Stable and unstable regions of the Lorenz system

The stable and unstable regions of the Lorenz system are studied. We discuss the relationship between the initial conditions and both these regions, specifically, the preference for the trajectory of the Lorenz system to move towards the left or right equilibrium-point region from an initial point and the residence time of a trajectory in an equilibrium-point region. For this purpose, the four-rank Runge–Kutta algorithms and mathematical derivations are used, whereas a statistical method for the residence times is used. We conclude that the stable and unstable regions are intrinsic to the Lorenz system and have no correlation with the initial conditions; indeed, these regions do not change given different initial conditions. The trajectory of the Lorenz system tends towards the left equilibrium-point region locally, with an average residence time of 8.74 but only 5.789 for the right equilibrium-point region. In general, the system prefers the right equilibrium-point region for which the jump frequency of trajectories to the right region is 535 but only 465 to the left region from the initial conditions for the first time.