Nonlinearity 4 (1992) 189-209. Printed in the U K Three-dimensional NavierStokes equations truncated on a torust V Franceschini and R Zanasi Dipartimento di Matematica Pura ed Applicata, Universitd di Modena, via Campi 213 B, 41100 Modena, Italy Received 20 December 1990, in final form 19 July 1991 Accepted by D Meiron Abstract. N-mode truncations of the three-dimensional NavierStakes equations with periodic boundary conditions are considered. We show that the solution of truncated model exhibits, under some assumptions concerning the external force, general properties of symmetry and invariance. A particular sevenmode truncation is then derived and investigated in detail by miking use of numerical tod~ 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. AMS classification scheme numbers: 34A34, 76D05, 58F13 1. Introduction The advent and rapid development of computers has made it possible to simulate numerical mathematical models which were absolutely unthinkable a few decades ago. Today, in particular, it is possible to simulate nonlinear systems, even very large ones, of ordinary differential equations representing the truncation (‘modes’) of suitable Fourier expansions of the partial differential equations that rule fluid flows !o a finite number of components. A numerical investigation can have various objectives. For instance, while one may he describing some specific behaviour in detail, another may be the characterization of a phenomenology from a statistical point of view. Clearly, the objectives of the study, besides the specific mathematical model, can determine the type of truncation which is adopted and the numerical tools which are used. The first important work in this perspective was given by Lorenz [l], in the already distant 1963, with a study of a three-mode truncation of the equations proposed by Saltzman for convection between plates. Since 1978, several papers have followed, each of them contributing to the understanding of fluid phenomena, sometimes, however, from only the qualitative point of view of the dynamical systems (see, for instance, [2-81). t Work supported in part by MURST (40%), in part by GNFM. 0951-7715/92/010189+21%03.50 0 1992 IOP Publishing Ltd and LMS Publishing Ltd 189