Hopf bifurcations to quasi-periodic solutions for the two-dimensional   plane Poiseuille flow
  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. 
Pablo S. Casas and Angel Jorba
math.DS nlin.CD physics.flu-dyn