  We report a novel route to active turbulence, observed in numerical
simulations of a polar active fluid model under confinement. To deal with
large-scale computations with arbitrary geometries, we developed a GPU-based
scheme that can be used for any boundary shape in a unified manner. For the
circular confinement, as the radius was increased, we found a series of
transitions first from a single stationary vortex to an oscillating pair of
vortices, then through reentrant transitions between oscillatory and chaotic
dynamics before finally reaching the active turbulence. The first transition
turned out to be hysteretic, with the emergence of the oscillatory state
consistent with the subcritical Hopf bifurcation. In dumbbell-shaped boundaries
composed of two overlapping circles, we observed a transition comparable to the
ferromagnetic-antiferromagnetic vortex-order transition reported in previous
experiments, but the transition point turned out to show a qualitatively
different geometry dependence.
