The Lorenz System

    The Lorenz system is a model proposed by Edward Lorenz in 1963 to give a simplified modelization of convection rolls in the atmosphere, the  three dimensional system proposed by Lorenz follows :

    The Rayleigh-Benard convection problem is described the velocity field U(x,t) and temperature field T(x,t), where x represents the point location, t the time, U the velocity at time t and location x and T the temperature at time t and location x.
    In the Lorenz system, the x coordinate is proportional to the velocity of the circulating flux, while the z coordinate represents the distortion of the temperature with respect to a linear profile between the upper and the lower temperature. The r parameter is the ratio between the Rayleigh number and the Critical Rayleigh number, s is the Prandt number and b is the ratio between the height of a roll and the length of two neighbour rolls.

    The standard parameters are s=10.0, r=28.0 and b=2.66, they yield to a chaotic regime. This regime corresponds to a "strange attractor" called the Lorenz attractor.

    I have written a matlab program to calculate and plot in the three dimensional space the location of successive points that evolve accordingly to the Lorenz system. This program can be found here. Some of the plots obtained using this program follow (the blue point represents the starting point, the black one is the endding point.
 


On the upper image we can see that, for r=1, the point (0,0,0) is a stable equilibrium point.

On the upper image we can see the Lorenz strange attractor, with its two attracting points.