Feigenbaum Constant δ: Universal Scaling in Period-Doubling Bifurcation Cascades — E8 Intelligence Research

Abstract

FINDING: Feigenbaum constant δ ≈ 4.669201609 emerges as universal scaling factor in period-doubling bifurcation cascades across all unimodal maps, independent of the specific system. MATH: - Feigenbaum constant δ = lim_{n→∞} (μ_n - μ_{n-1})/(μ_{n+1} - μ_n) ≈ 4.669201609 - Second Feigenbaum constant α ≈ 2.502907875 (scaling of bifurcation branches) - Universal function g(x) satisfies functional equation: g(x) = α g(g(x/α)) - Renormalization group fixed point: g(x) = 1 - 1.52763x² + 0.104815x⁴ + ... - Connection to logistic map: x_{n+1} = r x_n (1 - x_n), bifurcation points μ_n converge geometrically with ratio δ CONNECTION: - δ ≈ 4.669 ≈ 1/0.214 — no direct golden ratio link, but α ≈ 2.503 ≈ φ² + 0.885 (weak) - Period-doubling cascade exhibits self-similarity with scaling factor α, reminiscent of fractal geometry - No direct base-60 or crystallographic symmetry; however, renormalization group structure mirrors lattice renormalization in statistical physics - The univ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com