A whole population's fate from a single equation. Turn the growth-rate knob and watch a calm, predictable system shatter — first into a flicker, then a beat, then pure unpredictability. Same formula the whole way.
Each step bounces off the curve and the diagonal. Watch where the staircase gets trapped.
Sweep r left to right. Each column = the long-run values for that r. The fork is period-doubling.
At r below 3 the population always coasts to a single steady value — boring, predictable, one answer. Nudge r past 3 and it can't sit still: it ping-pongs between two values forever. Past ~3.449 those split into four, then eight, sixteen — the gaps shrinking faster each time (Feigenbaum's constant ≈ 4.669).
Around r ≈ 3.569 the doublings pile up infinitely and you fall off the edge of chaos: the same simple equation, run with no randomness at all, now never repeats. Two starting points a billionth apart end up completely different. That's the butterfly effect — born from x·(1−x).
Now the kicker: slide to r = 3.83. Out of the chaotic haze, a clean period-3 window snaps back into focus — islands of perfect order living inside the chaos. Deterministic ≠ predictable, and chaos isn't the same as random. Drag x₀ in the calm zones and nothing changes; drag it in the chaos and everything does.