The stability analysis and numerical simulation of a five-mode truncation incompressible Navier-Stokes equations on a torus

The incompressible Navier-Stokes equations are expanded in a Fourier series on a torus, and a five-mode Lorenz equation is obtained. The existence of the attractor is proved, and the stability of the equations is discussed. The whole process from bifurcation to chaos is simulated numerically by a computer with the changing of the Reynolds number changes. Based on numerical simulation results of the bifurcation diagram, Lyapunov exponent spectrum, Poincaré section, power spectrum, and return map, some basic dynamical behavior of the new chaos system is revealed.