A New Seven-modes Truncation of the Plane Incompressible Navier-Stokes Equations

A novel seven-mode truncation scheme for the Navier-Stokes equations governing two-dimensional incompressible flow on a torus is presented in this work. Rigorous proofs are provided for the stationary solutions of this truncated system, the existence of its attractor, and its global stability properties. Concurrently, numerical simulations are employed to investigate several key aspects—including fundamental dynamical behaviors and pathways to chaos—by varying the Reynolds number. It is found that the system displays stochastic characteristics, which emerge via a complex sequence of bifurcation events.The numerical simulation results show that this new seven-mode truncation system exhibits persistent chaos and a stable attractor structure, a characteristic that is more consistent with the essential feature of actual turbulence, i.e., "persistent chaos".