Elliptic Islands on the Elliptical Stadium

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.