What question did this study set out to answer?

To explore how Feigenbaum constants δ and α illustrate universal scaling laws in chaotic systems like the logistic map.

July 26, 2026Open Access

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

Key Points

To explore how Feigenbaum constants δ and α illustrate universal scaling laws in chaotic systems like the logistic map.
Analyzed the logistic map equation to identify constants δ and α governing period-doubling chaos.
Utilized mathematical limits to derive Feigenbaum constants and their implications in nonlinear systems.
Examined connections with chaotic behaviors and their independence from known geometric ratios.
Feigenbaum constant δ was found to be approximately 4.669, indicating the universal period-doubling route to chaos.
Feigenbaum constant α approximately equals 2.503, serving as a scaling factor for bifurcation branches in chaotic systems.
Chaos emerges through universal scaling laws that apply across diverse fields such as biology, fluid dynamics, and electronics.

Abstract

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₍+₁ = r xₙ (1 - xₙ) \). Feigenbaum δ = \ (₍ r₍ - ₑ_₍-₁r₍+₁ - rₙ = 4. 669201609102990. . . \). Feigenbaum α = \ (₍ dₙd₍+₁ = 2. 502907875095892. . . \) where \ (dₙ\) is branch width at bifurcation. Renormalization group fixed-point function \ (g (x) = g (g (x/) ) \) 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

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