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Bifurcation Hoodie

SPLIT, SPLIT, SPLIT, CHAOS | Pixelated Physics

\[x_{n+1} = r\,x_n(1-x_n)\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

One line of arithmetic: multiply by r, multiply by one minus itself, repeat. The logistic map x → r x(1 − x) from r = 2.8 to 4: period doubling at a rate set by δ = 4.669…, then chaos with windows of order. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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The physics

The logistic map is a single line of arithmetic, yet as r increases its long-term value splits into 2, 4, 8 values and then becomes chaotic. The splittings come closer together by a factor that approaches Feigenbaum's constant, the same for a whole family of maps.

One line of arithmetic: multiply by r, multiply by one minus itself, repeat.

For small r the value settles on a single number. At r = 3 it starts alternating between two, then four, then eight. Each split comes sooner than the last, by a factor that tends to δ = 4.669…, the same number for a whole class of maps. Mitchell Feigenbaum found that in 1978.

Past r ≈ 3.5699 the orbit never settles. The art plots where it lands after a long transient, for each r, shaded by how often. Inside the chaos are thin windows where order returns, the widest a period-3 window at r = 1 + √8.

Robert May brought the map to ecologists in 1976 as a warning: very simple models can do very complicated things.

Split, split, split, chaos.

Equations

\[x_{n+1} = r\,x_n(1-x_n)\]
\[\delta = \lim_{k\to\infty}\frac{r_k - r_{k-1}}{r_{k+1}-r_k} = 4.6692\ldots\]
SymbolMeaningUnit
\(x_n\) population as a fraction of its maximum, step n \(\mathrm{1}\)
\(r\) growth parameter \(\mathrm{1}\)
\(r_k\) value of r at the k-th period doubling \(\mathrm{1}\)
\(\delta\) Feigenbaum constant \(\mathrm{1}\)

Sources

  1. May (1976) Simple mathematical models with very complicated dynamics, Nature 261, 459
  2. Feigenbaum (1978) Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19, 25
  3. Strogatz, Nonlinear Dynamics and Chaos, Ch. 10

Source May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature 261, 459–467.doi.org/10.1038/261459a0

Field notes

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