We consider the billiard motion inside a C2-small perturbation of a ndimensional
ellipsoid Q with a unique major axis. The diameter of the ellipsoid Q is a
hyperbolic two-periodic trajectory whose stable and unstable invariant manifolds are
doubled, so that there is a n-dimensional invariant set W of homoclinic orbits for the
unperturbed billiard map. The set W is a stratified set with a complicated structure.
For the perturbed billiard map the set W generically breaks down into isolated
homoclinic orbits. We provide lower bounds for the number of primary homoclinic
orbits of the perturbed billiard which are close to unperturbed homoclinic orbits in
certain strata of W.
The lower bound for the number of persisting primary homoclinic billiard orbits
is deduced from a more general lower bound for exact perturbations of twist maps
possessing a manifold of homoclinic orbits.