Logistic Edge-of-Chaos
chaos · 36 views
chaos
What It Is
Logistic map at the Feigenbaum point r=3.5699 --- the boundary between order and chaos, where the period-doubling cascade accumulates
Interpretation
Standard analysis sees: bounded / light-tailed; anti-correlated (alternating); anti-persistent; strongly periodic; monofractal; low-dimensional; stationary. The atlas finds no named structure, but the source is distinctively extreme on Hyperbolic (Poincaré):curvature_structure (+6.5z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.08
asymmetry0.19
occupancy0.27
short-range corr0.01
long-range memory0.01
spectral colour0.38
periodicity0.98
complexity0.21
time-irreversibility0.19
volatility clustering0.34
multifractality0.02
dimensionality0.04
nonstationarity0.02
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Hyperbolic (Poincaré):curvature_structure | +6.5z | bank-miss 1.7σ |
Heisenberg (Nil) (centered):xy_aspect | +6.0z | bank-miss 1.2σ |
Multi-Scale Wasserstein:w_slope | +5.8z | bank-miss 2.6σ |
Heisenberg (Nil) (centered):xy_spread | +5.5z | bank-miss 1.1σ |
Projective ℙ²:cross_ratio_std | +3.8z | bank-miss 1.1σ |
Modular Residue:cycle_fraction | +3.2z | bank-miss 1.2σ |
Boltzmann:spectral_gap_J | -2.8z | bank-miss 1.2σ |
Wavelet Cascade:intermittency_slope | +2.8z | bank-miss 1.1σ |
Composition
dtypefloat64
range[0.3426, 0.8925]
unique values6422 / 16384
mean ± std0.648 ± 0.216
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Logistic_Edge-of-Chaos.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Logistic_Edge-of-Chaos.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Logistic_Edge-of-Chaos.png)
Inflation (Substitution) > barcode
/d_curve/Logistic_Edge-of-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_Edge-of-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_Edge-of-Chaos.png)
Ulam Spiral (Sacks) > default
/default/Logistic_Edge-of-Chaos.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Quartic Map (Feigenbaum) | 2.58 | |
| Sine Map (Feigenbaum) | 3.59 | |
| Logistic r=3.5 (Period-4) | 4.29 |
Which Geometries Light Up
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 4/298 | 1.3315 |
Catch24 › Catch24:CO_f1ecac | rank 295/298 | 0.3308 |
Cayley › Cayley:local_linearity | rank 5/298 | 0.9999 |
Cayley › Cayley:growth_exponent | rank 298/298 | 0.3907 |
Chladni › Chladni:modal_nodal_cascade | rank 5/298 | 1.9655 |
E8 Lattice › E8 Lattice:std_profile | rank 295/298 | 0.1163 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughness | rank 3/298 | 3.5438 |
H4 600-Cell › H4 600-Cell:edge_walk_fraction | rank 3/298 | 0.6687 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_aspect | rank 2/298 | 10.1699 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratio | rank 4/298 | 0.3744 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_spread | rank 4/298 | 4507.4926 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):z_rate_spectral_entropy | rank 4/298 | 0.8840 |
Higher-Order Statistics › Higher-Order Statistics:bicoherence_max | rank 3/298 | 1.0000 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structure | rank 2/298 | 1496.3932 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_area | rank 4/298 | 2.1761 |
Julia Set › Julia Set:potential_smoothness | rank 4/298 | 6945.6577 |
Julia Set › Julia Set:potential_variance | rank 4/298 | 46092238.5543 |
Laplacian › Laplacian:laplacian_spectral_ratio | rank 295/298 | 0.0000 |
Lorentzian › Lorentzian:spacelike_fraction | rank 2/298 | 0.1824 |
Moiré › Moiré:moire_integer_excess | rank 2/298 | 0.6238 |
Moiré › Moiré:moire_max_coherence | rank 5/298 | 0.9919 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_slope | rank 4/298 | 4.6558 |
Multifractal Spectrum › Multifractal Spectrum:spectrum_width | rank 5/298 | 0.4732 |
Nonstationarity › Nonstationarity:dynamic_coupling | rank 296/298 | 0.9768 |
Projective ℙ² › Projective ℙ²:cross_ratio_std | rank 1/298 | 45.4487 |
Sol (Thurston) › Sol (Thurston):path_length | rank 1/298 | 33301994.5346 |
Spectral Analysis › Spectral Analysis:spectral_flatness | rank 296/298 | 0.0000 |
Spherical S² › Spherical S²:spectral_gini | rank 3/298 | 0.9997 |
Spherical S² › Spherical S²:hemisphere_balance | rank 298/298 | 0.0000 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersion | rank 4/298 | 2.5529 |
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variance | rank 294/298 | 0.0096 |
Wavelet Cascade › Wavelet Cascade:intermittency_slope | rank 2/298 | 1.4015 |
Wavelet Cascade › Wavelet Cascade:scale_gini_dispersion | rank 4/298 | 0.2944 |
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