# 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](https://www.wavelace.com/app#p=180) · [This page](https://www.wavelace.com/presets/four-wing-attractor)

### 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.

### Read more

- [Chaos theory](https://en.wikipedia.org/wiki/Chaos_theory)
- [Attractor](https://en.wikipedia.org/wiki/Attractor)
- [Lyapunov exponent](https://en.wikipedia.org/wiki/Lyapunov_exponent)
- [Lorenz system](https://en.wikipedia.org/wiki/Lorenz_system)