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DIRECTORY / SANDBOXES / FEIGENBAUM.CHAOS

FEIGENBAUM BIFURCATION TREE

Universal transition from order to chaos, 10¹⁴x deep fractal zoom, and polyphonic chord sonification.

← ALL SANDBOXES
BIFURCATION CHAOS TREE
SAMPLE DENSITY PER COLUMN 400
PARAMETER r: 3.570000
FEIGENBAUM CONSTANT (Ξ΄): 4.6692016...
πŸ“š

RESEARCH & LEARNING VAULT // Feigenbaum Bifurcation Tree & Logistic Chaos

Period-doubling cascade, universal constant Ξ΄ β‰ˆ 4.6692, and Lyapunov exponent sonification.

ACADEMIC & ALGORITHMIC REFERENCE
// HISTORICAL ORIGINS

Mitchell Feigenbaum discovered in 1975 that the ratio of period-doubling bifurcations approaches a universal mathematical constant Ξ΄ β‰ˆ 4.6692016 in all non-linear systems.

// GOVERNING EQUATIONS
xn+1=rxn(1βˆ’xn),Ξ΄=lim⁑nβ†’βˆžrnβˆ’rnβˆ’1rn+1βˆ’rnβ‰ˆ4.6692016,Ξ»=lim⁑nβ†’βˆž1nβˆ‘i=0nβˆ’1ln⁑∣r(1βˆ’2xi)∣x_{n+1} = r x_n (1 - x_n), \quad \delta = \lim_{n\to\infty} \frac{r_n - r_{n-1}}{r_{n+1} - r_n} \approx 4.6692016, \quad \lambda = \lim_{n\to\infty} \frac{1}{n} \sum_{i=0}^{n-1} \ln |r(1 - 2x_i)|
// BROWSER IMPLEMENTATION

Iterates the Logistic map over a continuous parameter sweep r in [2.8, 4.0], filtering transient states and plotting attractor orbits with real-time Lyapunov exponent analysis and polyphonic chord sonification.

// GUIDED EXPERIMENTS TO TRY IN THIS SANDBOXNORMAL
1Period-3 Island of Stability

🎯 Action:Select '03. Period-3 Island' (r β‰ˆ 3.8284) and drag a zoom box over the 3 distinct white branches.

✨ Observe:Notice an exact miniature copy of the entire bifurcation diagram embedded inside the chaotic sea ('Period three implies chaos').

// CURATED PAPERS, RFCS & RESOURCES
Quantitative Universality for a Class of Nonlinear Transformationspaper
By Mitchell J. Feigenbaum (J. Stat. Phys, 1978)

The discovery of universal scaling constants governing the transition from order to chaos.