Logistic r=3.74 (Period-5 Window)

iterated-maps · 37 views

iterated-mapsperiodic

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 additionally detects Time Reversibility:ordinal_reversal_distance, Multi-Scale Wasserstein:w_max_ratio. It sits beside Pell Word in the atlas (standard-bank rank 27) — a neighbor conventional features miss.

What standard analysis sees

tail heaviness0.33

asymmetry0.11

occupancy0.16

short-range corr0.04

long-range memory0.02

spectral colour0.94

periodicity0.83

complexity0.26

time-irreversibility0.08

volatility clustering0.05

multifractality0.04

dimensionality0.00

nonstationarity0.07

What the atlas adds

Time Reversibility:ordinal_reversal_distance+3.0z

time-reversal symmetry breaking via Bandt-Pompe ordinal patterns

Multi-Scale Wasserstein:w_max_ratio+2.6z

scale-coherent dispersion mismatch (max/min Wasserstein across scales)

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

`Fisher Information:velocity_spectral_gini`	-4.2z	bank-miss 1.9σ
`Klein Bottle:linear_complexity`	-4.0z	bank-miss 1.4σ
`Attractor Reconstruction:d2_saturation`	-3.4z	bank-miss 2.5σ
`Boltzmann:frustration`	+2.6z	bank-miss 2.5σ
`Isochronicity:frequency_shear`	+2.5z	bank-miss 1.7σ
`Hyperbolic (Poincaré):spatio_temporal_corr`	+2.5z	bank-miss 2.9σ
`D4 Triality:triplet_temporal`	+2.4z	bank-miss 3.3σ

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

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
Tribonacci Word	4.75	cross-origin
Periodic Word	5.05	cross-origin
Pell Word	5.13	cross-origin

Open in Atlas →

Which Geometries Light Up

Attractor Reconstruction › `Attractor Reconstruction:filling_ratio`	rank 303/307	0.0049
AutoRegressive › `AutoRegressive:ar_residual_frac`	rank 304/307	0.0000
Bispectrum › `Bispectrum:coupling_frequency_centroid`	rank 2/307	0.7160
Catch24 › `Catch24:CO_trev_1_num`	rank 304/307	-2.8000
Catch24 › `Catch24:SP_Summaries_welch_rect_area_5_1`	rank 304/307	0.0000
Chladni › `Chladni:modal_nodal_cascade`	rank 5/307	1.9719
Chladni › `Chladni:plate_low_mode_fraction`	rank 305/307	0.0026
D4 Triality › `D4 Triality:triplet_temporal`	rank 3/307	0.8100
Heisenberg (Nil) (centered) › `Heisenberg (Nil) (centered):xy_spread`	rank 303/307	1.0759
Hyperbolic (Poincaré) › `Hyperbolic (Poincaré):spatio_temporal_corr`	rank 1/307	0.9217
Inflation (Substitution) › `Inflation (Substitution):complexity_subexponential`	rank 307/307	0.6166
Isochronicity › `Isochronicity:frequency_shear`	rank 2/307	0.8659
Moiré › `Moiré:moire_dilation_entropy`	rank 304/307	1.4188
Nonstationarity › `Nonstationarity:vol_of_vol`	rank 305/307	0.2402
Predictability › `Predictability:cond_entropy_k1`	rank 306/307	0.0000
SL(2,ℝ) (Thurston) › `SL(2,ℝ) (Thurston):cocycle_growth_burstiness`	rank 305/307	0.0000
Symplectic › `Symplectic:windowed_area_cv`	rank 305/307	0.0000

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