Why the Lorenz attractor looks like a butterfly

The animation on the home page is not a drawing of a butterfly. It is the Lorenz system, a set of three ordinary differential equations that Edward Lorenz wrote down in 1963 as a stripped-down model of convection in the atmosphere:

dx/dt = σ (y − x)
dy/dt = x (ρ − z) − y
dz/dt = x y − β z

With the classic parameters σ = 10, ρ = 28 and β = 8/3, every trajectory is pulled onto the same thin, folded surface, and once there it never settles down and never repeats. That surface is the strange attractor, and seen from the right angle it has two wings.

Where the wings come from

The system has three fixed points. The origin is a saddle: trajectories approach it along one direction and are flung away along another. The other two, at x = y = ±√(β(ρ−1)) and z = ρ − 1, sit at the centre of each wing. For ρ = 28 both are unstable spirals: a trajectory circles one of them, drifting slowly outward, until it crosses the surface that separates the two basins and is handed over to the other wing. How many loops it makes before switching is, for practical purposes, unpredictable. Two trajectories that start a millimetre apart will be on different wings within a dozen loops.

What the animation does

Eighty trajectories are integrated in the browser with a fourth-order Runge–Kutta step, each dragging a tail of about a thousand points behind it. Every few seconds the integration freezes, the points on the left lobe are matched to points sampled on a left wing and the points on the right lobe to a right wing, ordered by their angle around the lobe centre so the swirl is preserved, and the cloud is relaxed onto the wing surfaces. A solid butterfly fades in, flaps, and dissolves back into the attractor it came from.

The mapping from lobe to wing is the whole point. The attractor was already the butterfly; the morph just makes the resemblance literal for a moment.