# Whirlpool

Flow · (ẋ, ẏ, ż) = f(x, y, z, t)

`(−z − 0.05x,  −0.2y,  x − 0.05z)`

[Open in the app](https://www.wavelace.com/app#p=81) · [This page](https://www.wavelace.com/presets/whirlpool)

### The system

Every term here is linear, and the three lines split into two independent halves. The app is y-up, so `y` is the height and the turning happens in the floor plane, the one spanned by `x` and `z`, the plane a textbook would call `x, y`.

In that plane the pair `(−z, x)` is a rotation at one radian per second: a full turn every `2π ≈ 6.28` seconds at `Speed` 1, carrying a seed from the `+x` axis toward `+z`. The `−0.05` on each of those lines pulls the radius in, `r = r₀·e^(−0.05·t)`: an e-fold in 20 seconds, a factor `0.73` per turn. The height obeys the second line alone, `y = y₀·e^(−0.2·t)`, halving every `ln 2 / 0.2 ≈ 3.47` seconds.

### Why it winds in

The three eigenvalues are `−0.05 ± i` and `−0.2`. All three have negative real part, so the origin is a stable spiral and every seed reaches it. The imaginary pair is what makes the approach a turn rather than a fall. The four-to-one ratio between `0.2` and `0.05` is the whole picture: the seeds sink onto the floor plane long before they have wound in, so the cloud flattens into a turning plate and only then closes on the centre. After 20 seconds a seed that started at radius 3 and height 4 is at radius `3/e ≈ 1.10` and height `4·e^(−4) ≈ 0.073`.

Being linear, the flow has no other behaviour to offer: no second rest point, no cycle, no chaos. Each thread is a copy of every other, rotated and scaled, which is why the drain looks so clean.

### Try

- Change both `−0.05`s to `+0.05`: the eigenvalues cross to the right and the spiral runs outward instead, radius 3 reaching 8.16 in 20 seconds, while the height still collapses.
- Set the middle line to `−0.02·y`: the height now outlives the turning, and the threads are long helices winding down a funnel.
- Set the middle line to `0`: every seed keeps its height, and the plate becomes a stack of flat spirals.
- `Top` in the deck for the spiral alone, with the sinking heights projected away.
- `Front` instead, to watch the cloud settle onto the floor before it winds in.

### Read more

- [Phase portrait](https://en.wikipedia.org/wiki/Phase_portrait)
- [Equilibrium point](https://en.wikipedia.org/wiki/Equilibrium_point)
- [Eigenvalues and eigenvectors](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors)
- [Exponential decay](https://en.wikipedia.org/wiki/Exponential_decay)