  Animal motion and flocking are ubiquitous nonequilibrium phenomena that are
often studied within active matter. In examples such as insect swarms,
macroscopic quantities exhibit power laws with measurable critical exponents
and ideas from phase transitions and statistical mechanics have been explored
to explain them. The widely used Vicsek model with periodic boundary conditions
has an ordering phase transition but the corresponding homogeneous ordered or
disordered phases are different from observations of natural swarms. If a
harmonic potential (instead of a periodic box) is used to confine particles,
the numerical simulations of the Vicsek model display periodic, quasiperiodic
and chaotic attractors. The latter are scale free on critical curves that
produce power laws and critical exponents. Here we investigate the
scale-free-chaos phase transition in two space dimensions. We show that the
shape of the chaotic swarm on the critical curve reflects the split between
core and vapor of insects observed in midge swarms and that the dynamic
correlation function collapses only for a finite interval of small scaled
times, as also observed. We explain the algorithms used to calculate the
largest Lyapunov exponents, the static and dynamic critical exponents and
compare them to those of the three-dimensional model.
