Journal of Statistical Physics, Vol. 27, No. 2, 1982 Truncated Navier-Stokes Equations: Continuous Transition from a Five-Mode to a Seven-Mode Model Laura Tedeschini-Lalli ~ Received June 12, 1981 A two-parameter family of nonlinear differential equations ~ = F(x, R, e) is studied, which allows one to connect continuously, as e varies from zero to one, the different phenomenologies exhibited by a model of 5-mode truncated Navier-Stokes equations and by a 7-mode one extending it. A critical value is found for c, at which the most significant phenomena of the 5-mode system either vanish or go to infinity. New phenomena arise then, leading to the 7-mode model. KEY WORDS: Navier-Stokes equations; truncations of the Navier- Stokes equations; stationary bifurcation; Hopf bifurcation; period-doubling bifurcation; bifurcation of a periodic orbit into a two-toms; turbulence; strange attractors. 1. INTRODUCTION In order to give a mathematical interpretation to the phenomenon of turbulence in fluids, many numerical investigations were performed on models of simple nonlinear equations which, although deterministic, dis- play a chaotic behavior as one or more parameters increase beyond certain critical values. We refer to Ref. 1 for a wide review of studies in this line, while Ref. 2 provides the theoretical framework to understand the different phenomena that occur in such models. Directly connected with the study of fluid motion are the models obtained by truncating to a finite number of modes the Fourier series expansion of the bidimensional Navier-Stokes equations for an incom- pressible fluid on a torus. In such a way one obtains a one-parameter Supported by G.N.F.M., C.N.R. 1 Istituto Matematico "G. Castelnuovo," Universit~ di Roma, Rome, Italy 10085. 365 0022-4715/82/0200-0365503.00/0 9 1982 Plenum Publishing Corporation