The Fermi–Ulam model (FUM) is a dynamical system that was introduced by Polish mathematician Stanislaw Ulam in 1961.
The system consists of a particle that collides elastically between a fixed wall and a moving one, each of infinite mass.
A. J. Lichtenberg and M. A. Lieberman provided a simplified version of FUM (SFUM) that derives from the Poincaré surface of section
If the velocity law of the moving wall is differentiable enough, according to KAM theorem invariant curves in the phase space
Over the years FUM became a prototype model for studying non-linear dynamics and coupled mappings.
The rigorous solution of the Fermi-Ulam problem (the velocity and energy of the particle are bounded) was given first by L. D. Pustyl'nikov in [1] (see also [2] and references therein).
In spite of these negative results, if one considers the Fermi–Ulam model in the framework of the special theory of relativity, then under some general conditions the energy of the particle tends to infinity for an open set of initial data.
[9] In the experimental arena this topic arises in the theory of nuclear friction,[10][11] and more recently in the studies of cold atoms that are trapped in optical billiards.