  The parameter dependence of the various attractive solutions of the three
variable nonlinear Lorenz model equations for thermal convection in
Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been
investigated as a function of r, the normalized Rayleigh number, at fixed
Prandtl number \sigma. The present work extends the analysis to the entire
(r,\sigma) parameter plane. An onion like periodic pattern is found which is
due to the alternating stability of symmetric and non-symmetric periodic
orbits. This periodic pattern is explained by considering non-trivial limits of
large r and \sigma. In addition to the limit which was previously analyzed by
Sparrow, we identify two more distinct asymptotic regimes in which either
\sigma/r or \sigma^2/r is constant. In both limits the dynamics is
approximately described by Airy functions whence the periodicity in parameter
space can be calculated analytically. Furthermore, some observations about
sequences of bifurcations and coexistence of attractors, periodic as well as
chaotic, are reported.
