  Chaotic properties of symmetrical two-dimensional stadium-like billiards with
elliptical arcs are studied numerically and analytically. For the two-parameter
truncated elliptical billiard the existence and linear stability of several
lowest-order periodic orbits are investigated in the full parameter space.
Poincar\' e plots are computed and used for evaluation of the degree of
chaoticity with the box-counting method. The limit of the fully chaotic
behavior is identified with circular arcs. Above this limit, for flattened
elliptical arcs, mixed dynamics with numerous stable elliptic islands is
present, similarly as in the elliptical stadium billiards. Below this limit the
full chaos extends over the whole region of elongated shapes and the existing
orbits are either unstable or neutral. This is conspicuously different from the
behavior in the elliptical stadium billiards, where the chaotic region is
strictly bounded from both sides. To examine the mechanism of this difference,
a generalization to a novel three-parameter family of boundary shapes is
proposed and suggested for further evaluation.
