Local and global transitions to chaos and hysteresis in a porous layer heated from below

The routes to chaos in a fluid saturated porous layer heated from below are investigated by using the weak nonlinear theory as well as Adomian's decom position method to solve a system of ordinary differential equations which result from a truncated Galerkin representation of the governing equations. This representation is equivalent to the familiar Lorenz equations with di fferent coefficients which correspond to the porous media convection. While the weak nonlinear method of solution provides significant insight to the problem, to its solution and corresponding bifurcations and other transitio ns, it is limited because of its local domain of validity, which in the pre sent case is in the neighbourhood of any one of the two steady state convec tive solutions. On the other hand, the Adomian's decomposition method provi des an analytical solution to the problem in terms of infinite power series . The practical need to evaluate numerical values from the infinite power s eries, the consequent series truncation, and the practical procedure to acc omplish this task transform the otherwise analytical results into a computa tional solution achieved up to a finite accuracy. The transition from the s teady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the co nvective steady solutions. The analysis explains previously obtained comput ational results for low Prandtl number convection in porous media suggestin g a transition from steady convection to chaos via a Hopf bifurcation, repr esented by a solitary limit cycle at a sub-critical value of Rayleigh numbe r. A simple explanation of the well known experimental phenomenon of Hyster esis in the transition from steady convection to chaos and backwards from c haos to steady state is provided in terms of the present analysis results.