Feigenbaum Constant δ Links Logistic Map Period-Doubling to Mandelbrot Self-Similarity — E8 Intelligence Research

FINDING: The logistic map's period-doubling cascade converges at Feigenbaum constant δ ≈ 4.669, which is universal across unimodal maps and directly linked to the Mandelbrot set's self-similar structure. MATH: - Logistic map: \( xₙ₊₁ = r x_n (1 - x_n) \) - Period-doubling bifurcation points \( r_n \) converge: \( δ = limn → ∞ {r_n - rₙ₋₁}{rₙ₊₁ - r_n} ≈ 4.669201609 \) - Mandelbrot set: \( zₙ₊₁ = z_n^2 + c \), with period-doubling cascade along the real axis (c → -2) - Feigenbaum constant α ≈ 2.5029 (scaling of bifurcation branches) - Fibonacci numbers appear in Mandelbrot set's periodic orbit counts (e.g., period-3,5,8 bulbs) via rational rotation numbers. CONNECTION: - δ ≈ 4.669 is not a classical sacred ratio, but the cascade's scaling factor α ≈ 2.5029 relates to the golden ratio φ ≈ 1.618 via \( α ≈ φ + 1/φ = 2.618 \) (within 4.4% error). - The Mandelbrot set's main cardioid and bulbs exhibit φ-based rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com