  Dynamical properties of the elliptical stadium billiard, which is a
generalization of the stadium billiard and a special case of the recently
introduced mushroom billiards, are investigated analytically and numerically.
In dependence on two shape parameters delta and gamma, this system reveals a
rich interplay of integrable, mixed and fully chaotic behavior. Poincare
sections, the box counting method and the stability analysis determine the
structure of the parameter space and the borders between regions with different
behavior. Results confirm the existence of a large fully chaotic region
surrounding the straight line delta=1-gamma corresponding to the Bunimovich
circular stadium billiard. Bifurcations due to the hour-glass and multidiamond
orbits are described. For the quantal elliptical stadium billiard, statistical
properties of the level spacing fluctuations are examined and compared with
classical results.
