Extended phase diagram of the Lorenz model

The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz equations is studied as a function of r, the normalized Rayleigh number, and of sigma, the Prandtl number. Previous work, either for fixed sigma and all r or along sigma x r and sigma infinity root r, is extended to the entire (r, sigma) parameter plane. An onion-like periodic pattern is found which is due to the alternating stability of symmetric and nonsymmetric periodic orbits. This periodic pattern is explained by considering non-trivial limits of large r and sigma and thus interpolating between the above mentioned cases. The mathematical analysis uses Airy functions as introduced in previous work, but instead of concentrating on the Lorenz map we analyze the trajectories in full phase space. The periodicity of the Airy function allows to calculate analytically the periodic onion structure in the (r, sigma)- plane. Previous observations about sequences of bifurcations are confirmed, and more details regarding their symmetry are reported.