Abstract: A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained. And its stationary solutions, the existence of attractor and the global stability of the equations are firmly proved. At the same time, several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number. The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.

Key words: Navier-Stokes equations, the strange attractor, Lyapunov function, bifurcation; chaos