  We consider dissipative dynamics of atoms in a strong standing laser wave and
find a nonlinear dynamical effect of synchronization between center-of-mass
motion and internal Rabi oscillations. The synchronization manifests itself in
the phase space as limit cycles which may have different periods and riddled
basins of attraction. The effect can be detected in the fluorescence spectra of
atoms as equidistant sideband frequencies with the space between adjacent peaks
to be inversely proportional to the value of the period of the respective limit
cycle. With increasing the intensity of the laser field, we observe numerically
cascades of bifurcations that eventually end up in settling a strange chaotic
attractor. A broadband noise is shown to destroy a fine structure of the
bifurcation scenario, but prominent features of period-1 and period-3 limit
cycles survive under a weak noise. The character of the atomic motion is
analyzed with the help of the friction force whose zeroes are attractor or
repellor points in the velocity space. We find ranges of the laser parameters
where the atomic motion resembles a random but deterministic walking of atoms
erratically jumping between different wells of the optical potential. Such a
random walking is shown to be fractal in the sense that the measured
characteristic of the motion, time of exit of atoms from a given space of the
standing wave, is a complicated function that has a self-similar structure with
singularities on a Cantor set of values of one of the control parameters.
