FEIGENBAUM BIFURCATION TREE
Universal transition from order to chaos, 10ΒΉβ΄x deep fractal zoom, and polyphonic chord sonification.
RESEARCH & LEARNING VAULT // Feigenbaum Bifurcation Tree & Logistic Chaos
Period-doubling cascade, universal constant Ξ΄ β 4.6692, and Lyapunov exponent sonification.
Mitchell Feigenbaum discovered in 1975 that the ratio of period-doubling bifurcations approaches a universal mathematical constant Ξ΄ β 4.6692016 in all non-linear systems.
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)|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.
π― Action:Select '03. Period-3 Island' (r β 3.8284) and drag a zoom box over the 3 distinct white branches.
β¨ Observe:Notice an exact miniature copy of the entire bifurcation diagram embedded inside the chaotic sea ('Period three implies chaos').