Probability distribution for the number of cycles between successive regime transitions for the Lorenz model

The Lorenz model has been widely used for exploring many real world problems. In this paper we obtain, with the help of an invariant manifold technique, the return map for the maximum value of the variable x of the model and use this return map to derive the simple, empirically obtained, regime transition rules for forecasting regime changes and length in the new regime for the model. The probability distribution for number of cycles between successive regime transitions of the Lorenz model may be of interest in many disciplines. We apply the Perron–Frobenius algorithm over the return map to estimate the probability distribution for the number of cycles between successive regime transitions. These probabilities are also estimated for the forced Lorenz model, which is a conceptual model to explore the effects of sea surface temperature on seasonal rainfall.