The Lorenz system is a system of ODEs:

\[\begin{align*} \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y - x), \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. \end{align*}\]

Below is an example of the Lorenz attractor, a chaotic solution of the Lorenz system for parameters \( \sigma=10, \rho=42, \beta=8/3 \), and initial conditions \( x_0=[0,10,20], y_0=1, z_0=20 \).