Chaotic time-dependent billiards

A billiard in the form of a stadium with periodically perturbed boundary is considered. Two types of such billiards are studied : stadium with strong chaotic properties and a near-rectangle billiard. Phase portraits of such billiards are investigated. In the phase plane areas corresponding to decrease and increase of the velocity of billiard particles are found. Average velocities of the particle ensemble as functions of the number of collisions are obtained.