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PreprintCC BY 4.0arXiv

Arithmetic of the sync basin for pulse-coupled oscillators

  • K. P. O'Keeffe
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Abstract

A population of N identical pulse-coupled oscillators ultimately settles into one of two outcomes: full synchrony or a state of co-existing synchronized clusters. We show that which outcome occurs is controlled by the prime factorization of N. At the critical charging curve --- linear, the boundary between the synchronizing and clustering regimes --- the synchronization basin acquires exact arithmetic structure. The synchronization probability is Psync=AN,1/NN for all N, where AN,1 satisfies an exact recurrence relation. For prime N, AN,1=NN−1 giving the closed form Psync=1−1/NN; for composite N, the observed asymptotic scaling is 1−Psync∼ Cm N−(m−1), where m is the smallest prime divisor. The result adds a new member to the atlas of exotic basin geometries: alongside fractal, riddled, and tentacled basins, we now have a basin that is arithmetic.

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Cite

BibTeXRIS
BibTeX
@misc{okeeffe2026arithmetic,
  title = {{Arithmetic of the sync basin for pulse-coupled oscillators}},
  author = {K. P. O'Keeffe},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01668},
  eprint = {2609.01668},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01668},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Arithmetic of the sync basin for pulse-coupled oscillators
AU  - K. P. O'Keeffe
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.01668
UR  - https://arxiv.org/abs/2609.01668
AB  - A population of N identical pulse-coupled oscillators ultimately settles into one of two outcomes: full synchrony or a state of co-existing synchronized clusters. We show that which outcome occurs is controlled by the prime factorization of N. At the critical charging curve --- linear, the boundary between the synchronizing and clustering regimes --- the synchronization basin acquires exact arithmetic structure. The synchronization probability is Psync=A_N,1/N^N for all N, where A_N,1 satisfies an exact recurrence relation. For prime N, A_N,1=N^N−1 giving the closed form Psync=1−1/N^N; for composite N, the observed asymptotic scaling is 1−Psync∼ C_m N^−(m−1), where m is the smallest prime divisor. The result adds a new member to the atlas of exotic basin geometries: alongside fractal, riddled, and tentacled basins, we now have a basin that is arithmetic.
ER  -