Route to chaos for moderate Prandtl number convection in a porous layer heated from below

The route to chaos for moderate Prandtl number gravity driven convection in porous media is analysed by using Adomian's decomposition method which pro vides an accurate analytical solution in terms of infinite power series. Th e practical need to evaluate numerical values from the infinite power serie s, the consequent series truncation, and the practical procedure to accompl ish this task, transform the otherwise analytical results into a computatio nal solution achieved up to a desired but finite accuracy. The solution sho ws a transition to chaos via a period doubling sequence of bifurcations at a Rayleigh number value far beyond the critical value associated with the l oss of stability of the convection steady solution. This result is extremel y distinct from the sequence of events leading to chaos in low Prandtl numb er convection in porous media, where a sudden transition from steady convec tion to chaos associated with an homoclinic explosion occurs in the neighbo urhood of the critical Rayleigh number (unless mentioned otherwise by 'the critical Rayleigh number' we mean the value associated with the loss of sta bility of the convection steady solution). In the present case of moderate Prandtl number convection the homoclinic explosion leads to a transition fr om steady convection to a period-2 periodic solution in the neighbourhood o f the critical Rayleigh number. This occurs at a slightly sub-critical valu e of Rayleigh number via a transition associated with a period-1 limit cycl e which seem to belong to the sub-critical Hopf bifurcation around the poin t where the convection steady solution looses its stability. The different regimes are analysed and periodic windows within the chaotic regime are ide ntified. The significance of including a time derivative term in Darcy's eq uation when wave phenomena are being investigated becomes evident from the results.