Capturing analytically the transition to weak turbulence and its control in porous media convection

The transition point from steady to weak-turbulent convection in a porous layer heated from below is predicted analytically via a weak nonlinear analysis using an asymptotic expansion and compared to numerical results. Introducing feedback control is shown to be a possible way to promote or suppress this transition to weak turbulence. The analysis expands previous investigations showing that the transition from steady convection to weak-turbulent convection does not occur at the Hopf bifurcation point, but earlier depending on the initial conditions. The results are being compared to numerical solutions, indicating an excellent match as long as the asymptotic expansion is valid and consistent with the basic assumptions. It is further shown that the feedback control model can be transformed into a corresponding model without feedback control through a simple transformation of variables, implying that the main effect the feedback control has on the solution is equivalent to altering the initial conditions. Since the initial conditions affect the transition point to weak turbulence (chaos) and the controller acts to alter these initial conditions, one may use the controller to promote or suppress the transition point.