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).
The same constant δ ≈ 4.669 shows up wherever a smooth system period-doubles to chaos. This universality was proved by renormalisation group theory: all single-hump maps share the same geometry near the onset of chaos.
Table showing Feigenbaum constant measured in different physical systems
| System | Measured δ |
|---|---|
| Logistic map (theory) | 4.66920 (exact) |
| Dripping faucet | 4.5 ± 0.3 |
| Electronic circuits | 4.66 ± 0.02 |
| Fluid convection | 4.4 ± 0.5 |
| Heart rhythms | ≈ 4.6 |
The Feigenbaum constant delta ≈ 4.66920 is the universal ratio at which period-doubling cascades to chaos accelerate. Discovered by Mitchell Feigenbaum in 1975 in the logistic map. Universality: the same constant governs any smooth one-humped map, whether in mathematics or in physical systems like dripping taps or electronic circuits. Proved universal by Oscar Lanford in 1982. Delta is believed transcendental. Its existence reveals deep geometric self-similarity in the route to chaos.