Quartic Map (Feigenbaum)

chaos · 36 views
chaos

What It Is

Quartic map x(n+1) = 1 - r*x^4 at r=1.5949 --- accumulation of the period-doubling cascade for the z=4 family. DIFFERENT Feigenbaum universality class from logistic/sine (quartic-tip, delta=7.2847). Tests whether the atlas distinguishes universality classes: should be near Logistic Edge-of-Chaos and Sine Map (Feigenbaum) but discernibly separate

Interpretation

Standard analysis sees: bounded / light-tailed; left-skewed; anti-correlated (alternating); anti-persistent; strongly periodic; monofractal; low-dimensional; stationary. The atlas finds no named structure, but the source is distinctively extreme on Multi-Scale Wasserstein:w_slope (+6.9z) — beyond what the standard bank predicts for it.

What standard analysis sees
tail heaviness0.11
asymmetry0.12
occupancy0.27
short-range corr0.02
long-range memory0.02
spectral colour0.82
periodicity0.95
complexity0.21
time-irreversibility0.19
volatility clustering0.35
multifractality0.03
dimensionality0.06
nonstationarity0.02
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Multi-Scale Wasserstein:w_slope+6.9zbank-miss 1.1σ
Heisenberg (Nil) (centered):area_length_ratio+5.8zbank-miss 1.7σ
Heisenberg (Nil) (centered):xy_spread+5.2zbank-miss 2.3σ
Multifractal Spectrum:spectrum_width+4.9zbank-miss 2.1σ
Fisher Information:velocity_spectral_gini-4.2zbank-miss 1.1σ
Hyperbolic (Poincaré):curvature_structure+4.0zbank-miss 1.5σ
Boltzmann:coupling_strength+3.8zbank-miss 1.1σ
H² × ℝ (Thurston):hyperbolic_triangle_area+3.4zbank-miss 1.2σ

Composition

dtypefloat64
range[-0.5949, 1]
unique values16384 / 16384
mean ± std0.358 ± 0.625

Render Gallery

Native > phase_plot
Native > phase_plotraw
Native > timeseries
Native > timeseriesraw
Native > triple_lattice
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > lag_map
Boltzmann > spin_grid
Boltzmann > spin_grid
Cayley > default
Cayley > default
Chladni > plate_bottom
Chladni > plate_bottom
Chladni > plate_top
Chladni > plate_top
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > signed_log_z
Heisenberg (Nil) (centered) > xy_path
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
Hodge–Laplacian > default
Inflation (Substitution) > barcode
Inflation (Substitution) > barcode
Inflation (Substitution) > d_curve
Inflation (Substitution) > d_curve
Information Theory > default
Information Theory > default
Julia Set > escape_histogram
Julia Set > escape_histogram
Klein Bottle > default
Klein Bottle > default
Laplacian > default
Laplacian > default
Logarithmic Spiral > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Möbius-S³ > s2_base_map
Ordinal Partition > default
Ordinal Partition > default
Penrose (Quasicrystal) > phi_spectrum
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > barcode
Persistent Homology > h1_diagram
Persistent Homology > h1_diagram
Predictability > default
Predictability > default
Recurrence Quantification > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Analysis > default
Spectral Graph > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > flux_grid
Symplectic > phase_space
Symplectic > phase_space
Ulam Spiral (Sacks) > default
Ulam Spiral (Sacks) > default
Ulam Spiral (Square) > default
Ulam Spiral (Square) > default
Visibility Graph > default
Visibility Graph > default
Wavelet Cascade > default
Wavelet Cascade > default

Atlas Position

Nearest neighborDistance
Logistic Edge-of-Chaos2.58
Sine Map (Feigenbaum)4.01
Logistic r=3.5 (Period-4)4.93

Open in Atlas →

Which Geometries Light Up

Catch24 › Catch24:FC_LocalSimple_mean3_stderrrank 5/2981.3271
Cayley › Cayley:local_linearityrank 4/2980.9999
Cayley › Cayley:growth_exponentrank 297/2980.6832
D4 Triality › D4 Triality:gram_consistencyrank 295/2980.4244
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughnessrank 5/2983.3368
H4 600-Cell › H4 600-Cell:edge_walk_fractionrank 2/2980.6750
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_aspectrank 4/2989.6930
Higher-Order Statistics › Higher-Order Statistics:bicoherence_maxrank 2/2981.0000
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structurerank 3/298959.2443
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):spatio_temporal_corrrank 294/298-0.2879
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_arearank 1/2982.5129
Julia Set › Julia Set:potential_smoothnessrank 5/2986864.1125
Julia Set › Julia Set:potential_variancerank 5/29845864307.6406
Lorentzian › Lorentzian:spacelike_fractionrank 4/2980.1758
Moiré › Moiré:moire_integer_excessrank 5/2980.5423
Moiré › Moiré:moire_dilation_entropyrank 294/2981.4408
Mostow Rigidity › Mostow Rigidity:mean_turn_anglerank 297/2980.0535
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_sloperank 3/2985.4995
Multifractal Spectrum › Multifractal Spectrum:spectrum_widthrank 2/2980.6540
Navier-Stokes › Navier-Stokes:sl_fit_qualityrank 5/2980.9212
Projective ℙ² › Projective ℙ²:cross_ratio_stdrank 3/29841.0959
Sol (Thurston) › Sol (Thurston):path_lengthrank 5/29830909591.3916
Spectral Analysis › Spectral Analysis:spectral_flatnessrank 295/2980.0000
Spherical S² › Spherical S²:spectral_ginirank 4/2980.9995
Wavelet Cascade › Wavelet Cascade:intermittency_sloperank 4/2980.8176
Wavelet Cascade › Wavelet Cascade:cascade_couplingrank 295/298-0.5368
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