Animals having a trend to align their velocities to an average of their neighbors' may flock as illustrated by the Vicsek model and its variants. If, in addition, they feel a systematic contrarian trend, the result may be a time periodic adjustment of the flock or period doubling in time. This is demonstrated by analyzing a modified Vicsek model of self-propelled particles and its corresponding kinetic equation valid for a large number of particles. We have carried out a stability and bifurcation analysis of the order-disorder transition to spatially uniform stationary or time periodic solutions that are characterized by their complex order parameters. Direct numerical simulations differing from theoretical predictions indicate the formation of spatiotemporal structures. Strikingly, we have found that increasing the usual alignment noise may favor flocking and an optimum noise produces the strongest possible order parameter.