Logistic r=3.74 (Period-5 Window)
chaos · 36 views
chaos
What It Is
Logistic map inside the period-5 stability window at r=3.74 --- an island of order embedded in the chaotic regime, intermittent near its boundary
Interpretation
Standard analysis sees: left-skewed; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; low-dimensional; stationary. The atlas finds no named structure, but the source is distinctively extreme on Julia Set:potential_variance (+4.8z) — beyond what the standard bank predicts for it. It sits beside Pell Word in the atlas (standard-bank rank 28) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.33
asymmetry0.12
occupancy0.15
short-range corr0.05
long-range memory0.02
spectral colour0.93
periodicity0.83
complexity0.25
time-irreversibility0.08
volatility clustering0.05
multifractality0.04
dimensionality0.00
nonstationarity0.07
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Julia Set:potential_variance | +4.8z | bank-miss 1.1σ |
Julia Set:potential_smoothness | +4.7z | bank-miss 1.0σ |
Fisher Information:velocity_spectral_gini | -4.2z | bank-miss 2.4σ |
Klein Bottle:linear_complexity | -3.9z | bank-miss 1.1σ |
Attractor Reconstruction:d2_saturation | -3.4z | bank-miss 2.7σ |
Julia Set:escape_entropy | -2.9z | bank-miss 1.1σ |
AutoRegressive:ar_coef_10 | +2.8z | bank-miss 1.5σ |
Isochronicity:frequency_shear | +2.6z | bank-miss 1.9σ |
Composition
dtypefloat64
range[0.2275, 0.9349]
unique values10 / 16384
mean ± std0.632 ± 0.253
Fixed alphabet — only 10 distinct symbols across 16384 samples.
Render Gallery
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Native > phase_plotraw
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Native > timeseriesraw
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Native > triple_latticeraw
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Boltzmann > lag_map
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Boltzmann > spin_grid
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Cayley > default
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Chladni > plate_bottom
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Chladni > plate_top
_(centered)/signed_log_z/Logistic_r=3.74_(Period-5_Window).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Logistic_r=3.74_(Period-5_Window).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Logistic_r=3.74_(Period-5_Window).png)
Inflation (Substitution) > barcode
/d_curve/Logistic_r=3.74_(Period-5_Window).png)
Inflation (Substitution) > d_curve
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Information Theory > default
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Julia Set > escape_histogram
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Klein Bottle > default
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Laplacian > default
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Logarithmic Spiral > default
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Möbius-S³ > mobius_cylinder
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Möbius-S³ > s2_base_map
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Ordinal Partition > default
/phi_spectrum/Logistic_r=3.74_(Period-5_Window).png)
Penrose (Quasicrystal) > phi_spectrum
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Persistent Homology > barcode
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Persistent Homology > h1_diagram
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Predictability > default
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Recurrence Quantification > default
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Spectral Analysis > default
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Spectral Graph > default
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Spirograph > mode_spectrum
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Spirograph > phasor_trajectory
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Symplectic > flux_grid
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Symplectic > phase_space
/default/Logistic_r=3.74_(Period-5_Window).png)
Ulam Spiral (Sacks) > default
/default/Logistic_r=3.74_(Period-5_Window).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Tribonacci Word | 4.89 | cross-domain |
| Pell Word | 5.37 | cross-domain |
| Logistic r=3.83 (Period-3 Window) | 5.47 |
Which Geometries Light Up
Attractor Reconstruction › Attractor Reconstruction:filling_ratio | rank 294/298 | 0.0049 |
AutoRegressive › AutoRegressive:ar_residual_frac | rank 296/298 | 0.0000 |
Bispectrum › Bispectrum:coupling_frequency_centroid | rank 2/298 | 0.7160 |
Boltzmann › Boltzmann:frustration | rank 3/298 | 1.0000 |
Catch24 › Catch24:CO_trev_1_num | rank 294/298 | -2.8000 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 295/298 | 0.0000 |
Chladni › Chladni:modal_nodal_cascade | rank 4/298 | 1.9719 |
Chladni › Chladni:plate_low_mode_fraction | rank 296/298 | 0.0026 |
D4 Triality › D4 Triality:triplet_temporal | rank 3/298 | 0.8100 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_spread | rank 294/298 | 1.0759 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):spatio_temporal_corr | rank 1/298 | 0.9217 |
Inflation (Substitution) › Inflation (Substitution):complexity_linearity | rank 298/298 | 0.6166 |
Information Theory › Information Theory:compression_ratio | rank 294/298 | 0.0041 |
Isochronicity › Isochronicity:frequency_shear | rank 2/298 | 0.8659 |
Laplacian › Laplacian:laplacian_spectral_ratio | rank 294/298 | 0.0000 |
Moiré › Moiré:moire_dilation_entropy | rank 295/298 | 1.4188 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 296/298 | 0.2402 |
Predictability › Predictability:cond_entropy_k1 | rank 297/298 | 0.0000 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_std | rank 296/298 | 0.0000 |
Symplectic › Symplectic:windowed_area_cv | rank 297/298 | 0.0000 |
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