Dependence on Parameters

The Lorenz model shows a rich variety of behavior as the parameters a,b,c are changed. Usually b and c are held fixed and a is varied. This corresponds to increasing the nonlinearity, driving the system further away from equilibrium, or in the convection context increasing the temperature difference across the layer of fluid.

Some examples of the behavior:

• For a < 1 the solution rapidly decays to the origin X=Y=Z=0. This corresponds to no motion in the fluid context.
• For a > 1 (e.g. a=5) the orbit approaches one of two fixed points (depending on the initial values) away from the origin. The fixed points are at X ²=Y ²=Z=a-1. In the convection context this corresponds to nonzero but steady fluid flow (in a circulating "roll" configuration).
• At larger values of a, for example a=24.1, the long time dynamics may either approach one of the fixed points or a strange attractor (depending on the choice of initial values), which coexist at these values of a. (Choose nearby initial values to find solutions that converge to the fixed points.)
• For a>24.74 the strange attractor collides with the fixed points, which become unstable so that practically all initial values lead to the familiar butterfly dynamics.
• a=28 gives the usual picture.