Logistic r=3.83 (Period-3 Window)

chaos · 36 views

chaos

What It Is

Logistic map in the period-3 window --- a stable island inside the chaotic sea. By Sharkovskii's theorem, period-3 implies all periods exist nearby

Interpretation

Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on AutoRegressive:ar_coef_10 (+13.7z) — beyond what the standard bank predicts for it.

What standard analysis sees

tail heaviness0.11

asymmetry0.66

occupancy0.10

short-range corr0.07

long-range memory0.01

spectral colour0.99

periodicity0.92

complexity0.16

time-irreversibility0.05

volatility clustering0.02

multifractality0.03

dimensionality0.37

nonstationarity0.06

What the atlas adds

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

`AutoRegressive:ar_coef_10`	+13.7z	bank-miss 1.9σ
`Mostow Rigidity:spectral_rigidity`	-9.4z	bank-miss 3.9σ
`D4 Triality:triplet_temporal`	+6.5z	bank-miss 3.4σ
`Mostow Rigidity:distance_rigidity`	-5.2z	bank-miss 4.6σ
`Zariski:heyting_gap`	+5.0z	bank-miss 1.5σ
`Wavelet Cascade:cascade_coupling`	-4.3z	bank-miss 1.2σ
`Fisher Information:velocity_spectral_gini`	-4.2z	bank-miss 1.9σ
`Bispectrum:off_diagonal_ratio`	-4.1z	bank-miss 1.8σ

Composition

dtypefloat64

range[0.1561, 0.9574]

unique values3 / 16384

mean ± std0.539 ± 0.328

Fixed alphabet — only 3 distinct symbols across 16384 samples.

Render Gallery

Native > phase_plotraw

Native > timeseriesraw

Native > triple_latticeraw

Boltzmann > lag_map

Boltzmann > spin_grid

Cayley > default

Chladni > plate_bottom

Chladni > plate_top

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

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

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

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

Nearest neighbor	Distance
Logistic r=3.74 (Period-5 Window)	5.47
Pell Word	5.96	cross-domain
Tribonacci Word	6.49	cross-domain

Open in Atlas →

Which Geometries Light Up

2-adic › `2-adic:multiscale_markov_predictability`	rank 2/298	1.0000
2-adic › `2-adic:valuation_spectral_concentration`	rank 2/298	1.0000
Attractor Reconstruction › `Attractor Reconstruction:filling_ratio`	rank 296/298	0.0029
AutoRegressive › `AutoRegressive:ar_coef_1`	rank 298/298	-5.0000
Bispectrum › `Bispectrum:coupling_frequency_centroid`	rank 1/298	0.8113
Bispectrum › `Bispectrum:bicoherence_concentration`	rank 2/298	0.9786
Bispectrum › `Bispectrum:off_diagonal_ratio`	rank 298/298	0.0849
Boltzmann › `Boltzmann:coupling_strength`	rank 1/298	15.2634
Catch24 › `Catch24:SP_Summaries_welch_rect_area_5_1`	rank 294/298	0.0000
Catch24 › `Catch24:CO_trev_1_num`	rank 298/298	-3.5814
Cayley › `Cayley:delta_hyp_norm`	rank 296/298	0.0000
Chladni › `Chladni:plate_low_mode_fraction`	rank 297/298	0.0012
D4 Triality › `D4 Triality:triplet_temporal`	rank 1/298	2.0000
Heisenberg (Nil) (centered) › `Heisenberg (Nil) (centered):xy_spread`	rank 296/298	0.8014
Higher-Order Statistics › `Higher-Order Statistics:perm_forbidden`	rank 2/298	0.9750
H² × ℝ (Thurston) › `H² × ℝ (Thurston):hyperbolic_triangle_area`	rank 297/298	0.0000
H² × ℝ (Thurston) › `H² × ℝ (Thurston):hyperbolic_step_dispersion`	rank 298/298	0.0000
Information Theory › `Information Theory:compression_ratio`	rank 296/298	0.0028
Klein Bottle › `Klein Bottle:linear_complexity`	rank 294/298	0.0093
Laplacian › `Laplacian:curvature_autocorrelation`	rank 298/298	-0.5000
Laplacian › `Laplacian:laplacian_evolutionary_index`	rank 298/298	-0.4983
Lorentzian › `Lorentzian:spacelike_fraction`	rank 1/298	0.1888
Moiré › `Moiré:moire_dilation_entropy`	rank 297/298	1.1005
Mostow Rigidity › `Mostow Rigidity:mean_turn_angle`	rank 1/298	1.0000
Mostow Rigidity › `Mostow Rigidity:distance_rigidity`	rank 298/298	0.0000
Mostow Rigidity › `Mostow Rigidity:margulis_ratio`	rank 298/298	0.0000
Mostow Rigidity › `Mostow Rigidity:spectral_rigidity`	rank 298/298	0.3392
Möbius-S³ › `Möbius-S³:phi_vertex_excess`	rank 2/298	0.2000
Möbius-S³ › `Möbius-S³:hopf_fiber_coherence`	rank 294/298	0.2500
Navier-Stokes › `Navier-Stokes:intermittency_autocorr`	rank 297/298	-0.5000
Predictability › `Predictability:cond_entropy_k1`	rank 296/298	0.0000
Projective ℙ² › `Projective ℙ²:cross_ratio_std`	rank 298/298	0.0000
Projective ℙ² › `Projective ℙ²:distance_std`	rank 298/298	0.0000
Projective ℙ² › `Projective ℙ²:mean_distance`	rank 298/298	0.0000
SL(2,ℝ) (Thurston) › `SL(2,ℝ) (Thurston):lyapunov_std`	rank 297/298	0.0000
SL(2,ℝ) (Thurston) › `SL(2,ℝ) (Thurston):mean_trace`	rank 298/298	-5.1102
Spirograph › `Spirograph:petal_symmetry`	rank 4/298	0.6667
Symplectic › `Symplectic:windowed_area_cv`	rank 294/298	0.0477
S² × ℝ (Thurston) › `S² × ℝ (Thurston):sphere_concentration`	rank 1/298	1.0000
S² × ℝ (Thurston) › `S² × ℝ (Thurston):great_circle_dispersion`	rank 298/298	0.0000
Wavelet Cascade › `Wavelet Cascade:scale_gini_dispersion`	rank 297/298	0.0002
Wavelet Cascade › `Wavelet Cascade:cascade_coupling`	rank 298/298	-1.0000
Zariski › `Zariski:heyting_gap`	rank 3/298	0.9048
Zipf–Mandelbrot (16-bit) › `Zipf–Mandelbrot (16-bit):gini_coefficient`	rank 298/298	0.0000

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