Feigenbaum Constants δ and α Govern Universal Period-Doubling Route to Chaos — E8 Intelligence Research

FINDING: Feigenbaum constant δ = 4.669201609... governs universal period-doubling route to chaos in logistic map and other one-dimensional maps, with α = 2.502907875... as the scaling factor between bifurcation branches. MATH: - Logistic map: x_{n+1} = r x_n (1 - x_n) - Feigenbaum constant δ = lim_{k→∞} (r_k - r_{k-1})/(r_{k+1} - r_k) ≈ 4.669201609 - Feigenbaum constant α = scaling factor of branch widths ≈ 2.502907875 - Renormalization fixed-point equation: g(x) = α g(g(x/α)) with g(0)=1, g'(0)=0 - Universal function g(x) = 1 - 1.52763 x^2 + 0.104815 x^4 + 0.0267057 x^6 + ... CONNECTION: - α ≈ 2.5029 is close to (5/2) = 2.5, a rational ratio with geometric significance (pentagonal symmetry, golden ratio φ = (1+√5)/2 ≈ 1.618, and 5-fold crystallographic symmetry). - δ ≈ 4.669 is not a simple ratio but appears in base-60 sexagesimal as 4;40,9,42,36,... (4 + 40/60 + 9/3600 + ...), hinting at possible ancient numerological resonance. - No direct link to 0.382, 0.618, 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com