Instability in Fermi–Ulam ‘ping-pong’ problem

The motion of a classical particle bouncing elastically between two parallel walls, with one of the walls undergoing a periodic motion is considered. This problem, called Fermi–Ulam ‘ping-pong’, is known to possess only bounded solutions if the motion of the wall is sufficiently smooth p(t) ∈ C4+ , where p(t) is the position of the wall. It is shown that the stability result does not hold if p(t) is just a continuous function by providing two examples of instability. The second example also answers the question posed in Levi M and Zehnder E (1995 Boundedness of solutions for quasiperiodic potentials SIAM J. Math. Anal. 26 1233–56) about instability in the ‘squash player’s’ problem. Both examples are constructed for an equivalent system with motionless walls. The reduced system is obtained using the transformation, developed in the heat equation theory to solve the moving boundary problem. AMS classification scheme numbers: 34D99, 58F05