Skills that automate
the academic grind.

Reading brief

Why does a one-line model go chaotic, and what is universal about it?

1. Sharkovskii (1964), Coexistence of cycles of a continuous map of the line into itself. Ukrainian Mathematical Journal 16, 61–71. Orders the integers so that a continuous interval map with a cycle of period m must have cycles of every period that comes after m. Period 3 sits at the top: it forces all the others.
2. Li & Yorke (1975), Period three implies chaos. The American Mathematical Monthly 82(10), 985–992. Independently shows that period 3 forces every period, plus an uncountable set of non-periodic orbits. This is where "chaos" got its mathematical name.
3. May (1976), Simple mathematical models with very complicated dynamics. Nature 261, 459–467. Brings the logistic map to a wide audience, walks through the period-doubling route, and argues it should be taught early.
4. Feigenbaum (1978), Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics 19(1), 25–52. The gaps between successive period doublings shrink by a universal factor δ ≈ 4.6692, the same for any smooth map with a quadratic maximum.

How they connect

Sharkovskii and Li–Yorke say which periods must coexist. May shows the route a real model takes. Feigenbaum explains why that route looks the same across models.

Read next

Lanford (1982), A computer-assisted proof of the Feigenbaum conjectures, Bulletin of the AMS 6(3), 427–434: the step from numerical observation to proof. Open thread: δ depends on the order of the map's maximum, so a non-quadratic peak lands in a different universality class.

Analysis note

Estimating δ from superstable parameters

Result. δ ≈ 4.669202, agreeing with the reference value 4.669201609… to seven significant figures (error 7 × 10⁻⁸ at n = 11).

Method. R_n is the parameter at which x = ½ lies on a cycle of period 2ⁿ. Each one is found by Newton's method on f_r²ⁿ(½) − ½, starting from the Feigenbaum-scaled guess. Then δ_n = (R_n−1 − R_n−2) / (R_n − R_n−1).

Caveats. The table stops at n = 11 on purpose. Beyond it the estimate gets worse in double precision: the error grows to 10⁻⁶ at n = 12, and by n = 16 the third decimal is wrong. Each R_n carries round-off from 2ⁿ iterations, while the gaps it is divided by shrink like δ⁻ⁿ. Going further needs extended precision. Superstable points and bifurcation points converge to the same δ. Superstable ones are used here because Newton converges on them cleanly.

# Newton step for R_n, carrying d/dr along the orbit
x, dx = 0.5, 0.0
for _ in range(2**n):
    x, dx = r*x*(1 - x), x*(1 - x) + r*(1 - 2*x)*dx
r -= (x - 0.5) / dx