# Lorenz attractor

File:Lorenz system r28 s10 b2-6666.png
A plot of the trajectory Lorenz system for values r=28, σ = 10, b = 8/3
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The Lorenz attractor, introduced by Edward Lorenz in 1963, is a non-linear three-dimensional deterministic dynamical system derived from the simplified equations of convection rolls arising in the dynamical equations of the atmosphere. For a certain set of parameters the system exhibits chaotic behavior and displays what is today called a strange attractor; this was proven by W. Tucker in 2001. The strange attractor in this case is a fractal of Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± 0.01.

The system arises in lasers, dynamos, and specific waterwheels.

$\displaystyle \frac{dx}{dt} = \sigma (y - x)$
$\displaystyle \frac{dy}{dt} = x (r - z) - y$
$\displaystyle \frac{dz}{dt} = xy - b z$

where $\displaystyle \sigma$ is called the Prandtl number and r is called the Reynolds number. $\displaystyle \sigma,r,b>0$ , but usually $\displaystyle \sigma=10$ , $\displaystyle b=8/3$ and r is varied. The system exhibits chaotic behavior for r = 28 but displays knotted periodic orbits for other values of r. For example, with r = 99.96 it becomes a T(3,2) torus knot.

The butterfly-like shape of the Lorenz attractor may have inspired the name of the butterfly effect in chaos theory.