Nonlinear stability analysis of the Lorenz model by Lyapunov's direct method

The preturbulent region of the Lorenz model is investigated with the aim to determine the attractive region around the stable fixed points as a function of the Rayleigh parameterr. Close to the turbulent threshold, i.e. for small deviations ?=r _T?r, the attractive region is a highly anisotropic ellipsoid in the phase space. Its volumeV obays the power lawV(ε)=const ?^α with the critical exponent α=5/2. A Landau type approximation, on the other hand, leads to α-3/2. The reason for this discrepancy is discussed. In addition we develop a series expression which allows to calculate the maximal domain of attraction also for large deviations ε. Numerical results from the two lowest terms of the series are represented for 0<ε?6.