THE LOGISTIC MAP one knob from a fixed point to chaos — x → r·x·(1−x)
A population model so simple it fits on a napkin: x → r·x·(1−x). Turn the one knob r and watch the fixed point come apart. Low r: a single stable fixed point (the population settles). Past r≈3: the fixed point goes unstable and splits into a 2-cycle, then 4, then 8 — period-doubling — and past r≈3.57: chaos, never repeating, exquisitely sensitive to the start. Same one line, deterministic throughout — yet it crosses from a still point to unpredictability by turning a dial. The doublings arrive at a universal rate (Feigenbaum's δ≈4.669), the same for a huge class of maps.
THE TECHNIQUE x → r x(1−x); fixed point → period-doubling → chaos
The demo runs the map at r=2 (settles to a fixed point 0.5) and r=3.2 (a 2-cycle, no fixed point): live demo
HISTORY & CREDIT Robert May, 1976 · Feigenbaum 1978
“Simple deterministic rules give simple behaviour.” — one quadratic line goes from a stable point to full chaos as you turn a single knob. Complexity from nothing. cited
1838 · Verhulst — the logistic equation (continuous). 1976 · Robert May — “Simple mathematical models with very complicated dynamics” (the discrete map, in Nature). 1978 · Mitchell Feigenbaum — the universal period-doubling constant δ≈4.669.
A still point, then two, then four, then chaos — all from turning one dial on one line. The fixed point, dissolving. May 1976
RECOMMEND FOR I-13 fixed point vs 2-cycle, on the compiler
On the canonical compiler, r=2 settles on the fixed point 0.5; r=3.2 has no fixed point but a 2-cycle (0.799 ↔ 0.513):
$ i13 run f_logisticmap.i13 # x -> r x(1-x)
RUN OK · 4050 step(s) · peak stack 4 · call depth 82
fp2 = 0.49999999999999994 -- r=2 : a single stable fixed point
a = 0.7994... bb = 0.5130... period2 = 1 -- r=3.2 : the point split into a 2-cycle
Recommend: the logistic map is the fixed point coming apart — one knob from a still point through period-doubling to chaos — and i13 shows the split (fixed point at r=2, 2-cycle at r=3.2). Not a keeper (the bifurcation cascade is a theorem the map obeys — witnessed, and universal via Feigenbaum). The dart that shows a fixed point is not always the end of the story: turn the dial and it fractures into everything.