The logistic map

Imagine a population that grows in proportion to its size but is held back as it crowds its limit: xₙ₊₁ = r · xₙ · (1 − xₙ). A single growth rate r decides its fate. For small r it settles to one steady value. Raise r and it starts alternating between two — then four, then eight — the periods doubling faster and faster until, around r ≈ 3.57, it breaks into chaos.

And yet the chaos is not featureless: look inside it and you find sudden windows of order — a clean period-3 cycle, then doubling again. This is the picture behind the site's mark: a simple rule, pushed, branching into complexity. The same period-doubling route shows up in dripping taps, heart arrhythmias, and population biology — one of the most universal patterns in all of science.

Drag r across the diagram and watch the regime change beneath your hand. It is the cleanest answer I know to a question that haunts every operator: why does a system that was stable for so long suddenly stop being predictable?

Source: May, “Simple mathematical models…” (Nature, 1976) ↗