Subcritical transitions to chaos and hysteresis in a ̄uid layer heated from below

The route to chaos in a ̄uid layer heated from below is investigated by using the weak non-linear theory as well as Adomian's decomposition method to solve a system of ordinary di€erential equations which result from a truncated Galerkin representation of the governing equations. This representation yields the familiar Lorenz equations. While the weak non-linear method of solution provides signi®cant insight to the problem, to its solution and corresponding bifurcations and other transitions, it is limited because of its local domain of validity, which in the present case is in the neighbourhood of any one (but only one) of the two steady state convective solutions. This method is expected to loose accuracy and gradually breakdown as one moves away from this neighbourhood. On the other hand, Adomian's decomposition method provides an analytical solution to the problem in terms of an in®nite power series. The practical need to evaluate numerical values from the in®nite power series, the consequent series truncation, and the practical procedure to accomplish this task transform the otherwise analytical results into a computational solution achieved up to a ®nite accuracy. The transition from the steady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the convective steady solutions. The analysis explains the computational results, which indicate a transition from steady convection to chaos via a solitary limit cycle followed by a homoclinic explosion at a subcritical value of a Rayleigh number. A transient analysis of the amplitude equation obtained from the weak nonlinear solution reveals the mechanism by which the Hopf bifurcation becomes subcritical. A simple explanation of the well-known experimental phenomenon of hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is provided in terms of the present analysis results. # 1999 Elsevier Science Ltd. All rights reserved.