  We investigate numerically the clustering behavior of a system of phase
oscillators with positive and negative couplings under a periodic external
driving field with a bimodal distribution of driving phases. The phase
distribution and the mean speed of the traveling state, as well as the order
parameter for synchronization, are computed as the driving amplitude is varied.
We observe that the periodically-driven system can also host traveling states
for parameters in the same range as those for the case of a system without a
driving field. The traveling speed is found to depend non-monotonically on the
driving amplitude. In particular, oscillators divide into four clusters and
move in pairs. Further, depending on the driving amplitude, two kinds of
traveling mode arise: pairs of clusters traveling in the same direction
(symmetric mode) and in opposite directions (antisymmetric mode). In the latter
case (antisymmetric traveling mode), the average phase speed of the whole
system apparently vanishes. A phenomenological argument for such behavior is
given.
