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 held to unity, since the aim of this paper is to obtain a realistic model compared with the Lorenz model but still maintaining comparable simplicity. Examination of bifurcation structures of the present model obtained by increasing Rayleigh number as a parameter shows that there is a definite parameter region 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 newly found only in the case of (no-slip/no-slip) conditions.