# Lorenz Attractor

Consider a rectangular box (parallel sides) containing a gaseous substance. If the box is heated, e.g. from the bottom, the warm gases rise while while the cooler gases fall. At certain temperatures, adjacent cylindrical rolls form along the length of the box. If the temperature is constant, and the interior of the box is smooth (so there is no drag) one might expect this circular motion would be stable and predictable. Edward Lorenz took a few Navier-Stokes equations from a field of physics called fluid dynamics. While these equations are very difficult, Lorenz simplified them to model the convection currents inside a heated box of a gaseous substance (i.e., air). The result was a three-variable system: $\frac{dx}{dt} = s(y-x)$ $\frac{dy}{dt} = x(r-z)-y$ $\frac{dz}{dt} = xy - bz$ Here: $s$ is the Prandtl number which represents the ratio of the fluid viscosity of a substance to its thermal conductivity (Lorenz used the value $s=10$); $r$ is the difference in temperature between the top and bottom of the gaseous system (Lorenz used the value $r=28$); $b$ is the width to height ratio of the box which is being used (Lorenz used the value $b=8/3$); $x$ represents the rate of rotation of the cylinder; $y$ represents the difference in temperature at opposite sides of the cylinder; $z$ represents the deviation of the system from a linear temperature gradient. The left pane is the $xy$-plane, the right pane is the $xz$-plane.