Plate I: the Lorenz attractor

Plate I Pause Re-run

In the winter of 1961 Edward Lorenz was running a toy weather simulation on a Royal McBee LGP-30 — a machine slower than the chip in a modern doorbell. He wanted to re-examine a run, so he restarted it from the middle, typing in the numbers from a printout. The new run tracked the old one for a while and then diverged completely.

The printout had rounded to three decimal places. The machine held six.^[1]

That difference — one part in a thousand, far smaller than any real measurement error in atmospheric data — was enough to produce a different month of weather. Lorenz had not found a bug. He had found the thing that makes long-range weather forecasting impossible in principle rather than in practice, and he spent the rest of his career on it.

By 1963 he had boiled the convection model down to three equations, which is about as small as an interesting dynamical system gets:

$$\begin{array}{ccc}\frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}& =& \mathrm{\sigma }(\mathrm{y}-\mathrm{x})\\ \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{t}}& =& \mathrm{x}(\mathrm{\rho }-\mathrm{z})-\mathrm{y}\\ \frac{\mathrm{d}\mathrm{z}}{\mathrm{d}\mathrm{t}}& =& \mathrm{x}\mathrm{y}-\mathrm{\beta }\mathrm{z}\end{array}$$

With σ = 10, ρ = 28, β = 8/3 — Lorenz’s canonical values — the trajectory never repeats and never escapes. It winds around one lobe some number of times, crosses over, winds around the other, crosses back. The number of circuits before each crossing is, for practical purposes, unpredictable.^[2]

Three things are true at once, and holding all three is the whole lesson:

1. It is deterministic. No noise term. Same initial condition, same trajectory, every time, forever.
2. It is bounded. Every trajectory is pulled onto the same set of states and stays there. Nothing runs off to infinity.
3. It is unpredictable. Nearby states separate exponentially, at a rate set by the leading Lyapunov exponent — about 0.9 per unit time here, so errors grow by roughly e every unit and by a factor of a thousand in under eight.

Determinism does not buy you predictability. That was the surprise, and it took the rest of the field a decade to absorb it.

Plot the trajectory and you get the figure in the plate above, and behind every page of this library: two lobes, a surface that looks like it has area but has measured volume zero, a fractal dimension around 2.06. Orbits on it never cross — they cannot, since the system is deterministic and crossing would mean two futures from one state — so the apparent surface is really infinitely many sheets, packed arbitrarily close.

That it was a strange attractor — attracting, but not to a point or a cycle — was named later, by Ruelle and Takens in 1971. That Lorenz’s particular set really is one was not proved until Warwick Tucker did it in 1999, with a computer-assisted argument. Thirty-six years between the picture and the proof.

Because it is an honest picture of what this place is for.

Entries here get revised. A revision is a small perturbation — a sentence, a number, a source I had not read. Most of them change nothing. Some of them move the whole argument somewhere I could not have predicted from where it started, and there is no way to tell in advance which kind a given edit is. Writing in public, with the revision history exposed, is a way of admitting that.

It is also bounded, which is the consoling half. The trajectory wanders forever and never leaves the attractor. A library can be endlessly revised and still be about something.

Notes

1. 0.506 instead of 0.506127. Lorenz first assumed a vacuum tube had failed, which is the correct first assumption about a 1961 computer. ↩︎

2. You can encode a trajectory as the sequence of lobes it visits — LLRLRRRL… — and the resulting symbolic dynamics is essentially a shift map on random binary sequences. The unpredictability is not sloppiness in the model; it is the model’s content. ↩︎