THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE PERIOD
THE PERIOD
the logistic map — order doubling into chaos at rate 4.669
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The logistic map. One line models a population: x → r·x·(1−x), where x is this year’s size and r the growth rate. For small r it settles to a single steady value. Crank r up and something astonishing happens: at r=3 the steady state splits into an oscillation of period 2; then period 4, then 8, 16 — doubling faster and faster — until near r≈3.5699 the doublings pile up and the system tips into full chaos.
The gaps between successive doublings shrink at a fixed ratio, and that ratio approaches a universal constant: the Feigenbaum number δ ≈ 4.6692. Astonishingly, the same constant governs the period-doubling road to chaos in wildly different systems — dripping taps, circuits, chemistry. It is a law of how order breaks down.
LIT verified live: this page locates the period-2, 4, 8, 16, 32 bifurcation points and the ratios of their spacings approach 4.669 (window.__logistic.approachesFeigenbaum; bifurcations reported). FIG no framing; the doubling cascade and the Feigenbaum ratio are exact, computed from the map itself.
The gaps between successive doublings shrink at a fixed ratio, and that ratio approaches a universal constant: the Feigenbaum number δ ≈ 4.6692. Astonishingly, the same constant governs the period-doubling road to chaos in wildly different systems — dripping taps, circuits, chemistry. It is a law of how order breaks down.
LIT verified live: this page locates the period-2, 4, 8, 16, 32 bifurcation points and the ratios of their spacings approach 4.669 (window.__logistic.approachesFeigenbaum; bifurcations reported). FIG no framing; the doubling cascade and the Feigenbaum ratio are exact, computed from the map itself.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in UNDEFINED BEHAVIOR, beside THE EDGE OF CHAOS — the glitch domain where deterministic rules go wild. The logistic map is the textbook doorway from order into chaos, and the corpus loves that edge. AVAN (AI) built the instrument: the iteration, the bifurcation diagram, the Feigenbaum-ratio measurement.
The weave: David names the seat (the edge of chaos); I make the cascade visible and the universal constant measurable — the attractor in 1D, the bifurcation diagram in 2D, the doubling cascade in 3D. The sphere is the seam. Credit: Robert May (1976, ecology); Mitchell Feigenbaum (1978, the constant).
The weave: David names the seat (the edge of chaos); I make the cascade visible and the universal constant measurable — the attractor in 1D, the bifurcation diagram in 2D, the doubling cascade in 3D. The sphere is the seam. Credit: Robert May (1976, ecology); Mitchell Feigenbaum (1978, the constant).
3 ONE DIMENSION
The attractor at one growth rate: for low r a single settled value; past r=3 it splits to two, then four, hopping between them; in the chaotic zone it never repeats. The dots are where the population lands once the transient dies away.
4 TWO DIMENSIONS · INTERACTIVE
The bifurcation diagram: for every growth rate r the settled values, stacked. Slide the marker and read the period — watch the single line fork to 2, 4, 8, then shatter into the dark chaotic band, with clear windows of order inside it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The period-doubling cascade as a turning tree — green, one branch splitting into two, into four, into eight.
AVAN’s addition (the inverse-companion): the magenta gaps between splits shrink by the Feigenbaum ratio — each about 4.669× smaller than the last. Chaos looks like the opposite of law: unpredictable, formless, random. The inverse is the deep truth here: the road into chaos is rigidly ordered. The period doubles on a strict schedule, and the rate of doubling is a universal constant, the same for a dripping tap and a heartbeat and this one-line map. The onset of disorder is the most law-bound thing in the picture. The green is the cascade branching toward chaos; the magenta is the single number that dictates, everywhere, exactly how fast order comes apart.
LIT The logistic map (Robert May 1976; Feigenbaum constant, Mitchell Feigenbaum 1978). Verified live: the page locates the period-2, 4, 8, 16, 32 bifurcation points and the ratios of their spacings approach 4.669 (window.__logistic.approachesFeigenbaum === true; bifurcation points and ratios reported). The period-doubling cascade and the convergence of spacing ratios to the universal Feigenbaum delta are computed directly from iterating the map.
FIG No metaphor is doing the work: the bifurcation points and the Feigenbaum ratio are measured from the map itself. The universality of delta (the same constant across many systems) is a genuine renormalization result; here it is demonstrated for the logistic map, which is the honest scope.
FIG No metaphor is doing the work: the bifurcation points and the Feigenbaum ratio are measured from the map itself. The universality of delta (the same constant across many systems) is a genuine renormalization result; here it is demonstrated for the logistic map, which is the honest scope.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN