The Lorenz system (or "attractor") is a system of nonlinear ordinary differential equations in three variables, \( x \), \( y \), \( z \). \( x \), \( y \), \( z \) are not spatial coordinates for particles, but rather more abstract quantities in fluid flow. The equations are $$ \begin{align} \dot{x} &=\sigma\left(y-x\right), \\ \dot{y} &=x\left(\rho-z\right)-y, \\ \dot{z} &=x y - \beta z, \end{align} $$ where \( \sigma \), \( \rho \), \( \beta \) are constants.
Lorenz proposed the equations in his 1963 paper which described the difficulty of long-range weather prediction. They are the canonical "chaotic" dynamical system, exhibiting sensitive dependence of long-range trajectories on initial conditions. "Chaos" became a new branch of science in the 1980s as described in James Gleick's book.
Here you can explore the dynamical behavior of the Lorenz system depending on the values of the constants \( \sigma \), \( \rho \), \( \beta \). The default arguments (reload page for them) show the famous chaotic behavior, where closely-spaced initial conditions (blue dots) converge to the evolution defined by the "attractor," but become completely separated from each other after a couple of orbits around it, effectively losing their memory of their starting points. The behavior of the system changes qualitatively for different values of the constants \( \sigma \), \( \rho \), \( \beta \). Equilibria can appear or disappear, and become stable or unstable, depending on these values.