In Physics TodayAdilson Motter and David Campbell survey the field fifty years later and what we’ve learned about chaos and our limits to predictability. Here’s a description of the Lorenz system (and chaotic attractors in general):

A chaotic attractor is the example par excellence of a chaotic set. A chaotic set has uncountably many chaotic trajectories; on such a set, any point that lies in the neighborhood of a given point will also, with probability one, give rise to a chaotic trajectory. Yet no matter the proximity of those two points, in the region between them will lie points of infinitely many periodic orbits. In mathematical parlance, the periodic orbits constitute a countable, zero-measure, but dense set of points embedded in the chaotic set, analogous to the rational numbers embedded in the set of real numbers. Not only will trajectories that lie on the attractor behave chaotically, any point lying within the attractor’s basin of attraction will also give rise to chaotic trajectories that converge to the attractor.

Of course, chaos is found in many places far beyond weather models.