  We investigate the dynamical motion of particles on a two-dimensional
symmetric periodic substrate in the presence of both a dc drive along a
symmetry direction of the periodic substrate and an additional circular ac
drive. For large enough ac drives, the particle orbit encircles one or more
potential maxima of the periodic substrate. In this case, when an additional
increasing dc drive is applied in the longitudinal direction, the longitudinal
velocity increases in a series of discrete steps that are integer multiples of
the lattice constant of the substrate times the frequency. Fractional steps can
also occur. These integer and fractional steps correspond to distinct stable
dynamical orbits. A number of these phases also show a rectification in the
positive or negative transverse direction where a non-zero transverse velocity
occurs in the absence of a dc transverse drive. We map out the phase diagrams
of the regions of rectification as a function of ac amplitude, and find a
series of tongues. Most of the features, including the steps in the
longitudinal velocity and the transverse rectification, can be captured with a
simple toy model and by arguments from nonlinear maps. We have also
investigated the effects of thermal disorder and incommensuration on the
rectification phenomena, and find that for increasing disorder, the
rectification regions are gradually smeared and the longitudinal velocity steps
are no longer flat but show a linearly increasing velocity.
