The Lorenz Equations

In $1961,$ meteorologist Edward Lorenz was using early computer simulation to study weather prediction and came across the following interesting system of equations:

$x^{\text{'}}=10(y-x)$

$y^{\text{'}}=28x-y-xz$

$z^{\text{'}}=-\frac{8}{3}z+xy$

We will see in a moment why this system is interesting, but first we note that because the dependent variables $x,y,z$ are multiplied together this is a nonlinear system. Usually we can't solve these systems. But modern software is much more powerful than what was available in $1961.$ We can easily approximate solutions to systems like these in Mathematica by using NDSolve.

Suppose we have initial conditions $x(0)=0,y(0)=1,z(0)=0.$ Then a numerical solution to the Lorenz equations on the domain $\{t,0,50\}$ is given by the following code