[en] A number of problems, including the calculation of magnetic field line trajectories in three-dimensional systems, can be treated as a mapping of a plane onto itself. KAM theory has shown that these trajectories can be either periodic, restricted to a KAM curve, or ergodic and thus filling a finite fraction of the available phase space. The problem addressed here is finding when particular KAM surfaces exist. The method adopted is to study numerically the periodic orbits of a particular mapping. The guiding hypothesis is that the disappearance of a KAM surface is associated with a sudden change from stability to instability of all nearby periodic orbits