Critical Circle Map (Bronze Mean)
chaos · 36 views
chaos
What It Is
Sine circle map at K=1 critical coupling tuned to the bronze-mean winding number (sqrt(13)-3)/2 (continued fraction [0;3,3,3,...]) --- a third distinct winding-number universality class for the quasiperiodic route to chaos
Interpretation
Standard analysis sees: anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); time-irreversible (slow rise, sharp collapse); homoskedastic; monofractal; low-dimensional; stationary. The atlas finds no named structure, but the source is distinctively extreme on Laplacian:laplacian_evolutionary_index (-2.8z) — beyond what the standard bank predicts for it. It sits beside Critical Circle Map in the atlas (standard-bank rank 42) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.42
asymmetry0.70
occupancy0.62
short-range corr0.14
long-range memory0.08
spectral colour0.88
periodicity0.82
complexity0.19
time-irreversibility0.04
volatility clustering0.04
multifractality0.09
dimensionality0.10
nonstationarity0.07
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Laplacian:laplacian_evolutionary_index | -2.8z | bank-miss 1.2σ |
Navier-Stokes:intermittency_autocorr | -2.1z | bank-miss 1.6σ |
Composition
dtypefloat64
range[0.003474, 0.9984]
unique values16384 / 16384
mean ± std0.48 ± 0.229
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/Critical_Circle_Map_(Bronze_Mean).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Critical_Circle_Map_(Bronze_Mean).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Critical_Circle_Map_(Bronze_Mean).png)
Inflation (Substitution) > barcode
/d_curve/Critical_Circle_Map_(Bronze_Mean).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/Critical_Circle_Map_(Bronze_Mean).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/Critical_Circle_Map_(Bronze_Mean).png)
Ulam Spiral (Sacks) > default
/default/Critical_Circle_Map_(Bronze_Mean).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Critical Circle Map (Silver Mean) | 3.44 | |
| Critical Circle Map | 3.64 | |
| Phyllotaxis | 4.19 | cross-domain |
Which Geometries Light Up
Boltzmann › Boltzmann:nn_dominance | rank 296/298 | 0.0012 |
Catch24 › Catch24:CO_trev_1_num | rank 297/298 | -3.1434 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):potential_roughness | rank 4/298 | 3.4232 |
Hodge–Laplacian › Hodge–Laplacian:solenoidal_fraction | rank 294/298 | 0.2801 |
Hodge–Laplacian › Hodge–Laplacian:curl_div_ratio | rank 296/298 | -0.8954 |
Julia Set › Julia Set:escape_entropy | rank 5/298 | 4.4539 |
Klein Bottle › Klein Bottle:rank_deficit_max | rank 296/298 | 0.1019 |
Möbius-S³ › Möbius-S³:phi_vertex_excess | rank 3/298 | 0.2000 |
Navier-Stokes › Navier-Stokes:intermittency_autocorr | rank 296/298 | -0.4629 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 294/298 | -0.2336 |
← / → within domain · ⇧← / ⇧→ alphabetical · ⇧← / ⇧→ inside an open render = same view across sources