Crossover between parabolic and hyperbolic scaling, oscillatory modes and resonances near flocking

A stability and bifurcation analysis of a kinetic equation indicates that the flocking bifurcation of the two-dimensional Vicsek model exhibits an interplay between parabolic and hyperbolic behavior. For box sizes under a certain large value, flocking appears continuously from a uniform disordered state at a critical value of the noise. Bifurcation equations contain two time scales and, due to mass conservation, comprise a scalar equation for the density disturbance from the homogeneous state and a vector equation for a current density. At the shorter scale, they are a hyperbolic system in which time and space scale in the same way. The equations are diffusive at the longer time scale. The bifurcating solution depends on the angle and is uniform in space as in the normal form of the usual pitchfork bifurcation. We show that linearization about the latter solution is described by a Klein-Gordon equation in the hyperbolic time scale. Then there are persistent oscillations with many incommensurate frequencies about the bifurcating solution that may resonate with a periodic forcing of the alignment rule. These predictions are confirmed by direct numerical simulations of the Vicsek model.