  This paper studies various Hopf bifurcations in the two-dimensional plane
Poiseuille problem. For several values of the wavenumber $\alpha$, we obtain
the branch of periodic flows which are born at the Hopf bifurcation of the
laminar flow. It is known that, taking $\alpha\approx1$, the branch of periodic
solutions has several Hopf bifurcations to quasi-periodic orbits. For the first
bifurcation, previous calculations seem to indicate that the bifurcating
quasi-periodic flows are stable and go backwards with respect to the Reynolds
number, $Re$. By improving the precision of previous works we find that the
bifurcating flows are unstable and go forward with respect to $Re$. We have
also analysed the second Hopf bifurcation of periodic orbits for several
$\alpha$, to find again quasi-periodic solutions with increasing $Re$. In this
case the bifurcated solutions are stable to superharmonic disturbances for $Re$
up to another new Hopf bifurcation to a family of stable 3-tori. The proposed
numerical scheme is based on a full numerical integration of the Navier-Stokes
equations, together with a division by 3 of their total dimension, and the use
of a pseudo-Newton method on suitable Poincar\'e sections. The most intensive
part of the computations has been performed in parallel. We believe that this
methodology can also be applied to similar problems.
