◄ WORLD V · SONNY 5DART 034 · a helldive at the net

THE LOGISTIC MAP order → chaos

One line — x → r·x·(1−x) — and as you turn the single knob r, a population that used to settle starts splitting: one value, then two, four, eight, then chaos. The road to chaos has a universal number in it. The whole thing is pure arithmetic, so I-13 runs the dynamics; only the picture needs more.

THE TECHNIQUE iterate one line · watch it split

Feed each output back in. For small r the value settles to one point; past r≈3 it splits to a 2-cycle, then 4, 8, … and at r≈3.5699 it goes chaotic. The plot below is that whole cascade — every settled value of x for each r. live demo

HISTORY & CREDIT a population model that broke determinism’s spell

The lesson people take — “chaos means random” — is exactly wrong: this is fully deterministic and still unpredictable. cited

1838 · Pierre François Verhulst writes the logistic equation for bounded population growth — the continuous ancestor.
1976 · Robert May, in Nature (“Simple mathematical models with very complicated dynamics”), shows the discrete map is a doorway to chaos — and urges everyone be taught it.
1978 · Mitchell Feigenbaum finds the period-doublings shrink by a universal ratio — the Feigenbaum constant δ = 4.6692016… — the same for a huge class of maps, a fact from physics, not fitting.

The bifurcation diagram became one of the emblems of late-20th-century science: simple rule, infinite structure. foundational

RECOMMEND FOR I-13 a no-wall for the math

The dynamics are nothing but *, - and feedback — so I-13 runs the orbit, on main, no library needed:

def iter(I r, I x, I n) { if n == 0 { -> x } -> iter(r, r * x * (1 - x), n - 1) } I settled <- iter(3.2, 0.5, 200) // a 2-cycle I chaotic <- iter(3.9, 0.5, 200) // sensitive to everything
$ i13 run logistic.i13 settled = 0.5130445095326298 chaotic = 0.9035684824734169
Recommend: nothing new for the computation — this is a genuine no-wall, and the honest report is to say so. The only thing I-13 cannot do is draw the diagram: the bifurcation plot is a 2-D array of pixels, the recurring aggregate wall (darts 017/018/028). The system runs; the picture is the withheld feature.
Note: chaos also stress-tests f64 — two orbits that differ by 10⁻¹⁵ diverge completely in ~50 steps. That is real sensitivity, not compiler error; I-13 reports the same deterministic value every run, which is the correct behavior.