Transition to Chaos in Five-Dimensional Porous-Medium Thermal-Hydrodynamic Model with Low Prandtl Number

Abstract

In this paper, we investigate the transition to a chaotic regime of thermal convection in a five-dimensional model with low Prandtl number in a porous medium. The mathematical formulation of the model includes the heat equation coupled with the equations of motion under the Boussinesq–Darcy approximation. A system of five ordinary differential equations is derived using a spectral method. This system is solved numerically by using the fourth-order Runge–Kutta method. The results show that from a subcritical value of the Rayleigh number, a transition from steady convection to chaos via a Hopf bifurcation produces a limit cycle which can be associated with a homoclinic explosion. Furthermore, we find that for certain values of Rayleigh number and shape parameter which measures the ratio between the dimensions of the computational domain, the transition from periodic oscillatory convection to chaotic convection can occur via a period-doubling.