Does the oscillation's frequency depend on its amplitude — the KAM twist (dω/dA) of nonlinear oscillators?
An oscillator is isochronous when its period is independent of amplitude (the ideal harmonic oscillator, a pure sine). Nonlinear oscillators — pendulum, Duffing, Van der Pol — are anisochronous: the period drifts with amplitude. The geometry extracts individual cycles (bandpass, then Hilbert phase marking 2π advances) and plots each as a point in (cycle amplitude, cycle period) — the orbit of orbits. Anisochronous systems show amplitude↔period coupling; isochronous ones stack at a single period.
|Spearman ρ(cycle amplitude, cycle frequency)|. 0 = isochronous, 1 = strong amplitude↔frequency coupling. Highest on quasiperiodic tori (2-Torus 0.96, 3-Torus 0.86, 4-Torus 0.86) and chaotic windows (Period-5 0.87); 0 on steady limit cycles (Clipped Sine, Lotka-Volterra), where the amplitude never spreads so dω/dA stays hidden — fundamental, not a bug. NaN when the signal does not oscillate.
Coefficient of variation (std/mean) of the per-cycle amplitudes. 0 for a steady limit cycle; high for amplitude-modulated oscillators, chaotic flows, and exploring systems (Heisenberg Walk 2.11, Earthquake P-wave 2.10, Speech "Five" 1.44). Independent of frequency_shear by construction — an AM sine has high amplitude spread but zero shear.
| Source | Origin | Value |
|---|---|---|
| Earthquake P-wave | geophysical | 2.1025 |
| Heisenberg Walk | symbolic-dynamics | 2.0540 |
| Speech "Five" | acoustic | 1.4353 |
| ··· | ||
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.0019 |
| Triangle Wave | signal-synthesis | 0.0075 |
| Clipped Sine | signal-synthesis | 0.0092 |
| Source | Origin | Value |
|---|---|---|
| 2-Torus Quasiperiodic | coupled-oscillators | 0.9604 |
| Logistic r=3.74 (Period-5 Window) | iterated-maps | 0.8659 |
| 3-Torus Quasiperiodic | coupled-oscillators | 0.8607 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | 0.0000 |
| Triangle Wave | signal-synthesis | 0.0000 |
| Clipped Sine | signal-synthesis | 0.0000 |
The oscillator lens for amplitude–frequency coupling. It catches the Hamiltonian KAM twist and vocal-tract modulation (speech is an amplitude_exploration outlier). Both metrics return NaN on non-oscillatory signals by design — the lens only speaks about things that actually oscillate.