Magnetic Pendulum (3-Magnet)
What It Is
Pendulum tip in 2D plane above three magnets at triangle vertices, with linear gravity-restoring force --- conservative (no damping) 2-DOF Hamiltonian chaos. The damped version has the famous fractal basin boundaries; the undamped version sweeps the energy surface chaotically. Velocity-Verlet (symplectic) integration. Output: x(t).
Interpretation
Standard analysis sees: rich, high-entropy values; smooth / autocorrelated; long-range memory (persistent); low-complexity (predictable, not noise-like); volatility-clustering (bursty); multifractal. The atlas finds no named structure, but the source is distinctively extreme on Moiré:moire_invariance_breadth (+3.0z) — beyond what the standard bank predicts for it. It sits beside Duffing Oscillator in the atlas (standard-bank rank 20) — a neighbor conventional features miss.
Moiré:moire_invariance_breadth | +3.0z | bank-miss 1.3σ |
Ammann-Beenker (Octagonal):convergent_resonance | +2.4z | bank-miss 1.2σ |
Composition
Render Gallery
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_(centered)/signed_log_z/Magnetic_Pendulum_(3-Magnet).png)
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/phi_spectrum/Magnetic_Pendulum_(3-Magnet).png)
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/default/Magnetic_Pendulum_(3-Magnet).png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Chua's Circuit | 2.71 | |
| Spring Pendulum | 2.89 | |
| Duffing Oscillator | 2.92 |
Which Geometries Light Up
Ammann-Beenker (Octagonal) › Ammann-Beenker (Octagonal):convergent_resonance | rank 4/306 | 0.9000 |
Bispectrum › Bispectrum:coupling_frequency_centroid | rank 5/306 | 0.6192 |
Catch24 › Catch24:CO_Embed2_Dist_tau_d_expfit_meandiff | rank 2/306 | 56.9308 |
Catch24 › Catch24:SB_BinaryStats_diff_longstretch0 | rank 5/306 | 775.2000 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 302/306 | 0.0110 |
Chladni › Chladni:captured_power_fraction | rank 304/306 | 0.0006 |
Hodge–Laplacian › Hodge–Laplacian:curl_div_ratio | rank 3/306 | 5.8527 |
Hodge–Laplacian › Hodge–Laplacian:solenoidal_fraction | rank 4/306 | 0.9967 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):hyperbolic_triangle_area | rank 303/306 | 0.0001 |
Hölder Regularity › Hölder Regularity:holder_min | rank 5/306 | 0.9227 |
Laplacian › Laplacian:gradient_sign_persistence | rank 2/306 | 0.9976 |
Moiré › Moiré:moire_invariance_breadth | rank 2/306 | 0.0380 |
Möbius-S³ › Möbius-S³:hopf_fiber_coherence | rank 3/306 | 0.8039 |
Projective ℙ² › Projective ℙ²:distance_std | rank 302/306 | 0.0001 |
Spectral Analysis › Spectral Analysis:spectral_r2 | rank 4/306 | 0.9658 |
Symplectic › Symplectic:flux_sign_persistence | rank 5/306 | 277.9894 |
Ulam Spiral (Square) › Ulam Spiral (Square):radon_anisotropy | rank 2/306 | 0.0421 |
Visibility Graph › Visibility Graph:nvg_hvg_reach_divergence | rank 2/306 | 0.9974 |
p-Variation › p-Variation:volatility_clustering | rank 1/306 | 0.9998 |
p-Variation › p-Variation:variation_index | rank 303/306 | 1.0005 |