Circle Map Quasiperiodic
chaos · 36 views
chaos
What It Is
Sine circle map at K=0 with golden-mean rotation --- pure quasiperiodic motion on a torus, deterministic but never periodic, flat spectrum with dense peaks at golden-mean harmonics
Interpretation
Standard analysis sees: rich, high-entropy values; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; time-irreversible (sharp rises, slow decay); homoskedastic; monofractal; low-dimensional; stationary. The atlas additionally detects quasicrystalline order. It sits beside Critical Circle Map (Bronze Mean) in the atlas (standard-bank rank 23) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.24
asymmetry0.43
occupancy0.99
short-range corr0.08
long-range memory0.07
spectral colour0.87
periodicity0.88
complexity0.20
time-irreversibility0.90
volatility clustering0.00
multifractality0.05
dimensionality0.08
nonstationarity0.05
What the atlas adds
quasicrystalline order+6.0z
aperiodic long-range order at incommensurate (golden-ratio / Fibonacci) spacing — structure repeats at irrational ratios, not a fixed period
Atlas-extreme metrics the standard bank can’t predict for this source
Moiré:moire_phi_response | +7.5z | bank-miss 1.9σ |
D4 Triality:triplet_temporal | -2.9z | bank-miss 1.2σ |
Time Reversibility:increment_skewness_max | +2.3z | bank-miss 1.1σ |
Penrose (Quasicrystal):phi_tower | +2.1z | bank-miss 1.2σ |
Hodge–Laplacian:solenoidal_fraction | -2.1z | bank-miss 1.5σ |
Composition
dtypefloat64
range[5.48e-05, 0.9999]
unique values16384 / 16384
mean ± std0.5 ± 0.289
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/Circle_Map_Quasiperiodic.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Circle_Map_Quasiperiodic.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Circle_Map_Quasiperiodic.png)
Inflation (Substitution) > barcode
/d_curve/Circle_Map_Quasiperiodic.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/Circle_Map_Quasiperiodic.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/Circle_Map_Quasiperiodic.png)
Ulam Spiral (Sacks) > default
/default/Circle_Map_Quasiperiodic.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Phyllotaxis | 2.21 | cross-domain |
| Critical Circle Map | 4.20 | |
| Critical Circle Map (Bronze Mean) | 4.40 |
Which Geometries Light Up
Catch24 › Catch24:CO_trev_1_num | rank 3/298 | 2.3164 |
Cayley › Cayley:local_linearity | rank 1/298 | 1.0000 |
Fisher Information › Fisher Information:effective_dimension | rank 3/298 | 16.0000 |
Fisher Information › Fisher Information:log_det_fisher | rank 296/298 | 44.3614 |
Fisher Information › Fisher Information:trace_fisher | rank 296/298 | 256.0002 |
Hodge–Laplacian › Hodge–Laplacian:solenoidal_fraction | rank 297/298 | 0.2384 |
Hodge–Laplacian › Hodge–Laplacian:curl_div_ratio | rank 298/298 | -1.2723 |
Inflation (Substitution) › Inflation (Substitution):acf_geometric | rank 3/298 | 0.8888 |
Laplacian › Laplacian:gradient_curvature_anticorrelation | rank 4/298 | 1.0000 |
Laplacian › Laplacian:curvature_autocorrelation | rank 295/298 | -0.3090 |
Laplacian › Laplacian:cross_scale_curvature_coherence | rank 296/298 | -0.5878 |
Laplacian › Laplacian:laplacian_evolutionary_index | rank 296/298 | -0.3090 |
Modular Residue › Modular Residue:cycle_fraction | rank 1/298 | 1.0000 |
Modular Residue › Modular Residue:occupancy_entropy | rank 1/298 | 1.0000 |
Moiré › Moiré:moire_phi_response | rank 1/298 | 0.3666 |
Möbius-S³ › Möbius-S³:phi_vertex_excess | rank 5/298 | 0.1708 |
Nonstationarity › Nonstationarity:metric_volatility | rank 5/298 | 3.0933 |
Penrose (Quasicrystal) › Penrose (Quasicrystal):phi_tower | rank 2/298 | 0.8854 |
Septagonal (Danzer) › Septagonal (Danzer):cubic_coherence | rank 298/298 | -0.2779 |
Septagonal (Danzer) › Septagonal (Danzer):z_primary | rank 298/298 | -0.5634 |
Sol (Thurston) › Sol (Thurston):path_length | rank 297/298 | 7704.6942 |
Spectral Graph › Spectral Graph:weyl_exponent | rank 2/298 | 2.1665 |
Wasserstein › Wasserstein:entropy | rank 3/298 | 5.0000 |
Wasserstein › Wasserstein:concentration | rank 295/298 | 1.0035 |
Wasserstein › Wasserstein:dist_from_uniform | rank 295/298 | 0.0000 |
p-Variation › p-Variation:volatility_clustering | rank 296/298 | -0.6180 |
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