Visualizing the structure of chaos in the Lorenz system

The Lorenz attractor, with its characteristic butterfly shape, has become a much published symbol of chaos. It can be computed by simply integrating (virtually) any initial point. However, it is much more difficult to understand how the Lorenz attractor organizes the dynamics in a global way. We use a recently developed algorithm to compute a complicated two-dimensional surface called the stable manifold. Visualization tools are key to conveying its intricate geometry and beauty, which in turn provides insight into the structure of chaos in the Lorenz system.