  We propose the first least-order Galerkin model of an incompressible flow
undergoing two successive supercritical bifurcations of Hopf and pitchfork
type. A key enabler is a mean-field consideration exploiting the symmetry of
the mean flow and the asymmetry of the fluctuation. These symmetries generalize
mean-field theory, e.g. no assumption of slow growth-rate is needed. The
resulting 5-dimensional Galerkin model successfully describes the phenomenogram
of the fluidic pinball, a two-dimensional wake flow around a cluster of three
equidistantly spaced cylinders. The corresponding transition scenario is shown
to undergo two successive supercritical bifurcations, namely a Hopf and a
pitchfork bifurcations on the way to chaos. The generalized mean-field Galerkin
methodology may be employed to describe other transition scenarios.

Fluid Dynamics