Linear and weakly nonlinear stability analyses of Darcy Bénard convection with feedback control

In this paper, the effect of feedback control on the criterion for the onset of Darcy-Bénard convection [Formula: see text] in a horizontal Boussinesq Newtonian fluid is studied theoretically. The bounding isothermal lower and upper surfaces are considered to be rigid. The single term Galerkin method [Formula: see text] and the Maclaurin series expansion [Formula: see text] are combined with the Newton-Raphson method [Formula: see text] of three variables to perform a linear stability analysis [Formula: see text] in order to determine eigen value. To make a weakly nonlinear stability analysis [Formula: see text] of the system, a Vadasz Lorenz model [Formula: see text]  is constructed. The model's various properties are discovered to be identical to those of the standard Lorenz model. The [Formula: see text] exhibits both dissipative and conservative characteristics and the bounded nature of its solution is demonstrated by the trapping region, which takes the form of an ellipsoid. The Hopf-Rayleigh number determined from the autonomous dynamical system predicts the onset of chaos. The influence of the controller gain parameter and the Biot number on the onset of convection has been analyzed. Results from the study reveal that the controller gain parameter stabilizes the system and further delays the onset of chaos. Overall, the study establishes that an increase in the Biot number promotes long-term periodic motion over chaotic behavior, while an increase in the controller gain parameter enlarges the trapping region, thereby contributing to improved system stability.