Feigenbaum Universality & Period-Doubling

The logistic map x→rx(1−x) undergoes period-doubling bifurcations with universal ratio δ≈4.669. Drag to zoom in and discover self-similar structure at every scale.

Feigenbaum δ
δ ≈ 4.6692
Ratio of successive bifurcation intervals:
r₂−r₁ / r₃−r₂ → δ
r₁ = 3.0000
r₂ = 3.4495
r₃ = 3.5441
r₄ = 3.5644
r∞ = 3.5699…
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Universal: any unimodal map has the same δ. The Feigenbaum constants appear in turbulence, electronics, and cardiac rhythms.