Thought Toys · Exhibit 08

The logistic map

Here is a one-line rule for next year's population: it grows when there's room and crashes when it's crowded. It has a single dial — call it the boom rate. Turn the dial up slowly. The population settles to a steady number… then starts flipping between two… then four… then dissolves into noise that never repeats. A simple equation walking, step by step, into chaos.

The map of every outcome — boom rate r across the bottom, where the population lands up the side

The population year by year at this dial setting

Select a boom rate to see the long-run behaviour.

2.5 — calm4.0 — wild
Jump to a landmark

What you're seeing

The big picture up top is a map of endings. Slide along the bottom to a boom rate, look straight up, and the bright dots tell you every value the population eventually visits, once the early wobble has died away. The little chart below it plays out that same future the slow way — the population year after year — so you can watch it hold a level, or swing between a few, or scribble.

At gentle rates the population just glides to one comfortable number and stays — the curve up top is a single line. Push past r = 3 and that line forks: now the population can't sit still, it ping-pongs between a high year and a low year forever. Push a little more and each branch forks again — a four-year cycle — and the splits come faster and faster, eight, sixteen, thirty-two, piling up around r ≈ 3.57. Past there the dots smear into a haze: the population now wanders without ever repeating. Tiny rounding differences would blow up — this is the same sensitive, unrepeatable behaviour as the double pendulum, born from one humble equation.

Now the twist. The chaos isn't solid. Drag to about r = 3.83 and the spray suddenly snaps back to just three clean lines — a calm three-year cycle sitting in the middle of the storm. (That's the "order in chaos" button.) Zoom anywhere into the fuzzy region and you'd find the whole forking tree again, smaller, repeating forever. One dial, and a single tidy rule, contains steadiness, rhythm, chaos, and hidden order all at once.

The rule, exactly. Each step, the next population fraction is x → r · x · (1 − x), with x between 0 (extinct) and 1 (packed full), and r the boom rate. The r·x term is growth; the (1 − x) term is the brake from crowding. The doublings happen at r ≈ 3, 3.449, 3.544, 3.564, 3.569…, each gap about 4.669× shorter than the last (the Feigenbaum constant), accumulating near r ≈ 3.5699 where chaos begins; a period-3 window opens near r ≈ 3.83. (Checked offline before shipping: those split points match the known values, the gaps shrink toward 4.669, and the Lyapunov exponent — the growth rate of tiny errors — is negative in the ordered bands and positive in the chaotic ones, which is exactly how the verdict below is decided.)

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