Take a population, and a rule: it grows in proportion to its size, but it also starves in proportion to how crowded it is. That is the whole model β one line of arithmetic, no dice anywhere in it. Turn the growth rate up slowly and the population settles, then flips between two values, then four, then eight, and then does something that no amount of computing power will ever let you predict.
On the left, the rule drawn as a curve with the line y = x across it. Start anywhere, go up to the curve, across to the line, up to the curve again β that staircase is the population's future. On the right, the same numbers as a time series. Move r slowly upward from 2.8 and watch the staircase go from spiralling into a point, to a rectangle, to a mess.
Now do that for every growth rate at once. Each vertical slice shows where that rate eventually ends up, after the early wobbles have been thrown away. One line means it settles; two means it alternates; the black gaps inside the chaos are windows where order suddenly returns. Drag a box on the diagram to zoom in β and keep zooming, because the shape inside is the same shape again.
The splits come faster and faster, and the gaps between them shrink by very nearly the same factor every time. That factor is Feigenbaum's constant, Ξ΄ β 4.6692. It was found by measuring, not by proof β and it turns up in dripping taps and heart arrhythmias too, in systems that have nothing to do with this equation. The table below is computed when the page loads, by solving for the rate at which each cycle passes exactly through the top of the hump. Watch the last column close in on 4.669 from both sides.
| cycle length | at r = | gap since the last | ratio of gaps |
|---|
Two populations, identical except that one starts a hair's breadth higher β by default one part in a billion. For a while they are indistinguishable. Then the gap doubles, and doubles, and within a few dozen steps they are completely unrelated. Nothing is random: both runs are exact arithmetic. But any error at all, however small, grows to swamp the answer.
Everything on this page is exact arithmetic on one line of algebra. Run it twice from the same start and you get the same answer to the last digit, forever. And yet past about r = 3.57 there is no way to say where the population will be in a hundred steps without simply doing all hundred steps β no formula, no shortcut. Chaos is not noise, and it is not complexity. It is what happens when small differences grow exponentially.
That is the whole reason weather forecasts stop being useful after about ten days. The atmosphere is obeying physics exactly; we simply cannot measure today accurately enough, and the error doubles every few days. Lorenz found this in 1963 when he restarted a simulation from a printout rounded to three decimals instead of six, and got a completely different fortnight.