We begin our study of chaos with the Lorenz equations \[ \dot{x} = \sigma(y - x) \] \[ \dot{y} = r x - y - x z \] \[ \dot{z} = x y - b z. \] Here \(\sigma, r, b > 0\) are parameters, \(\sigma\) is the Prandtl number, \(r\) is the Rayleigh number, and \(b\) has no name (In the convection problem it is related to the aspect ratio of the rolls). Ed Lorenz (1963) derived this three-dimensional system from a drastically simplified model of convection rolls in the atmosphere

In Lorenz’s original model, \(x\) is proportional to the rate of convection in the atmosphere when heated from below by a vertical temperature gradient proportional to \(z\), with horizontal temperature variation characterised by \(y\)