Logistic r=3.5 (Period-4)

iterated-maps · 37 views

iterated-mapsperiodic

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 additionally detects Nonstationarity:adf_pvalue, Time Reversibility:ordinal_reversal_distance.

What standard analysis sees

tail heaviness0.05

asymmetry0.23

occupancy0.15

short-range corr0.00

long-range memory0.00

spectral colour0.26

periodicity0.99

complexity0.21

time-irreversibility0.21

volatility clustering0.24

multifractality0.01

dimensionality0.42

nonstationarity0.00

What the atlas adds

Nonstationarity:adf_pvalue+5.8z

unit-root nonstationarity (ADF cannot reject random-walk null)

Time Reversibility:ordinal_reversal_distance+3.0z

time-reversal symmetry breaking via Bandt-Pompe ordinal patterns

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

`Heisenberg (Nil) (centered):xy_aspect`	+6.1z	bank-miss 1.8σ
`Heisenberg (Nil) (centered):xy_spread`	+5.5z	bank-miss 1.1σ
`2-adic:valuation_transition_predictability`	+3.0z	bank-miss 1.6σ
`Boltzmann:spectral_gap_J`	-2.7z	bank-miss 1.7σ
`2-adic:multiscale_markov_predictability`	+2.3z	bank-miss 1.2σ

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

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

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
Sine Map (Feigenbaum)	4.02
Logistic Edge-of-Chaos	4.16
Logistic r=3.2 (Period-2)	4.57

Open in Atlas →

Which Geometries Light Up

2-adic › `2-adic:valuation_transition_predictability`	rank 1/307	1.0000
2-adic › `2-adic:multiscale_markov_predictability`	rank 2/307	1.0000
2-adic › `2-adic:valuation_spectral_concentration`	rank 3/307	0.9015
Ammann-Beenker (Octagonal) › `Ammann-Beenker (Octagonal):convergent_profile`	rank 307/307	-0.2021
Attractor Reconstruction › `Attractor Reconstruction:d2_saturation`	rank 1/307	0.9974
Attractor Reconstruction › `Attractor Reconstruction:filling_ratio`	rank 307/307	0.0020
Catch24 › `Catch24:FC_LocalSimple_mean3_stderr`	rank 1/307	1.3333
Catch24 › `Catch24:CO_f1ecac`	rank 306/307	0.3236
Catch24 › `Catch24:SP_Summaries_welch_rect_area_5_1`	rank 306/307	0.0000
Chladni › `Chladni:modal_nodal_cascade`	rank 4/307	2.0377
Chladni › `Chladni:plate_low_mode_fraction`	rank 303/307	0.0235
D4 Triality › `D4 Triality:gram_consistency`	rank 305/307	0.0000
E8 Lattice › `E8 Lattice:std_profile`	rank 307/307	0.0000
G2 Root System › `G2 Root System:pair_angle_kurtosis`	rank 307/307	-2.0000
Heisenberg (Nil) (centered) › `Heisenberg (Nil) (centered):area_length_ratio`	rank 2/307	0.3899
Heisenberg (Nil) (centered) › `Heisenberg (Nil) (centered):xy_spread`	rank 2/307	4618.9876
Higher-Order Statistics › `Higher-Order Statistics:skew_mean`	rank 304/307	0.0000
Higher-Order Statistics › `Higher-Order Statistics:bicoherence_max`	rank 307/307	0.0000
Hyperbolic (Poincaré) › `Hyperbolic (Poincaré):curvature_structure`	rank 306/307	1.0000
Hyperbolic (Poincaré) › `Hyperbolic (Poincaré):temporal_variance`	rank 306/307	0.0000
H² × ℝ (Thurston) › `H² × ℝ (Thurston):hyperbolic_step_dispersion`	rank 2/307	8.2805
H² × ℝ (Thurston) › `H² × ℝ (Thurston):hyperbolic_triangle_area`	rank 2/307	2.3647
Hölder Regularity › `Hölder Regularity:holder_mean`	rank 305/307	-1.8938
Information Theory › `Information Theory:compression_ratio`	rank 303/307	0.0029
Julia Set › `Julia Set:potential_smoothness`	rank 2/307	7001.6005
Julia Set › `Julia Set:potential_variance`	rank 2/307	46322561.8590
Julia Set › `Julia Set:escape_entropy`	rank 305/307	2.5856
Klein Bottle › `Klein Bottle:wht_spectral_kurtosis`	rank 2/307	9.4294
Klein Bottle › `Klein Bottle:linear_complexity`	rank 303/307	0.0112
Laplacian › `Laplacian:biharmonic_ratio`	rank 3/307	15.8541
Lorentzian › `Lorentzian:spacelike_fraction`	rank 5/307	0.1758
Moiré › `Moiré:moire_integer_excess`	rank 1/307	0.6457
Moiré › `Moiré:moire_max_coherence`	rank 2/307	1.0000
Moiré › `Moiré:moire_dilation_entropy`	rank 305/307	1.3905
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_std`	rank 305/307	0.0000
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_fine`	rank 307/307	0.0000
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_mean`	rank 307/307	0.0000
Nonstationarity › `Nonstationarity:adf_pvalue`	rank 2/307	0.9585
Nonstationarity › `Nonstationarity:change_quantiles_mid`	rank 5/307	0.6625
Nonstationarity › `Nonstationarity:dynamic_coupling`	rank 306/307	0.0000
Nonstationarity › `Nonstationarity:metric_volatility`	rank 306/307	0.0000
Nonstationarity › `Nonstationarity:vol_of_vol`	rank 306/307	0.0000
Nonstationarity › `Nonstationarity:trajectory_dim`	rank 307/307	0.0000
Sol (Thurston) › `Sol (Thurston):path_length`	rank 3/307	31729860.1845
Spectral Analysis › `Spectral Analysis:spectral_peakedness`	rank 2/307	0.9991
Spectral Analysis › `Spectral Analysis:spectral_r2`	rank 306/307	0.0000
Spherical S² › `Spherical S²:angular_spread`	rank 306/307	0.0000
Spherical S² › `Spherical S²:hemisphere_balance`	rank 307/307	0.0000
S² × ℝ (Thurston) › `S² × ℝ (Thurston):great_circle_dispersion`	rank 2/307	2.6714
Zipf–Mandelbrot (16-bit) › `Zipf–Mandelbrot (16-bit):gini_coefficient`	rank 305/307	0.0000
Zipf–Mandelbrot (8-bit) › `Zipf–Mandelbrot (8-bit):gini_coefficient`	rank 305/307	0.0000

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