2.) The Lorenz Equations

The Lorenz equations describe the dynamical behaviour of convection rolls in fluid layers that are heated from below:

The Lorenz equations are given as

x: roational speed of the convectional rolls (and the waterwheel) y: temperature difference between p and q z: deviation of linear temperature development in vertical direction : Prandtl number r: proportional to the Raleigh number b has no name

Exactly, r is R_a / R_c where R_a is the Raleigh number and R_c ist the critical value of R_a. For details see [Peit92].
The equations for the waterwheel are different to the Lorenz equations. But the waterwheel equations can be converted into the Lorenz equations. Only x, the rotational speed, is the same in both systems. The conversion between the two systems is described in detail in [Stro95].

Lorenz fixed the parameters with =10, b=8/3, r=28. For these parameters the system has chaotic behaviour. As for the chaotic waterwheel, the Raleigh number is the most significant parameter, that characterizes the Lorenz system .
For r>1 the Lorenz system has three fixed points C⁰, C⁺ and C^-, where C⁰ = (0, 0, 0) is the origin of the phase space that indicates no motion. C⁺ and C^- only differ in sign of x and y. The behaviour of the fixed points is described in detail in the next section.

These images show different views of some trajectories from the Lorenz system for the parameter values above. The trajectories as thickened with sphere sweeps to give a more three-dimensional impression of the trajectories in phase space. The trajectories are color-coded to indicate the nearest distance to the next fixed point C⁺ = (8.485, 8.485, 27) and C^- = (-8.845, -8.845, 27). The third fixed point C⁰=(0, 0, 0) is not considered in color encoding.

Different views of the lorenz system (19K, 54K, 30K) (blue and green parts are near to the fixed points, yellow and red part ar far away)

In the next section we will explore the parameter space of r.