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 \[P_2 = \left(-\sqrt{\beta(\rho -1)},-\sqrt{\beta(\rho -1)},\rho-1\right)\]