Strange Attractors

Chaos with a shape

Or drag the attractor to turn it.

The address of the page keeps the attractor, to share it.

Never the same place twice

Take a point, apply a simple formula to get a new point, and repeat. Most formulas soon settle: the point stops, or loops through the same few places forever. Some never settle and never escape either. The point wanders without repeating, yet stays on a shape with a very definite form: a strange attractor. Two starting points a hair apart soon end up in completely different places of it, which is what chaos means, but the shape they trace is the same.

In 1963 the meteorologist Edward Lorenz found one in a toy model of the weather with three equations. It looks like the wings of a butterfly, and it gave its name to the butterfly effect. The 3D attractors here (Lorenz, Aizawa, Thomas and Halvorsen) are such flows, followed in small time steps and seen from an angle you can turn. The 2D ones (Clifford Pickover's, Peter de Jong's and Johnny Svensson's) are just two lines of sines and cosines, applied over and over.

The picture is a long exposure. Each pixel counts how many times the orbit passed through it, tens of millions of points in all, and the counts are shown on a logarithmic scale so that both the dense folds and the faint veils show. Slide the parameters: most values give a dull loop, but some give lace, smoke or silk. Random keeps trying until it finds chaos.