One of the most famous pictures in science: as a single parameter grows, a simple map doubles its period again and again, then erupts into chaos.
Bifurcation DiagramLive
logistic map · route to chaos
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The logistic map xₙ₊₁ = r·xₙ(1−xₙ) doubles its period again and again as r grows, then dissolves into chaos near r ≈ 3.57 — with windows of order inside.
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How it works
The logistic map xₙ₊₁ = r·xₙ(1−xₙ) is iterated for each value of r and its long-term values plotted. The result is the bifurcation diagram — a period-doubling cascade to chaos, with the universal Feigenbaum constant governing the spacing.
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