Bifurcations in plane Kolmogorov flows were discovered by Meshalkin and Sinai. Subsequent calculations by Platt, Sirovich and Fitzmaurice and by She are concerned with transitions to chaos or turbulence in these flows. Here we show many steady bifurcations and several Hopf bifurcations in the plane case but we also find bifurcations into three dimensional flows. Our numerical techniques are based on a DAE (differential algebraic equations) view of the spatially discretized Navier-Stokes equations. Then we apply RPM (recursive projection methods) to solve for the steady state branches of solutions. We are able to show where the steady flows of Platt et al. lie on various steady bifurcation branches. When we follow the Hopf branches that we have discovered, they may shed some light on the "chaotic" or "turbulent" behavior reported in the previous works.
This is joint work with P. Love.