any small enough perturbation of a prolate ellipsoid there exist
at least six (primary) heteroclinic orbits close to the unperturbed separatrix.

Both results are obtained from a general theorem
about the persistence of heteroclinic orbits for twist maps.

KEYWORDS:
Twist maps, billiards, separatrix splitting, variational methods,
Melnikov potential

AMS Subject Classification (2000):
37J15, 37J20, 37J30, 37J35, 37J40, 37J45, 37N05