**Title:
Bifurcation theory of swarm formation**

Louis Bonilla, **Universidad Carlos III de Madrid**

Abstract.

In
nature, insects, fish, birds and other animals flock. A
simple two-dimensional model due to Vicsek et al treats
them as self-propelled particles that move with constant
speed and, at each time step, tend to align their
velocities to an average of those of their neighbors
except for an alignment noise (conformist rule). The
distribution function of these active particles
satisfies a kinetic equation. Flocking appears as a
bifurcation from an uniform distribution of particles
whose order parameter is the average of the directions
of their velocities (polarization). This bifurcation is
quite unusual: it is described by a system of partial
differential equations that are hyperbolic on the short
time scale and parabolic on a longer scale. Uniform
solutions provide the usual diagram of a pitchfork
bifurcation but disturbances about them obey the
Klein-Gordon equation in the hyperbolic time scale. 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. In addition, if the
active particles may choose with probability p at each
time step to follow the conformist Vicsek rule or to
align their velocity contrary or almost contrary to the
average one, the bifurcations are of either period
doubling or Hopf type and we find stable time dependent
solutions. Numerical simulations demonstrate striking
effects of alignment noise on the polarization order
parameter: maximum polarization length is achieved at an
optimal nonzero noise level. When contrarian compulsions
are more likely than conformist ones, non-uniform
polarized phases appear as the noise surpasses
threshold.