Chapter 2One rule into chaos

The rule from chapter one has a single knob. Turn it slowly and the same equation goes from dead calm, to a steady beat, to a flutter, to chaos — and it tells you exactly where the door is.

A rabbit rule

Robert May, a biologist, wrote this rule down in 1976 for a herd that breeds and then runs short of grass.⁷ The knob r is how fast they breed. Small r: the herd finds a size and holds it. Turn r up and something strange starts to happen.

2a. Low knob. The herd climbs to one size and sits there. Every path leads to the same corner. · computed here · Nan · hongdam.net · CC BY 4.0

2b. Higher knob. Now it can't sit still — it swings high year, low year, high year, forever. A two-beat. · computed here · Nan · hongdam.net · CC BY 4.0

2c. Higher still. The path never lands twice in the same place. It fills the box. · computed here · Nan · hongdam.net · CC BY 4.0

How to read these: the hill is the rule, the straight diagonal is “next year equals this year”. Bounce between them — up to the hill, across to the diagonal, up to the hill — and you are stepping the herd forward one year at a time.

The whole family, on one page

Instead of picking one knob setting, draw them all. Left to right is the knob. Top to bottom, for each setting, are the sizes the herd settles into after the fuss dies down.

Fig 3. One steady size, then a fork to two, then to four, then a smear where the count is beyond counting. The forks come faster and faster until they pile up at a wall — and past the wall is chaos. · computed here · Nan · hongdam.net · CC BY 4.0

Read it like a river seen from above. For a while there is one channel. It splits to two, the two split to four, four to eight — each split closer to the last — and then the banks dissolve. The place where they dissolve is not vague; it is a specific number, near 3.5699.

The number hiding in the forks

Measure where each fork happens and something turns up that nobody put there. The forks on this run land at:

1 → 2: r = 2.998
2 → 4: r = 3.4488
4 → 8: r = 3.5438
8 → 16: r = 3.5643

Now take the gaps between forks and divide each by the next. On this run that ratio comes out 4.743, then 4.64 — closing in on a fixed number:

δ = 4.669 201 609 …

Reading it: each fork sits about 4.669 times closer to the wall than the fork before it. Mitchell Feigenbaum found this same number in the logistic map, in a dripping tap, in a heart — in every rule shaped like a single hill. It does not care what the system is made of.

That is the quiet shock of chaos theory. The road into it has a shape, and the shape is the same for things that share nothing else.⁹

And inside the chaos, calm

Fig 4. Zoom into a slice of the fog and clear bands open up — here the herd falls into a clean three-year cycle before smearing out again. Chaos has windows of order inside it, at every scale. · computed here · Nan · hongdam.net · CC BY 4.0

The three-year window is famous. In 1975 Li and Yorke proved that any rule of this kind which can produce a three-beat can produce a cycle of every length, and a tangle that never repeats at all. Their title gave the field its name: Period Three Implies Chaos.⁸

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