Home Theory Data Log

Notes on Coupled Oscillator Coherence

Last updated: 2026-07-14

This is a working notebook on phase-locking behaviour in weakly-coupled oscillator arrays under non-stationary perturbation. The core question: under what conditions does a ring of N coupled Kuramoto-type oscillators maintain phase coherence when a subset of nodes experiences stochastic frequency drift?

I first looked at this in 2023 as a side problem while working on distributed clock recovery. The standard result (all-to-all coupling, identical natural frequencies) gives a clean bifurcation at K = Kc. What I'm interested in is the non-identical case with sparse coupling and bounded noise — which, as far as I can tell, doesn't have a closed-form solution in the literature.

Current status

Simulation framework (coupled-osc) is in a usable state. Most recent run: N=64, ring topology, K ∈ [0.5, 3.0], noise scale σ ∈ [0.01, 0.15]. Preliminary phase-slip distributions match the heuristic from Strogatz (2000) for the homogeneous limit, but deviate sharply once σ > 0.08.

See Data for the runs. See Theory for notation and derivations. See Log for chronological notes.

Simulation code

git clone https://git.sr.ht/~someone/coupled-osc

Requires Python 3.10+, numpy, scipy, matplotlib.

References (partial)

  1. Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. International Symposium on Mathematical Problems in Theoretical Physics, 420–422.
  2. Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143(1–4), 1–20.
  3. Pikovsky, A., Rosenblum, M., & Kurths, J. (2001). Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge.
  4. Acebrón, J. A. et al. (2005). The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics, 77(1), 137.

This is a personal research log. No affiliation. Corrections and comments welcome via the contact address in the commit logs.