Van der Pol Oscillator
continuous-flows · 37 views
continuous-flowsperiodic
What It Is
Van der Pol relaxation oscillations --- slow drift along cubic nullcline punctuated by fast jumps, the canonical model of nonlinear self-sustained oscillation
Interpretation
Standard analysis sees: bounded / light-tailed; smooth / autocorrelated; long-range memory (persistent); strongly periodic; low-complexity (predictable, not noise-like); volatility-clustering (bursty); low-dimensional. The atlas additionally detects Persistent Homology:max_h1_lifetime, Multifractal Spectrum:hurst_estimate, Multi-Scale Wasserstein:w_max_ratio. It sits beside Period-Doubled Wave in the atlas (standard-bank rank 30) — a neighbor conventional features miss.
What standard analysis sees
tail heaviness0.06
asymmetry0.35
occupancy0.70
short-range corr0.97
long-range memory0.98
spectral colour0.21
periodicity0.92
complexity0.04
time-irreversibility0.17
volatility clustering0.93
multifractality0.77
dimensionality0.05
nonstationarity0.35
What the atlas adds
Persistent Homology:max_h1_lifetime+4.6z
reconstructed phase space contains a persistent topological loop (S¹) — a smooth simple limit cycle
Multifractal Spectrum:hurst_estimate+2.5z
self-affinity H = τ(2)/2 (alternate to Hölder hurst)
Multi-Scale Wasserstein:w_max_ratio+2.3z
scale-coherent dispersion mismatch (max/min Wasserstein across scales)
Atlas-extreme metrics the standard bank can’t predict for this source
G2 Root System:short_long_ratio | -3.8z | bank-miss 2.5σ |
Fisher Information:geodesic_velocity | +2.5z | bank-miss 2.6σ |
Composition
dtypefloat64
range[-2.066, 2.066]
unique values16384 / 16384
mean ± std0.0297 ± 1.62
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Van_der_Pol_Oscillator.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Van_der_Pol_Oscillator.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Van_der_Pol_Oscillator.png)
Inflation (Substitution) > barcode
/d_curve/Van_der_Pol_Oscillator.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/Van_der_Pol_Oscillator.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/Van_der_Pol_Oscillator.png)
Ulam Spiral (Sacks) > default
/default/Van_der_Pol_Oscillator.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Duffing Oscillator | 3.74 | |
| Period-Doubled Wave | 3.74 | cross-origin |
| Magnetic Pendulum (3-Magnet) | 3.90 |
Which Geometries Light Up
2-adic › 2-adic:mean_distance | rank 4/307 | 0.6682 |
Catch24 › Catch24:FC_LocalSimple_mean1_tauresrat | rank 1/307 | 1.0084 |
D4 Triality › D4 Triality:edge_ortho_polarity | rank 5/307 | 0.7382 |
D4 Triality › D4 Triality:gram_consistency | rank 303/307 | 0.4575 |
E8 Lattice › E8 Lattice:std_profile | rank 303/307 | 0.1635 |
Fisher Information › Fisher Information:velocity_spectral_gini | rank 2/307 | 0.7915 |
G2 Root System › G2 Root System:short_long_ratio | rank 307/307 | 0.0662 |
Laplacian › Laplacian:gradient_sign_persistence | rank 1/307 | 0.9986 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_mean | rank 4/307 | 0.1353 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_std | rank 4/307 | 0.1483 |
Möbius-S³ › Möbius-S³:hopf_fiber_coherence | rank 5/307 | 0.8023 |
Predictability › Predictability:sample_entropy | rank 303/307 | 0.0302 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):cocycle_growth_burstiness | rank 5/307 | 0.5923 |
Symplectic › Symplectic:flux_sign_persistence | rank 1/307 | 345.3302 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 1/307 | 0.0454 |
Visibility Graph › Visibility Graph:nvg_hvg_reach_divergence | rank 3/307 | 0.9955 |
Visibility Graph › Visibility Graph:degree_r_squared | rank 305/307 | 0.2299 |
p-Variation › p-Variation:variation_index | rank 305/307 | 1.0005 |
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