Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

摘要

This paper studies various Hopf bifurcations in the two-dimensional planePoiseuille problem. For several values of the wavenumber $\alpha$, we obtainthe branch of periodic flows which are born at the Hopf bifurcation of thelaminar flow. It is known that, taking $\alpha\approx1$, the branch of periodicsolutions has several Hopf bifurcations to quasi-periodic orbits. For the firstbifurcation, previous calculations seem to indicate that the bifurcatingquasi-periodic flows are stable and go backwards with respect to the Reynoldsnumber, $Re$. By improving the precision of previous works we find that thebifurcating flows are unstable and go forward with respect to $Re$. We havealso analysed the second Hopf bifurcation of periodic orbits for several$\alpha$, to find again quasi-periodic solutions with increasing $Re$. In thiscase the bifurcated solutions are stable to superharmonic disturbances for $Re$up to another new Hopf bifurcation to a family of stable 3-tori. The proposednumerical scheme is based on a full numerical integration of the Navier-Stokesequations, together with a division by 3 of their total dimension, and the useof a pseudo-Newton method on suitable Poincar\'e sections. The most intensivepart of the computations has been performed in parallel. We believe that thismethodology can also be applied to similar problems.