The Lorenz Attractor is a set of solutions to the followng set of differential equations, called the Lorenz system: \[ \begin{align*} \frac{dx}{dt} &= \sigma(y-x)\\ \frac{dy}{dt} &= x(\rho - z) - y\\ \frac{dz}{dt} &= xy - \beta z \end{align*} \] where \(\sigma\), \(\rho\), \(\beta\) are real-valued parameters. The system is noteworthy because, for certain choices of the parameters, solutions exhibit chaotic behavior. If we think of a solution as a particle moving in 3D space, then there are two "attractors" around which the particle may orbit, but the orbit alternates back and forth in a seemingly random manner.
Below is a visualization of a solution of the system projected onto the \(xy\)-plane.
Here is a larger version of the program more suited to desktop browsers.