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

Abstract

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 smaller than a certain large value, flocking appears continuously from a uniform disordered state at a critical value of the noise. Because of mass conservation, the amplitude equations describing the flocking state consist of a scalar equation for the density disturbance from the homogeneous particle density (particle number divided by box area) and a vector equation for a current density. These two equations contain two timescales. At the shorter scale, they are a hyperbolic system in which time and space scale in the same way. At the longer, diffusive, timescale, the equations are parabolic. 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 timescale. Then there are persistent oscillations with many incommensurate frequencies about the bifurcating solution; they produce a shift in the critical noise and resonate with a periodic forcing of the alignment rule. These predictions are confirmed by direct numerical simulations of the Vicsek model.