Truncations to 12, 14 and 18 modes of the Navier-Stokes equations on a two-dimensional torus

Three truncations of the Navier-Stokes equations on a two-dimensional torus are numerically investigated by making use of techniques based on bifurcation theory. The three truncations, to 12, 14 and 18 modes, respectively, are obtained by taking into account all the modes contained in balls of increasing radius. While a comparison of the details of the phenomenologies is meaningless, the three models show common global qualitative features. In fact, the behavior of each model is described by three different stories which start from three distinct fixed points and develop in parallel. Two stories are characterized by the presence of fixed points and periodic orbits, while the third one also involves two-dimensional tori. The three truncations exhibit a surprisingly rich collection of bifurcations. Breaking of tori and disappearance of strange attractors by crisis seem to be the phenomena of greatest interest, particularly in the framework of dynamical systems.