Strange Attractors

In 1963, Edward Lorenz was trying to model atmospheric convection. He had a set of three differential equations — absurdly simplified, a caricature of real weather — and he ran them forward in time on a computer. What he found broke something in the way people thought about determinism. The system never settled into a fixed point, never locked into a periodic orbit. It traced an endless, intricate path through a three-dimensional space, folding back on itself again and again in a shape that looked, from a distance, like two wings of a butterfly.

The Lorenz attractor is deterministic. Given a starting point, the equations tell you exactly where the trajectory goes next, and the step after that, and every step forever. But start two trajectories from points that are nearly identical — separated by a distance a thousand times smaller than any measurement instrument could detect — and within a short time they diverge completely. They still trace the same butterfly shape, the same strange attractor. But which wing they're on, moment to moment, becomes impossible to predict. This is sensitive dependence on initial conditions. This is what chaos actually means.

The three attractors here are each strange in a different way. Lorenz (σ=10, ρ=28, β=8/3) has that bilateral symmetry — two lobes, the trajectory hopping unpredictably between them. Rössler (a=0.2, b=0.2, c=5.7) is asymmetric, a single scroll that builds slowly and then collapses, builds and collapses — a more organic restlessness. Thomas (b=0.208) is something else entirely: a triply symmetric labyrinth that fills space in three directions simultaneously, a mathematical object that looks like it was designed to confuse. All three are bounded — the trajectory stays within a finite region forever. All three are aperiodic — it never exactly repeats. The boundary between bounded and aperiodic is precisely where the interesting things live.

What strikes me about strange attractors is the way they collapse our usual intuitions about information. A periodic orbit carries almost no information — you know the future completely once you know the period. A random walk carries maximal entropy — the future is independent of the past. A strange attractor sits between: deterministic but unpredictable, structured but non-repeating, finite in extent but infinite in detail. The Lorenz attractor has a fractal dimension of about 2.06 — it is more than a surface but less than a volume. It occupies a slice of mathematical space that has no name in ordinary geometry.

These equations are not weather. They are not the brain, or the economy, or any other system people invoke when they want to sound appropriately humble about prediction. But they are a proof of concept: that determinism and unpredictability can coexist, that order and chaos are not opposites but a spectrum with an interior. The butterfly shape Lorenz found is not a metaphor for the butterfly effect. It is the butterfly effect — made visible, made navigable, made beautiful.

Drag to rotate the view. Each attractor is computed in 3D and projected; rotation reveals the full geometry. Switch attractors to compare their structures — notice how Lorenz's bilateral symmetry differs from Thomas's cubic symmetry.

Related: logistic map & chaos  ·  Turing reaction-diffusion  ·  Mandelbrot explorer