Logistic r=3.5 (Period-4)
chaos · 36 views
chaos
What It Is
Logistic map in period-4 --- four-value cycle after the second bifurcation, output concentrates on four narrow bands
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; strongly periodic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Heisenberg (Nil) (centered):xy_aspect (+6.2z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.05
asymmetry0.24
occupancy0.14
short-range corr0.00
long-range memory0.00
spectral colour0.26
periodicity0.99
complexity0.21
time-irreversibility0.22
volatility clustering0.24
multifractality0.01
dimensionality0.41
nonstationarity0.00
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Heisenberg (Nil) (centered):xy_aspect | +6.2z | bank-miss 1.8σ |
Nonstationarity:adf_pvalue | +5.8z | bank-miss 3.5σ |
Chladni:modal_nodal_cascade | +3.4z | bank-miss 1.0σ |
H² × ℝ (Thurston):hyperbolic_triangle_area | +3.1z | bank-miss 1.0σ |
2-adic:valuation_transition_predictability | +3.1z | bank-miss 2.1σ |
Nonstationarity:metric_volatility | -2.9z | bank-miss 1.8σ |
Boltzmann:spectral_gap_J | -2.8z | bank-miss 1.5σ |
AutoRegressive:ar_coef_10 | -2.7z | bank-miss 2.0σ |
Composition
dtypefloat64
range[0.3828, 0.875]
unique values4 / 16384
mean ± std0.646 ± 0.209
Fixed alphabet — only 4 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=3.5_(Period-4).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Logistic_r=3.5_(Period-4).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Logistic_r=3.5_(Period-4).png)
Inflation (Substitution) > barcode
/d_curve/Logistic_r=3.5_(Period-4).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=3.5_(Period-4).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=3.5_(Period-4).png)
Ulam Spiral (Sacks) > default
/default/Logistic_r=3.5_(Period-4).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Sine Map (Feigenbaum) | 4.18 | |
| Logistic Edge-of-Chaos | 4.29 | |
| Logistic r=3.2 (Period-2) | 4.82 |
Which Geometries Light Up
2-adic › 2-adic:multiscale_markov_predictability | rank 1/298 | 1.0000 |
2-adic › 2-adic:valuation_transition_predictability | rank 1/298 | 1.0000 |
2-adic › 2-adic:valuation_spectral_concentration | rank 3/298 | 0.9015 |
Ammann-Beenker (Octagonal) › Ammann-Beenker (Octagonal):convergent_profile | rank 298/298 | -0.2021 |
Attractor Reconstruction › Attractor Reconstruction:d2_saturation | rank 1/298 | 0.9974 |
Attractor Reconstruction › Attractor Reconstruction:filling_ratio | rank 297/298 | 0.0020 |
AutoRegressive › AutoRegressive:ar_residual_frac | rank 294/298 | 0.0000 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 1/298 | 1.3333 |
Catch24 › Catch24:CO_f1ecac | rank 297/298 | 0.3236 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 297/298 | 0.0000 |
Chladni › Chladni:modal_nodal_cascade | rank 3/298 | 2.0377 |
Chladni › Chladni:plate_low_mode_fraction | rank 294/298 | 0.0235 |
D4 Triality › D4 Triality:gram_consistency | rank 296/298 | 0.0000 |
E8 Lattice › E8 Lattice:std_profile | rank 298/298 | 0.0000 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughness | rank 1/298 | 4.0000 |
Gottwald-Melbourne › Gottwald-Melbourne:angular_spectral_structure | rank 4/298 | 0.9896 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_aspect | rank 1/298 | 10.4882 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratio | rank 2/298 | 0.3899 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_spread | rank 2/298 | 4618.9876 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):z_rate_spectral_entropy | rank 2/298 | 0.9020 |
Higher-Order Statistics › Higher-Order Statistics:perm_forbidden | rank 3/298 | 0.9667 |
Higher-Order Statistics › Higher-Order Statistics:skew_mean | rank 295/298 | 0.0000 |
Higher-Order Statistics › Higher-Order Statistics:bicoherence_max | rank 298/298 | 0.0000 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):temporal_variance | rank 297/298 | 0.0000 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structure | rank 298/298 | 1.0000 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_step_dispersion | rank 2/298 | 8.2805 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_area | rank 2/298 | 2.3647 |
Hölder Regularity › Hölder Regularity:holder_mean | rank 295/298 | -1.8938 |
Information Theory › Information Theory:compression_ratio | rank 295/298 | 0.0029 |
Julia Set › Julia Set:potential_smoothness | rank 2/298 | 7001.6005 |
Julia Set › Julia Set:potential_variance | rank 2/298 | 46322561.8590 |
Julia Set › Julia Set:escape_entropy | rank 295/298 | 2.5856 |
Klein Bottle › Klein Bottle:wht_spectral_kurtosis | rank 2/298 | 9.4294 |
Laplacian › Laplacian:biharmonic_ratio | rank 3/298 | 15.8541 |
Laplacian › Laplacian:laplacian_spectral_ratio | rank 297/298 | 0.0000 |
Lorentzian › Lorentzian:spacelike_fraction | rank 5/298 | 0.1758 |
Moiré › Moiré:moire_integer_excess | rank 1/298 | 0.6457 |
Moiré › Moiré:moire_max_coherence | rank 2/298 | 1.0000 |
Moiré › Moiré:moire_dilation_entropy | rank 296/298 | 1.3905 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_mean | rank 296/298 | 0.0000 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_std | rank 297/298 | 0.0000 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_fine | rank 298/298 | 0.0000 |
Nonstationarity › Nonstationarity:adf_pvalue | rank 2/298 | 0.9585 |
Nonstationarity › Nonstationarity:change_quantiles_mid | rank 4/298 | 0.6625 |
Nonstationarity › Nonstationarity:dynamic_coupling | rank 297/298 | 0.0000 |
Nonstationarity › Nonstationarity:metric_volatility | rank 297/298 | 0.0000 |
Nonstationarity › Nonstationarity:trajectory_dim | rank 297/298 | 0.0000 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 297/298 | 0.0000 |
Projective ℙ² › Projective ℙ²:distance_std | rank 5/298 | 0.2281 |
Sol (Thurston) › Sol (Thurston):path_length | rank 3/298 | 31729860.1845 |
Spectral Analysis › Spectral Analysis:phase_coherence | rank 2/298 | 0.9991 |
Spectral Analysis › Spectral Analysis:spectral_r2 | rank 298/298 | 0.0000 |
Spherical S² › Spherical S²:spectral_gini | rank 1/298 | 0.9998 |
Spherical S² › Spherical S²:angular_spread | rank 297/298 | 0.0000 |
Spherical S² › Spherical S²:hemisphere_balance | rank 297/298 | 0.0000 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersion | rank 2/298 | 2.6714 |
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variance | rank 295/298 | 0.0096 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):gini_coefficient | rank 296/298 | 0.0000 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):gini_coefficient | rank 298/298 | 0.0000 |
p-Variation › p-Variation:var_p2 | rank 4/298 | 0.7077 |
p-Variation › p-Variation:increment_persistence | rank 296/298 | -0.9763 |
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