  We study electron motion in electromagnetic (EM) fields in the
radiation-dominated regime. It is shown that the electron trajectories become
close to some asymptotic trajectories in the strong field limit. The
description of the electron dynamics by this asymptotic trajectories
significantly differs from the ponderomotive description that is barely
applicable in the radiation-dominated regime. The particle velocity on the
asymptotic trajectory is completely determined by the local and instant EM
field. The general properties of the asymptotic trajectories are discussed. In
most of standing EM waves (including identical tightly-focused
counter-propagating beams) the asymptotic trajectories are periodic with the
period of the wave field. Furthermore, for a certain model of the laser beam we
show that the asymptotic trajectories are periodic in the reference frame
moving along the beam with its group velocity that may explain the effect of
the radiation-reaction trapping.
