Symmetries, Dynamics, and Control of a Seventh-Order Reduced ODE System of the 2-d Navier-Stokes Equations

The symmetries, dynamics and control problem of the two dimensional (2-d) Navier-Stokes (N-S) equations with periodic boundary conditions and with a forcing in the mode (0, 2) known as 2-d Kolmogorov flow are addressed. First, using the Fourier Galerkin method, we obtain a seventh order system of nonlinear ordinary differential equations (ODE) which approximates the behavior of the Kolmogorov flow. The dynamics and symmetries of the reduced seventh-order system are analyzed through computer simulations. Extensive numerical simulations show that the obtained system is able to display different behaviors of the Kolmogorov flow. Then, Lyapunov based controllers are designed to control the dynamics of the system of ODEs to different attractors (e.g., a fixed point, a periodic orbit or a chaotic attractor). Finally, numerical simulations are undertaken to validate the theoretical developments.