A five-mode truncation of the Navier-Stokes equations on a three-dimensional torus

After defining a general N-mode truncation of the three-dimensional Navier-Stokes equations for an incompressible fluid with periodic boundary conditions, a particular 5-mode truncation is considered, with an external force independent of time and acting on one mode only. The truncated system, which originally consists of twenty first order nonlinear differential equations depending on four external parameters, thanks to its peculiar features, can be reduced to only five equations with two parameters, ϱ and R2. The study of this reduced model is performed in deep detail in the case ϱ = 1, and only roughly for ϱ varying. In the former case a behavior is found which is quite similar to the ones of the Lorenz model and of a 5-mode truncation of the two-dimensional Navier-Stokes equations previously studied.