Logistic r=3.2 (Period-2)
chaos · 36 views
chaos
What It Is
Logistic map in stable period-2 --- output alternates between two fixed values, the simplest oscillation after the first bifurcation
Interpretation
Standard analysis sees: bounded / light-tailed; few distinct values; anti-correlated (alternating); anti-persistent; strongly periodic; low-complexity (predictable, not noise-like); homoskedastic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:change_quantiles_mid (+8.4z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.00
asymmetry0.41
occupancy0.07
short-range corr0.00
long-range memory0.00
spectral colour0.25
periodicity1.00
complexity0.02
time-irreversibility0.48
volatility clustering0.02
multifractality0.00
dimensionality0.32
nonstationarity0.00
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Nonstationarity:change_quantiles_mid | +8.4z | bank-miss 1.3σ |
Hölder Regularity:holder_mean | -6.2z | bank-miss 1.3σ |
Hodge–Laplacian:source_fraction | -6.1z | bank-miss 1.1σ |
Heisenberg (Nil) (centered):xy_spread | +5.8z | bank-miss 1.6σ |
Julia Set:potential_variance | +5.1z | bank-miss 1.5σ |
Julia Set:potential_smoothness | +5.1z | bank-miss 1.5σ |
Ulam Spiral (Square):diagonal_alignment | +4.6z | bank-miss 1.1σ |
Klein Bottle:wht_spectral_kurtosis | +4.0z | bank-miss 1.1σ |
Composition
dtypefloat64
range[0.513, 0.7995]
unique values2 / 16384
mean ± std0.656 ± 0.143
Binary sequence — two distinct symbols.
Render Gallery
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Native > phase_plotraw
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Native > timeseriesraw
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Native > triple_latticeraw
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Boltzmann > lag_map
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Boltzmann > spin_grid
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Cayley > default
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Chladni > plate_bottom
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Chladni > plate_top
_(centered)/signed_log_z/Logistic_r=3.2_(Period-2).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Logistic_r=3.2_(Period-2).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Logistic_r=3.2_(Period-2).png)
Inflation (Substitution) > barcode
/d_curve/Logistic_r=3.2_(Period-2).png)
Inflation (Substitution) > d_curve
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Information Theory > default
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Julia Set > escape_histogram
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Klein Bottle > default
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Laplacian > default
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Logarithmic Spiral > default
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Möbius-S³ > mobius_cylinder
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Möbius-S³ > s2_base_map
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Ordinal Partition > default
/phi_spectrum/Logistic_r=3.2_(Period-2).png)
Penrose (Quasicrystal) > phi_spectrum
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Persistent Homology > barcode
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Persistent Homology > h1_diagram
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Predictability > default
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Recurrence Quantification > default
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Spectral Analysis > default
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Spectral Graph > default
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Spirograph > mode_spectrum
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Spirograph > phasor_trajectory
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Symplectic > flux_grid
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Symplectic > phase_space
/default/Logistic_r=3.2_(Period-2).png)
Ulam Spiral (Sacks) > default
/default/Logistic_r=3.2_(Period-2).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Logistic r=3.5 (Period-4) | 4.82 | |
| Sine Map (Feigenbaum) | 6.35 | |
| Logistic Edge-of-Chaos | 6.37 |
Which Geometries Light Up
2-adic › 2-adic:valuation_spectral_concentration | rank 1/298 | 1.0000 |
2-adic › 2-adic:valuation_transition_predictability | rank 2/298 | 1.0000 |
2-adic › 2-adic:multiscale_markov_predictability | rank 3/298 | 1.0000 |
Attractor Reconstruction › Attractor Reconstruction:filling_ratio | rank 298/298 | 0.0020 |
AutoRegressive › AutoRegressive:ar_residual_frac | rank 298/298 | 0.0000 |
Cantor Set › Cantor Set:bit_plane_autocorrelation | rank 1/298 | 1.0000 |
Catch24 › Catch24:DN_Spread_Std | rank 1/298 | 0.5000 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 2/298 | 1.3333 |
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff | rank 296/298 | 0.0000 |
Catch24 › Catch24:CO_f1ecac | rank 298/298 | 0.3161 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 298/298 | 0.0000 |
Cayley › Cayley:delta_hyp_norm | rank 298/298 | 0.0000 |
Chladni › Chladni:plate_low_mode_fraction | rank 298/298 | 0.0000 |
D4 Triality › D4 Triality:gram_consistency | rank 297/298 | 0.0000 |
E8 Lattice › E8 Lattice:std_profile | rank 297/298 | 0.0000 |
Gottwald-Melbourne › Gottwald-Melbourne:angular_spectral_structure | rank 1/298 | 0.9976 |
