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; blue spectrum (high-frequency power); 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 heaviness0.12
asymmetry0.31
occupancy0.79
short-range corr0.20
long-range memory0.33
spectral colour0.85
periodicity0.16
complexity0.43
time-irreversibility0.05
volatility clustering0.07
multifractality0.32
dimensionality0.84
nonstationarity0.19
What the atlas adds
discrete-map sensitive dependence+3.8z
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 estimateTime 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 | +5.5z | bank-miss 1.8σ |
Composition
dtypefloat64
range[5.996e-09, 1]
unique values16384 / 16384
mean ± std0.5 ± 0.354
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Logistic_Chaos.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Logistic_Chaos.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Logistic_Chaos.png)
Inflation (Substitution) > barcode
/d_curve/Logistic_Chaos.png)
Inflation (Substitution) > d_curve
Information Theory > default
Julia Set > escape_histogram
Klein Bottle > default
Laplacian > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Ordinal Partition > default
/phi_spectrum/Logistic_Chaos.png)
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > h1_diagram
Predictability > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > phase_space
/default/Logistic_Chaos.png)
Ulam Spiral (Sacks) > default
/default/Logistic_Chaos.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Uniform Chaos (Logistic Scramble) | 2.79 | |
| Tent Map | 3.02 | |
| Baker Map | 3.18 |
Which Geometries Light Up
Cayley › Cayley:delta_hyp_norm | rank 5/307 | 0.2611 |
Hölder Regularity › Hölder Regularity:alpha_autocorrelation | rank 2/307 | 0.4235 |
Symplectic › Symplectic:phase_reflection_symmetry | rank 303/307 | -0.0337 |
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