Dynamics of a jumping particle on a staircase profile

We perform a detailed analysis of the dynamics of the descent of a particle bouncing down a staircase profile under the action of gravity. In order to get interesting dynamics we make a detail analysis of the case which the particle loses momentum in the direction orthogonal to the collision plane but preserves the tangential component of the momentum. We prove that in this case all orbits are bounded and show the existence and stability of periodic solutions. The interplay between loss and gain of energy due to impacts and free falling respectively generates a rich dynamics.