Logistic Edge-of-Chaos

iterated-maps · 37 views

iterated-mapschaotic

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 additionally detects Time Reversibility:ordinal_reversal_distance.

What standard analysis sees

tail heaviness0.08

asymmetry0.19

occupancy0.27

short-range corr0.01

long-range memory0.01

spectral colour0.39

periodicity0.98

complexity0.21

time-irreversibility0.19

volatility clustering0.34

multifractality0.02

dimensionality0.04

nonstationarity0.02

What the atlas adds

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

`Hyperbolic (Poincaré):curvature_structure`	+6.6z	bank-miss 2.0σ
`Heisenberg (Nil) (centered):xy_aspect`	+5.9z	bank-miss 1.5σ
`Multi-Scale Wasserstein:w_slope`	+5.8z	bank-miss 2.8σ
`Heisenberg (Nil) (centered):xy_spread`	+5.3z	bank-miss 1.6σ
`Multifractal Spectrum:spectrum_width`	+3.5z	bank-miss 1.4σ
`H4 600-Cell:edge_walk_fraction`	+2.9z	bank-miss 1.2σ
`Boltzmann:spectral_gap_J`	-2.7z	bank-miss 1.0σ
`Nonstationarity:variance_trend`	+2.5z	bank-miss 2.1σ

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 > 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
Quartic Map (Feigenbaum)	2.45
Sine Map (Feigenbaum)	3.46
Logistic r=3.5 (Period-4)	4.16

Open in Atlas →

Which Geometries Light Up

Catch24 › `Catch24:FC_LocalSimple_mean3_stderr`	rank 4/307	1.3315
Catch24 › `Catch24:CO_f1ecac`	rank 304/307	0.3308
Cayley › `Cayley:growth_exponent`	rank 307/307	0.3907
E8 Lattice › `E8 Lattice:std_profile`	rank 304/307	0.1163
H4 600-Cell › `H4 600-Cell:edge_walk_fraction`	rank 3/307	0.6687
Heisenberg (Nil) (centered) › `Heisenberg (Nil) (centered):area_length_ratio`	rank 4/307	0.3744
Heisenberg (Nil) (centered) › `Heisenberg (Nil) (centered):xy_spread`	rank 4/307	4507.4926
Higher-Order Statistics › `Higher-Order Statistics:bicoherence_max`	rank 3/307	1.0000
Hyperbolic (Poincaré) › `Hyperbolic (Poincaré):curvature_structure`	rank 2/307	1496.3932
H² × ℝ (Thurston) › `H² × ℝ (Thurston):hyperbolic_triangle_area`	rank 4/307	2.1761
Julia Set › `Julia Set:potential_smoothness`	rank 4/307	6945.6577
Julia Set › `Julia Set:potential_variance`	rank 4/307	46092238.5543
Julia Set › `Julia Set:escape_entropy`	rank 303/307	2.6039
Lorentzian › `Lorentzian:spacelike_fraction`	rank 2/307	0.1824
Moiré › `Moiré:moire_integer_excess`	rank 2/307	0.6238
Moiré › `Moiré:moire_max_coherence`	rank 5/307	0.9919
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_slope`	rank 4/307	4.6558
Multifractal Spectrum › `Multifractal Spectrum:spectrum_width`	rank 5/307	0.4732
Nonstationarity › `Nonstationarity:dynamic_coupling`	rank 305/307	0.9768
Projective ℙ² › `Projective ℙ²:cross_ratio_std`	rank 1/307	45.4487
Sol (Thurston) › `Sol (Thurston):path_length`	rank 1/307	33301994.5346
Spherical S² › `Spherical S²:hemisphere_balance`	rank 305/307	0.0000
S² × ℝ (Thurston) › `S² × ℝ (Thurston):great_circle_dispersion`	rank 4/307	2.5529
Wavelet Cascade › `Wavelet Cascade:coarse_scale_kurtosis_slope`	rank 2/307	1.4015
Wavelet Cascade › `Wavelet Cascade:scale_gini_dispersion`	rank 4/307	0.2944

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