Four-Wing Attractor
Flow · (ẋ, ẏ, ż) = f(x, y, z, t)
(4(0.2x + y · z), 4(−0.01x − 0.4y − x · z), 4(−z − x · y))
Open in the app The dials and keys named below are the app's.
The system
Each seed moves with velocity ẋ = 4(0.2x + yz), ẏ = 4(−0.01x − 0.4y − xz) and ż = 4(−z − xy). Inside the brackets are the published equations, ax + cyz, bx + dy − xz and ez + fxy. Their constants are a = 0.2, b = −0.01, c = 1, d = −0.4, e = −1 and f = −1.
The 4 in front runs the published clock four times faster. The seeds follow exactly the same paths, only sooner, so the wings fill in within seconds at Speed 1.
The system has no up and down of its own, so nothing is swapped. Its four wings stand in the x–y plane, two above the ground and two below.
Why four wings
The origin is a saddle. Near it, x grows at the rate 0.8 while y and z shrink at 1.6 and 4, so a seed is pushed out along the x axis, in either direction.
Out there sit four more equilibria, near (0.64, 0.45, −0.29), (−0.62, 0.45, 0.28) and their opposites below the ground. Each is the eye of one wing, and a seed spirals out round it until it is thrown back towards the middle and into another wing.
Turning every point half a turn about the z axis, (x, y, z) → (−x, −y, z), leaves all three equations unchanged. So the wings come in opposite pairs, each the other turned over.
Why it never settles
The three rates on the diagonal sum to 0.8 − 1.6 − 4 = −4.8, so any blob of seeds shrinks in volume by a factor of about 120 each second at Speed 1. The seeds are squeezed onto a thin set, the attractor.
On it, nearby seeds still drift apart. The largest Lyapunov exponent is about 0.26 here, four times the published 0.065, so a gap grows e-fold in about 4 seconds.
History
Z. Wang and colleagues published the system in 2009, in a paper titled A 3-D four-wing attractor and its analysis. The Lorenz butterfly, by comparison, has two wings.
Try
- Front: the x–y plane faces you, and the four wings sit one in each quarter.
- Change 0.2 to 0.1 inside the first bracket: the seeds stop drifting apart and trace a smaller tangle round all four wings, within about ±1.5.
- Change 0.2 to 0.05: each seed settles into a single wing, within about ±0.9.
- Set Seeds to 1: one thread alone, wandering from wing to wing in no pattern.