  We provide general sufficient conditions for branching out of a time-periodic
family of solutions from steady-state solutions to the two-dimensional
Navier-Stokes equations in the exterior of a cylinder. To this end, we first
show that the problem can be formulated as a coupled elliptic-parabolic
nonlinear system in appropriate function spaces. This is obtained by separating
the time-independent averaged component of the velocity field from its "purely
periodic" one. We then prove that time-periodic bifurcation occurs, provided
the linearized time-independent operator of the parabolic problem possess a
simple eigenvalue that crosses the imaginary axis when the Reynolds number
passes through a (suitably defined) critical value. We also show that only
supercritical or subcritical bifurcation may occur. Our approach is different
and, we believe, more direct than those used by previous authors in similar,
but distinct, context.
