Logistic Chaos

iterated-maps · 37 views

iterated-mapschaotic

What It Is

The simplest chaotic system --- one-line recurrence x(n+1) = 4x(1-x) at r=4, filling the unit interval ergodically with a beta(½,½) distribution

Interpretation

Standard analysis sees: bounded / light-tailed; aperiodic / broadband; time-irreversible (slow rise, sharp collapse); homoskedastic. The atlas additionally detects discrete-map sensitive dependence, Time Reversibility:ordinal_reversal_distance, Attractor Reconstruction:lyap_entropy.

What standard analysis sees

tail heaviness0.09

asymmetry0.37

occupancy0.79

short-range corr0.33

long-range memory0.35

spectral colour0.68

periodicity0.08

complexity0.44

time-irreversibility0.05

volatility clustering0.07

multifractality0.21

dimensionality0.83

nonstationarity0.18

What the atlas adds

discrete-map sensitive dependence+3.9z

deterministic chaos (positive λ_max, sensitive dependence)

names a discrete-map-scoped estimate, NOT chaos in general — continuous-flow chaos (Lorenz, Rössler) reads weak/neutral here despite being genuinely chaotic; spiky arithmetic sources can false-positive on the finite-time estimate

Time Reversibility:ordinal_reversal_distance+2.8z

time-reversal symmetry breaking via Bandt-Pompe ordinal patterns

Attractor Reconstruction:lyap_entropy+2.5z

Kolmogorov-Sinai entropy production rate (sum of positive λ)

Atlas-extreme metrics the standard bank can’t predict for this source

Hölder Regularity:alpha_autocorrelation +5.6z bank-miss 2.2σ