The differential equations describe the rate of change of a particle's position: \((x(t),y(t),z(t))\) with respect to time \(t\):
\begin{align} \frac{dx}{dt} &= \sigma (y-x) \\ \frac{dy}{dt} &= x(\rho-z)-y \\ \frac{dz}{dt} &= xy-\beta z \\ \end{align}
In these differential equations \(\sigma, \rho, \beta \) are constants which when set to be \(\sigma = 10, \rho = 28, \beta = \frac{8}{3} \) produce the spectacular demonstration of chaotic behaviour seen above.
You may notice that the solution set appears to be orbiting around two critical points (drawn in grey). These have coordinates in terms of \(\sigma, \rho, \beta \) :
\[P_1 = \left(\sqrt{\beta(\rho -1)},\sqrt{\beta(\rho -1)},\rho-1\right)\] and \[P_2 = \left(-\sqrt{\beta(\rho -1)},-\sqrt{\beta(\rho -1)},\rho-1\right)\]