Feigenbaum Constants δ and α: Universal Scaling in Period-Doubling Chaos — E8 Intelligence Research

FINDING: Feigenbaum constant δ = 4.6692016... governs universal period-doubling route to chaos in logistic map and many nonlinear systems; α = 2.502907875... is the scaling factor for bifurcation branches. | MATH: Logistic map: \( x_{n+1} = r x_n (1 - x_n) \). Feigenbaum δ = \(\lim_{n\to\infty} \frac{r_{n} - r_{n-1}}{r_{n+1} - r_n} = 4.669201609102990...\). Feigenbaum α = \(\lim_{n\to\infty} \frac{d_n}{d_{n+1}} = 2.502907875095892...\) where \(d_n\) is branch width at bifurcation. Renormalization group fixed-point function \( g(x) = \alpha g(g(x/\alpha)) \) with \(g(0)=1, g'(0)=0\). | CONNECTION: No direct geometric ratio (0.382, 0.618, 1.618) appears. However, α ≈ 2.5029 is close to 2.5, a rational ratio, but not a known sacred constant. δ is transcendental, unrelated to golden ratio. No base-60 or crystallographic symmetry link. | DEPTH: 9 — Profound universality across fluid dynamics, biology, electronics, quantum systems. Reveals that chaos emerges via a universal scaling law indep Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com