Gottwald-Melbourne › Gottwald-Melbourne:radial_spectral_structure | rank 1/298 | 0.9997 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratio | rank 1/298 | 0.4054 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_spread | rank 1/298 | 4729.7977 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):z_rate_spectral_entropy | rank 1/298 | 0.9226 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):xy_aspect | rank 297/298 | 0.0000 |
Higher-Order Statistics › Higher-Order Statistics:perm_forbidden | rank 1/298 | 0.9833 |
Higher-Order Statistics › Higher-Order Statistics:c3_energy | rank 296/298 | 0.0000 |
Higher-Order Statistics › Higher-Order Statistics:bicoherence_max | rank 297/298 | 0.0000 |
Higher-Order Statistics › Higher-Order Statistics:kurt_max | rank 298/298 | 0.0000 |
Higher-Order Statistics › Higher-Order Statistics:skew_mean | rank 298/298 | 0.0000 |
Hodge–Laplacian › Hodge–Laplacian:source_fraction | rank 298/298 | 0.0156 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structure | rank 297/298 | 1.0000 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):temporal_variance | rank 298/298 | 0.0000 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_step_dispersion | rank 1/298 | 9.4349 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_area | rank 296/298 | 0.0000 |
Hölder Regularity › Hölder Regularity:holder_mean | rank 298/298 | -3.3219 |
Hölder Regularity › Hölder Regularity:holder_min | rank 298/298 | -3.3219 |
Information Theory › Information Theory:compression_ratio | rank 298/298 | 0.0022 |
Julia Set › Julia Set:potential_smoothness | rank 1/298 | 7045.5463 |
Julia Set › Julia Set:potential_variance | rank 1/298 | 46408596.8745 |
Julia Set › Julia Set:escape_entropy | rank 296/298 | 2.5671 |
Klein Bottle › Klein Bottle:wht_spectral_kurtosis | rank 1/298 | 9.7041 |
Klein Bottle › Klein Bottle:linear_complexity | rank 295/298 | 0.0078 |
Laplacian › Laplacian:biharmonic_ratio | rank 1/298 | 15.9980 |
Laplacian › Laplacian:laplacian_spectral_ratio | rank 298/298 | 0.0000 |
Moiré › Moiré:moire_max_coherence | rank 1/298 | 1.0000 |
Moiré › Moiré:moire_integer_excess | rank 4/298 | 0.5434 |
Moiré › Moiré:moire_dilation_entropy | rank 298/298 | 0.9496 |
Mostow Rigidity › Mostow Rigidity:margulis_ratio | rank 1/298 | 1.0000 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_fine | rank 296/298 | 0.0000 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_std | rank 296/298 | 0.0000 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_mean | rank 298/298 | 0.0000 |
Möbius-S³ › Möbius-S³:hopf_fiber_coherence | rank 298/298 | 0.0000 |
Navier-Stokes › Navier-Stokes:ess_quality | rank 296/298 | 0.0000 |
Nonstationarity › Nonstationarity:change_quantiles_mid | rank 1/298 | 1.0000 |
Nonstationarity › Nonstationarity:dynamic_coupling | rank 298/298 | 0.0000 |
Nonstationarity › Nonstationarity:metric_volatility | rank 298/298 | 0.0000 |
Nonstationarity › Nonstationarity:trajectory_dim | rank 298/298 | 0.0000 |
Nonstationarity › Nonstationarity:vol_of_vol | rank 298/298 | 0.0000 |
Persistent Homology › Persistent Homology:persistence_entropy | rank 294/298 | 1.0000 |
Predictability › Predictability:cond_entropy_k1 | rank 298/298 | 0.0000 |
Projective ℙ² › Projective ℙ²:distance_std | rank 1/298 | 0.3073 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_exponent | rank 298/298 | 0.0000 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_std | rank 298/298 | 0.0000 |
Sol (Thurston) › Sol (Thurston):sol_step_persistence | rank 297/298 | -0.2000 |
Sol (Thurston) › Sol (Thurston):path_length | rank 298/298 | 6350.4459 |
Spectral Analysis › Spectral Analysis:phase_coherence | rank 1/298 | 0.9998 |
Spectral Analysis › Spectral Analysis:spectral_r2 | rank 297/298 | 0.0000 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 298/298 | 0.0000 |
Spectral Analysis › Spectral Analysis:spectral_flatness | rank 298/298 | 0.0000 |
Spherical S² › Spherical S²:concentration | rank 1/298 | 1.0000 |
Spherical S² › Spherical S²:hemisphere_balance | rank 296/298 | 0.0000 |
Spherical S² › Spherical S²:angular_spread | rank 298/298 | 0.0000 |
Spherical S² › Spherical S²:spectral_gini | rank 298/298 | 0.0000 |
Symplectic › Symplectic:phase_volume_explored | rank 298/298 | 0.0020 |
Symplectic › Symplectic:windowed_area_cv | rank 298/298 | 0.0000 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):great_circle_dispersion | rank 1/298 | 3.1416 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):sphere_concentration | rank 298/298 | 0.0002 |
Time Reversibility › Time Reversibility:increment_skewness_max | rank 297/298 | 0.0005 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 294/298 | 0.0004 |
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variance | rank 297/298 | 0.0096 |
Visibility Graph › Visibility Graph:assortativity | rank 298/298 | -0.4966 |
Wavelet Cascade › Wavelet Cascade:scale_gini_dispersion | rank 298/298 | 0.0000 |
Zariski › Zariski:residual_slope | rank 1/298 | -0.0406 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):gini_coefficient | rank 297/298 | 0.0000 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):gini_coefficient | rank 297/298 | 0.0000 |
p-Variation › p-Variation:var_p2 | rank 1/298 | 0.9999 |
p-Variation › p-Variation:increment_persistence | rank 298/298 | -1.0000 |
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