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Weak-Coupling Ramp Trial

Validation process — instantiates Decentralized Phase Locking

Brings coupling up slowly from near zero to find the threshold where the population captures into lock — and the detuning, noise, and delay it tolerates.

You do not want to discover where a population locks by turning coupling all the way up and hoping. Weak-Coupling Ramp Trial starts from near-zero coupling and raises it slowly, watching at each level for the moment the population captures into a stable phase relation — and it holds there to map how far the units' rates may differ, and how much noise and delay they can bear, while still locking. Its defining trait is direction: it approaches lock from the weak side, so it finds the onset of capture — the least coupling that works — rather than the behavior of an already-locked system. That lets you operate just above the threshold instead of over-stiffening the coupling into something brittle, and it publishes the disturbance budget the lock is actually warranted for.

Example

Two pendulum clocks hung on a shared beam will, famously, drift into anti-phase step through the beam's faint coupling — the effect Huygens noticed three centuries ago. Suppose you are designing such a coupled set and can vary the beam's stiffness, which is the coupling strength. A ramp trial begins with the beam almost decoupled and stiffens it in small increments, checking at each: do the clocks capture into a stable relation, at what stiffness do they first lock, how far apart can their natural rates be and still capture, and do they pass through an unstable slipping phase on the way. It reports that capture begins at ≈stiffness k*, tolerates a rate mismatch up to ≈X, and that beyond about 2·k* the relation turns over-stiff and brittle — so the design should sit just above k*.

How it works

What distinguishes the ramp from other validations:

  • Monotonic approach from weak. Coupling rises slowly from near zero; the trial reads the first level at which capture appears — the threshold, not the steady state.
  • Detuning sweep at the edge. Holding coupling, it varies the units' rate mismatch to map the range over which capture holds, not a single point.
  • Realistic disturbance injected. Noise and delay are added during the ramp so the tolerated budget reflects operating conditions, not an idealized bench.
  • Abort on premature clustering. If partial clusters or overshoot appear on the way up, the ramp stops before they can harden.

Tuning parameters

  • Ramp rate — slow reveals the true quasi-static threshold; fast is quicker but can overshoot and mis-locate capture.
  • Start/stop coupling bounds — how weak to begin and how strong to push before calling it.
  • Detuning sweep width — how far to vary rate mismatch when mapping the capture range.
  • Noise/delay injection — how much realistic disturbance to add, so the budget reflects the field and not the lab.
  • Abort criteria — the clustering or overshoot thresholds that halt the ramp before harm.

When it helps, and when it misleads

Its strength is setting coupling to the least value that captures — avoiding the over-stiff, brittle lock that maximal coupling produces — and publishing the detuning, noise, and delay budget under which lock is guaranteed before anything depends on it.

It misleads when its path-dependence is forgotten. A ramp explores capture from the weak side, and the state reached that way can differ from what the system settles into from other initial conditions — capture and release thresholds are often not the same, a hysteresis that a single upward ramp will miss.[1] Idealized trials, run without realistic noise, overstate the budget. The classic misuse is ramping fast to "confirm it locks" and reading a threshold that is really an overshoot artifact. The discipline is to ramp slowly, sweep detuning across the whole range, inject operating-grade noise and delay, and test both up and down to expose hysteresis.

How it implements the components

  • capture_range_and_entrainment_basin — the trial's core output: the coupling threshold at which lock forms and the range of detuning and initial conditions from which the target relation is actually reachable.
  • delay_noise_and_detuning_budget — by injecting realistic disturbance during the ramp, it bounds the detuning, noise, and delay under which capture holds — the conditions the lock is warranted for.

It does not shock an already-locked population to test recovery — that is Perturb-and-Relock Drill — nor measure a single unit's response curve, which is Phase-Response Curve Calibration; it sweeps the whole population's coupling from cold.

Notes

Capture is often hysteretic: the coupling at which a cold population locks (ramping up) is not the coupling at which a locked one breaks (ramping down). A single upward ramp measures only the first. Run the ramp in both directions when the gap matters — operating between the two thresholds means the lock will hold once formed but would not re-form if lost.

References

[1] The set of frequency-mismatch-and-coupling combinations for which two oscillators lock is classically an Arnold tongue — a wedge that widens with coupling. Mapping it, rather than confirming a single locking point, is what turns a ramp trial into a usable capture range and disturbance budget. ↩