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

The route to chaos in a fluid layer heated from below is investigated by us ing the weak non-linear theory as well as Adomian's decomposition method to solve a system of ordinary differential equations which result from a trun cated Galerkin representation of the governing equations. This representati on yields the familiar Lorenz equations. While the weak non-linear method o f solution provides significant 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 neighbou rhood of any one (but only one) of the two steady state convective solution s. This method is expected to loose accuracy and gradually breakdown as one moves away from this neighbourhood. On the other hand, Adomian's decomposi tion method provides an analytical solution to the problem in terms of an i nfinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practi cal procedure to accomplish this task transform the otherwise analytical re sults into a computational solution achieved up to a finite accuracy. The t ransition from the steady solution to chaos is analysed by using both metho ds and their results are compared, showing a very good agreement in the nei ghbourhood of the convective steady solutions. The analysis explains the co mputational results, which indicate a transition from steady convection to chaos via a solitary limit cycle followed by a homoclinic explosion at a su bcritical value of a Rayleigh number. A transient analysis of the amplitude equation obtained from the weak nonlinear solution reveals the mechanism b y which the Hopf bifurcation becomes subcritical. A simple explanation of t he well-known experimental phenomenon of hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is prov ided in terms of the present analysis results. (C) 1999 Elsevier Science Lt d. All rights reserved.