  We investigate the existence of elliptic islands for a special family of
periodic orbits of a two-parameter family of maps corresponding to the billiard
problem on the elliptical stadium. The hyperbolic or elliptical character of
these orbits is also investigated. Depending on the parameters, we obtain upper
bounds of ellipticity for this special family as a lower bound for chaos. On a
different region of the parameter space, we can prove that there is no upper
bound for the existence of elliptic islands. The main results we use are
Birkhoff Normal Form and Moser's Twist Theorem.
