Arnold Cat Map
chaos · 36 views
chaos
What It Is
Hyperbolic toral automorphism [[2,1],[1,1]] on the 2-torus --- uniformly hyperbolic, Anosov diffeomorphism, mixing with Lyapunov exponent ln((3+√5)/2)
Interpretation
Standard analysis sees: rich, high-entropy values; high-dimensional / space-filling. The atlas finds no named structure, but the source is distinctively extreme on Ordinal Partition:memory_order (+3.7z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.29
asymmetry0.46
occupancy0.92
short-range corr0.29
long-range memory0.24
spectral colour0.83
periodicity0.20
complexity0.84
time-irreversibility0.38
volatility clustering0.25
multifractality0.23
dimensionality0.91
nonstationarity0.21
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Ordinal Partition:memory_order | +3.7z | bank-miss 2.2σ |
Zipf–Mandelbrot (8-bit):zipf_r_squared | -2.9z | bank-miss 1.0σ |
Composition
dtypefloat64
range[4.18e-05, 1]
unique values16384 / 16384
mean ± std0.505 ± 0.285
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/Arnold_Cat_Map.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Arnold_Cat_Map.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Arnold_Cat_Map.png)
Inflation (Substitution) > barcode
/d_curve/Arnold_Cat_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/Arnold_Cat_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/Arnold_Cat_Map.png)
Ulam Spiral (Sacks) > default
/default/Arnold_Cat_Map.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| glibc LCG | 2.54 | cross-domain |
| ChaCha20 | 2.61 | cross-domain |
| BSL Residues | 2.63 | cross-domain |
Which Geometries Light Up
Catch24 › Catch24:SB_TransitionMatrix_3ac_sumdiagcov | rank 296/298 | 0.0000 |
Möbius-S³ › Möbius-S³:phase_profile_deviation | rank 294/298 | 0.1126 |
Predictability › Predictability:cond_entropy_k1 | rank 1/298 | 2.9977 |
Predictability › Predictability:cond_entropy_k8 | rank 4/298 | 2.9998 |
Symplectic › Symplectic:recurrence_rate | rank 295/298 | 0.0279 |
Zariski › Zariski:nonsep_fraction | rank 298/298 | 0.0065 |
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