  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.
