Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori

Abstract

We consider Kolmogorov's problem of viscous incompressible fluid motions on two dimensional tori. The problem is a bifurcation problem with two parameters, the Reynolds number and the aspect ratio. Varying the aspect ratio as a splitting parameter, we compute numerically bifurcation diagrams with the Reynolds number as a bifurcation parameter. As the aspect ratio changes, we observe turning points and secondary bifurcation points appear or disappear. Furthermore, Hopf bifurcation points are also found when the aspect ratio of the torus satisfies a certain condition. This paper is an improved and enlarged version of the report [26]. Some errors in [17, 26] are corrected here.