Three-dimensional Navier-Stokes equations truncated on a torus

N-mode truncations of the three-dimensional Navier-Stokes equations with periodic boundary conditions are considered. The authors show that the solution of the truncated model exhibits, under some assumptions concerning the external force, general properties of symmetry and invariance. A particular seven-mode truncation is then derived and investigated in detail by making use of numerical tools typical of dynamical systems. An intricate phenomenon is found with the significant presence of two- and three-dimensional tori. Among different transitions the one from quasiperiodicity to chaos in a three-torus appears of particular interest.