strange attractor

Drop many different starting states into a chaotic system and, after transients die away, they all converge onto the same intricate, filigreed set, never a simple point or loop, but an object that looks like an infinitely layered pastry, the same folded structure repeating at every zoom level. Trajectories are trapped on it forever yet never repeat and never cross. This paradoxical object, attracting yet endlessly complex, is a strange attractor.

An attractor is a set of states toward which nearby trajectories converge as time goes on; it is strange when it has two properties at once. First, it is geometrically strange: it is a fractal, with non-integer (fractal) dimension and self-similar structure at all scales, rather than a smooth point, curve, or surface. Second, the dynamics on it are chaotic: it has at least one positive Lyapunov exponent, so nearby trajectories on the attractor diverge exponentially. The two coexist because the flow simultaneously stretches (to separate trajectories, giving sensitivity) and folds (to keep them bounded), a stretch-and-fold action that, repeated forever, generates the fractal layering, like a baker repeatedly rolling out and folding dough.

Strange attractors are the geometric home of sustained chaos in dissipative systems: the Lorenz attractor, the Rössler attractor, and the Hénon attractor are the textbook examples. They can exist only in a phase space of dimension three or more for a continuous flow, since the no-crossing rule and the Poincaré-Bendixson theorem forbid chaos in the plane. A precise-usage caveat: strange refers strictly to the fractal geometry and chaotic to the positive Lyapunov exponent; in the vast majority of physical examples the two go together, but mathematicians have constructed strange-but-not-chaotic and chaotic-but-not-strange sets, so the terms are not synonyms.