In the thesis we describe the dynamics of two variations of the Fermi acceleration models.

The first model consists of a rectangular billiard with two periodically vertically oscillating slits. A point particle bounces elastically against the billiard table and the slits. We assume that the horizontal motion of the particle is in resonance with those of the slits. In this case, we have found a mechanism of trapping regions which provides the exponential acceleration for almost all initial conditions with sufficiently high initial energy. Under an additional hyperbolicity assumption on the parameters of the system, we estimate the waiting time after which most high-energy orbits start to gain energy exponentially fast.

The second model depicts a point particle bouncing elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion is piecewise smooth with one singularity. If the second derivative of the platform motion behaves well, i.e. it is either always positive or always less than the negative of the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits coexist with bounded orbits at arbitrarily high energy levels.