Sine Map (Feigenbaum)

chaos · 36 views
chaos

What It Is

Sine map x(n+1) = r*sin(pi*x) at r=0.8640 --- empirically-determined accumulation of the period-doubling cascade for this functional form (bisected between period-16 at r<=0.8639 and wide chaos at r>=0.8641). Functionally distinct from the logistic map but in the SAME Feigenbaum universality class (quadratic-tip, z=2, delta=4.6692). Tests whether the atlas rediscovers period-doubling universality unsupervised: should cluster tightly with Logistic Edge-of-Chaos despite the different functional form

Interpretation

Standard analysis sees: bounded / light-tailed; anti-correlated (alternating); anti-persistent; strongly periodic; monofractal; low-dimensional; stationary. The atlas finds no named structure, but the source is distinctively extreme on Hyperbolic (Poincaré):curvature_structure (+14.6z) — beyond what the standard bank predicts for it.

What standard analysis sees
tail heaviness0.08
asymmetry0.21
occupancy0.17
short-range corr0.01
long-range memory0.01
spectral colour0.41
periodicity0.98
complexity0.22
time-irreversibility0.18
volatility clustering0.33
multifractality0.02
dimensionality0.01
nonstationarity0.02
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Hyperbolic (Poincaré):curvature_structure+14.6zbank-miss 1.6σ
Nonstationarity:vol_of_vol+10.2zbank-miss 3.7σ
Heisenberg (Nil) (centered):xy_spread+5.5zbank-miss 1.3σ
Nonstationarity:variance_trend-5.1zbank-miss 2.4σ
Julia Set:potential_variance+5.1zbank-miss 1.1σ
Julia Set:potential_smoothness+5.0zbank-miss 1.1σ
AutoRegressive:ar_coef_6-3.9zbank-miss 1.3σ
AutoRegressive:ar_coef_4-3.5zbank-miss 1.2σ

Composition

dtypefloat64
range[0.3588, 0.8637]
unique values4855 / 16384
mean ± std0.636 ± 0.199

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-Chaos3.59
Quartic Map (Feigenbaum)4.01
Logistic r=3.5 (Period-4)4.18

Open in Atlas →

Which Geometries Light Up

2-adic › 2-adic:valuation_spectral_concentrationrank 4/2980.8244
Attractor Reconstruction › Attractor Reconstruction:filling_ratiorank 295/2980.0039
Catch24 › Catch24:FC_LocalSimple_mean3_stderrrank 3/2981.3316
Catch24 › Catch24:CO_f1ecacrank 296/2980.3307
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1rank 296/2980.0000
Chladni › Chladni:modal_nodal_cascaderank 2/2982.1481
E8 Lattice › E8 Lattice:std_profilerank 296/2980.0000
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughnessrank 2/2983.6066
H4 600-Cell › H4 600-Cell:edge_walk_fractionrank 1/2980.7000
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratiorank 3/2980.3746
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_aspectrank 3/29810.1276
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_spreadrank 3/2984509.4061
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):z_rate_spectral_entropyrank 3/2980.8879
Higher-Order Statistics › Higher-Order Statistics:bicoherence_maxrank 4/2981.0000
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structurerank 1/2983276.8316
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_arearank 3/2982.2428
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_step_dispersionrank 4/2987.7173
Julia Set › Julia Set:potential_smoothnessrank 3/2986957.6571
Julia Set › Julia Set:potential_variancerank 3/29846099674.8410
Julia Set › Julia Set:escape_entropyrank 294/2982.5983
Klein Bottle › Klein Bottle:wht_spectral_kurtosisrank 5/2988.7810
Laplacian › Laplacian:biharmonic_ratiorank 5/29815.7101
Laplacian › Laplacian:laplacian_spectral_ratiorank 296/2980.0000
Lorentzian › Lorentzian:spacelike_fractionrank 3/2980.1812
Modular Residue › Modular Residue:cycle_fractionrank 3/2981.0000
Moiré › Moiré:moire_integer_excessrank 3/2980.6233
Moiré › Moiré:moire_max_coherencerank 4/2980.9923
Mostow Rigidity › Mostow Rigidity:margulis_ratiorank 297/2980.0000
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_finerank 297/2980.0000
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_meanrank 297/2980.0000
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_stdrank 298/2980.0000
Nonstationarity › Nonstationarity:vol_of_volrank 1/2985.7662
Nonstationarity › Nonstationarity:metric_volatilityrank 296/2980.0373
Nonstationarity › Nonstationarity:trajectory_dimrank 296/2980.2000
Nonstationarity › Nonstationarity:variance_trendrank 298/298-0.5545
Persistent Homology › Persistent Homology:max_lifetimerank 295/2980.6784
Projective ℙ² › Projective ℙ²:cross_ratio_stdrank 2/29844.7827
Sol (Thurston) › Sol (Thurston):path_lengthrank 2/29832828213.2679
Spectral Analysis › Spectral Analysis:spectral_flatnessrank 297/2980.0000
Spherical S² › Spherical S²:spectral_ginirank 2/2980.9997
Spherical S² › Spherical S²:hemisphere_balancerank 295/2980.0000
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersionrank 3/2982.5537
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variancerank 296/2980.0096
Wavelet Cascade › Wavelet Cascade:intermittency_sloperank 1/2981.4313
Wavelet Cascade › Wavelet Cascade:scale_gini_dispersionrank 1/2980.4416
Zariski › Zariski:residual_sloperank 295/298-4.9448
p-Variation › p-Variation:increment_persistencerank 294/298-0.9544
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