
CNCS Seminar series
To subscribe to the CNCS seminar announcement emaillist,
click here
Upcoming Seminars:
 Tuesday, April 16, 2019, 3:00pm, 119 Physics, CNCS Seminar
Bifurcation theory of swarm formation
Luis Bonilla (Universidad Carlos III de Madrid)
 In nature, insects, fish, birds and other animals flock. A simple
twodimensional model due to Vicsek et al treats them as selfpropelled
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 KleinGordon
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, nonuniform polarized phases appear as the noise
surpasses threshold.
