  We study the contrarian voter model for opinion formation in a society under
the influence of an external oscillating propaganda and stochastic noise. Each
agent of the population can hold one of two possible opinions on a given issue
--against or in favor, and interacts with its neighbors following either an
imitation dynamics (voter behavior) or an anti-alignment dynamics (contrarian
behavior): each agent adopts the opinion of a random neighbor with a
time-dependent probability $p(t)$, or takes the opposite opinion with
probability $1-p(t)$. The imitation probability $p(t)$ is controlled by the
social temperature $T$, and varies in time according to a periodic field that
mimics the influence of an external propaganda, so that a voter is more prone
to adopt an opinion aligned with the field. We simulate the model in complete
graph and in lattices, and find that the system exhibits a rich variety of
behaviors as $T$ is varied: opinion consensus for $T=0$, a bimodal behavior for
$T<T_c$, an oscillatory behavior where the mean opinion oscillates in time with
the field for $T>T_c$, and full disorder for $T \gg 1$. The transition
temperature $T_c$ vanishes with the population size $N$ as $T_c \simeq 2/\ln N$
in complete graph. Besides, the distribution of residence times $t_r$ in the
bimodal phase decays approximately as $t_r^{-3/2}$. Within the oscillatory
regime, we find a stochastic resonance-like phenomenon at a given temperature
$T^*$. Also, mean-field analytical results show that the opinion oscillations
reach a maximum amplitude at an intermediate temperature, and that exhibit a
lag respect to the field that decreases with $T$.
