Critical Circle Map
chaos · 36 views
chaos
What It Is
Sine circle map at K=1 critical coupling with golden-mean bare frequency --- sits on the boundary of mode-locking, a different universality class from Feigenbaum's period-doubling
Interpretation
Standard analysis sees: anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; time-irreversible (sharp rises, slow decay); homoskedastic; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Navier-Stokes:cascade_asymmetry (-3.2z) — beyond what the standard bank predicts for it. It sits beside Critical Circle Map (Bronze Mean) in the atlas (standard-bank rank 27) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.40
asymmetry0.20
occupancy0.56
short-range corr0.11
long-range memory0.09
spectral colour0.87
periodicity0.86
complexity0.18
time-irreversibility0.91
volatility clustering0.01
multifractality0.09
dimensionality0.17
nonstationarity0.09
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Navier-Stokes:cascade_asymmetry | -3.2z | bank-miss 3.3σ |
D4 Triality:triplet_temporal | -3.0z | bank-miss 1.6σ |
Sol (Thurston):sol_step_persistence | -2.4z | bank-miss 1.0σ |
p-Variation:volatility_clustering | -2.2z | bank-miss 1.3σ |
Cayley:local_linearity | +2.0z | bank-miss 1.3σ |
Composition
dtypefloat64
range[0.0004602, 0.9949]
unique values16384 / 16384
mean ± std0.529 ± 0.228
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Critical_Circle_Map.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Critical_Circle_Map.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Critical_Circle_Map.png)
Inflation (Substitution) > barcode
/d_curve/Critical_Circle_Map.png)
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
/phi_spectrum/Critical_Circle_Map.png)
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
/default/Critical_Circle_Map.png)
Ulam Spiral (Sacks) > default
/default/Critical_Circle_Map.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Critical Circle Map (Silver Mean) | 3.39 | |
| Critical Circle Map (Bronze Mean) | 3.64 | |
| Circle Map Quasiperiodic | 4.20 |
Which Geometries Light Up
Catch24 › Catch24:CO_trev_1_num | rank 2/298 | 2.3464 |
Cayley › Cayley:growth_exponent | rank 294/298 | 0.7459 |
D4 Triality › D4 Triality:triplet_temporal | rank 294/298 | -0.7815 |
Fractal (Mandelbrot) › Fractal (Mandelbrot):escape_entropy | rank 4/298 | 3.3300 |
Klein Bottle › Klein Bottle:rank_deficit | rank 298/298 | 0.0471 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 295/298 | -0.5832 |
Laplacian › Laplacian:curvature_autocorrelation | rank 297/298 | -0.3451 |
Laplacian › Laplacian:laplacian_evolutionary_index | rank 297/298 | -0.3380 |
Modular Residue › Modular Residue:cycle_fraction | rank 4/298 | 0.9867 |
Möbius-S³ › Möbius-S³:phi_vertex_excess | rank 4/298 | 0.2000 |
Navier-Stokes › Navier-Stokes:cascade_asymmetry | rank 294/298 | -0.7348 |
Projective ℙ² › Projective ℙ²:cross_ratio_std | rank 4/298 | 40.7796 |
Sol (Thurston) › Sol (Thurston):sol_step_persistence | rank 294/298 | -0.0955 |
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