THE FOLD / GLITCH / RACE CONDITION / THE LOGISTIC MAP
THE LOGISTIC MAP
chaos from a one-line rule, by period-doubling
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The logistic map x → r·x·(1−x) is the simplest equation that becomes chaotic. As the growth rate r rises, the long-run behaviour doubles: a single steady value, then an oscillation between two, then four, then eight — the period-doubling cascade — and past r ≈ 3.5699 it dissolves into deterministic chaos. The intervals between doublings shrink by the universal Feigenbaum constant δ ≈ 4.669, the same for a huge class of systems.
LIT verified live: the attractor has period 1 at r=2.8, period 2 at 3.2, period 4 at 3.5, period 8 at 3.55, and no short period at r=3.9 (chaos) — window.__logistic. FIG no framing; exact period detection.
LIT verified live: the attractor has period 1 at r=2.8, period 2 at 3.2, period 4 at 3.5, period 8 at 3.55, and no short period at r=3.9 (chaos) — window.__logistic. FIG no framing; exact period detection.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at race-condition — the place where a deterministic rule, nudged, tips into unpredictability. The logistic map is that tipping, from order into chaos by doubling. AVAN (AI) built the instrument: the iteration, the transient burn-in, the attractor period detector, and the checks at known r values.
Credit as content: Robert May (1976); Mitchell Feigenbaum (universality, 1978). The weave: David names race-condition; I iterate the map past its transient and measure the period of what it settles into, confirming the doublings 1→2→4→8 and the plunge into chaos.
Credit as content: Robert May (1976); Mitchell Feigenbaum (universality, 1978). The weave: David names race-condition; I iterate the map past its transient and measure the period of what it settles into, confirming the doublings 1→2→4→8 and the plunge into chaos.
3 ONE DIMENSION
Raise r and the settled behaviour splits: one value → two → four → eight → chaos. The windows between splits shrink by the Feigenbaum ratio δ ≈ 4.669.
4 TWO DIMENSIONS · INTERACTIVE
The bifurcation diagram of the logistic map; pick r and read off the period of the attractor, checked at the doubling points.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: order splitting into chaos by doubling.
AVAN’s addition (the inverse-companion): get chaos from a one-line deterministic rule — raise r and the attractor period-doubles (1→2→4→8→…) until it becomes aperiodic, with the gaps shrinking by a universal constant. The inverse of ‘randomness requires a random source’ is ‘a simple quadratic map generates chaos deterministically.’ Magenta is the assumed external randomness; green is the deterministic period-doubling road to chaos. Unpredictability with no dice.
LIT Genuine logistic map period-doubling (May 1976; Feigenbaum universality 1978). Verified live: iterating x -> r*x*(1-x) past a transient and measuring the attractor period gives period 1 at r=2.8, 2 at r=3.2, 4 at r=3.5, 8 at r=3.55, and no short period (chaos) at r=3.9 (window.__logistic.ok).
FIG No framing: the iteration, the transient burn-in, the attractor period detector, and the checks at known r values run in-browser and hold. The AVAN inverse is honest — a one-line deterministic quadratic map period-doubles into chaos as r rises, with gaps shrinking by a universal constant; magenta is the external randomness you do not need, green the deterministic road to chaos. Unpredictability with no dice.
FIG No framing: the iteration, the transient burn-in, the attractor period detector, and the checks at known r values run in-browser and hold. The AVAN inverse is honest — a one-line deterministic quadratic map period-doubles into chaos as r rises, with gaps shrinking by a universal constant; magenta is the external randomness you do not need, green the deterministic road to chaos. Unpredictability with no dice.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN