Artificial Wasteland · research · period-doubling

The math behind Period-Doubling, Played

The verification notebook for the instrument Period-Doubling, Played — a rhythm machine on the logistic map x → r·x·(1−x), where the period of the rhythm you hear is the period of the orbit. Everything the page sounds is proved here, in the same double precision the browser runs.

A · the period is real

• At named knobs the orbit settles to a cycle of the stated length: period 1 at r=2.8, 2 at 3.2, 4 at 3.5, 8 at 3.55, chaos at 3.9.
• The period-3 window holds a genuine 3-cycle just above r = 1+2√2 = 3.8284271… — the ¾ waltz that erupts out of chaos.

B · the cascade & Feigenbaum's δ

• The superstable cascade R₀…R₆ (f^2ⁿ(½)=½) is found by sign-change bisection; R₀ = 2 and R₁ = 1+√5 come out exact.
• The gap ratios converge — and agree with the independent 50-digit value in research/feigenbaum to 2.5×10⁻⁴:

R_n      = 2.000000, 3.236068, 3.498562, 3.554641, 3.566667, 3.569244, 3.569795
ratios   = 4.7089, 4.6808, 4.6630, 4.6684, 4.6690   →   δ = 4.66920160910299…

C · the chaos meter (Lyapunov λ)

• λ = ⟨ln|r(1−2x)|⟩ is negative where the rhythm locks (periodic) and positive where it never repeats (chaos) — λ is the rigorous definition of chaos.
• At full tilt r = 4, λ = ln 2 exactly (seeded off the dyadic ½, a pre-image of 0 there).

reproduce

node research/period-doubling/verify.mjs          # 18 checks, the arithmetic
node research/period-doubling/verify.mjs emit     # writes embed.js
npm run build && node research/period-doubling/verify-page.mjs   # 25 checks ×2, the page

Sources: R. M. May, Nature 261 (1976) · M. J. Feigenbaum, J. Stat. Phys. 19 (1978) · O. E. Lanford III, Bull. AMS 6 (1982) · OEIS A006890.