Rules for predicting regime change in the Lorenz chaotic system based on the Lorenz map

Corresponding to two strange Lorenz attractors, in the Lorenz model there exist two opposite regimes which can be called as positive and negative regimes. Despite the trajectory of the Lorenz system changing between the two regimes back and forth with an unfixed period, the regime change is predictable. In this paper, with the help of the Lorenz map, three rules for predicting regime change are obtained. In particular, besides two generic predictable rules for the condition of regime transition and duration in new regime, a new rule about length for reaching transition condition, which has not been reported in previous work, is also very important. It provides another approach to forecasting the evolution of the nonlinear dynamical system. The results show that the position for highest point in cusps is the critical value for regime change. When the value of variable z is greater than the corresponding critical value, the current regime is about to end, and the Lorenz model will move to other regime in the next cycle. The length for reaching transition condition in the current regime decreases monotonically with local maximum value zmax, and the smaller zmax in current status implies the bigger length for reaching transition condition. The duration in new regime increases monotonically with the maximum value zM in the previous regime, and the bigger the value of zM, the larger the range for the duration increase is. In addition, the forcing is also associated with the prediction rules for regime change. It not only makes transition conditions for positive and negative regimes different, but also determines the speed of decrease in length for reaching transition condition and the range of increase for duration in new regime.