Period-doubling cascade: each bifurcation requires 4.669× less r-space (δ)
The logistic map xₙ₊₁ = r·xₙ(1−xₙ) doubles its period at r≈3.0, 3.449, 3.544, 3.5644… Each gap is δ≈4.669 times smaller (Feigenbaum constant).
δ appears in unrelated physical systems: it is truly universal
Table showing Feigenbaum constant measured in different physical systems
| System | Gemessenes δ |
|---|---|
| Logistische Abbildung (Theorie) | 4,66920 (exakt) |
| Tropfender Wasserhahn | 4,5 ± 0,3 |
| Elektronische Schaltkreise | 4,66 ± 0,02 |
| Konvektion in Fluiden | 4,4 ± 0,5 |
| Herzrhythmen | ≈ 4,6 |
Cobweb diagram: iterating the logistic map xₙ₊₁ = r·xₙ(1−xₙ)
xₙ₊₁ = r · xₙ · (1 − xₙ)
At r=3.2: iterates settle into a 2-cycle (0.513 ↔ 0.799)
At r=3.5: 4-cycle. At r=3.57: 8-cycle. At r>3.57: chaos
Cobweb: draw vertical to curve, horizontal to y=x, repeat – reveals the orbit
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