Bifurcation Structure of Lorenz-Type Five-Equation Model in Thermal Convection

Abstract

The 5-equation model which represents weakly nonlinear two-dimensional thermal convection with various boundary conditions at the top and bottom walls has been constructed. Although the present system includes two control parameters of competing instabilities, the second parameter is fixed at unity, since the aim of this work is to obtain a model which is realistic compared with the Lorenz model but which maintains comparable simplicity. Examination of bifurcation structures of the present model obtained by increasing Rayleigh number as a parameter shows that there is a definite range of parameters in which the structures of orbits are strongly dependent on the boundary conditions such as slip/slip, slip/no-slip and no-slip/no-slip at respective top/bottom walls. In particular, a unique pattern for an inverse transitional region, which seems to be a "twin period-doubling bifurcation, " has been found for the first time in the case of no-slip/no-slip conditions.