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PreprintCC BY 4.0arXiv

A solvable normal form for coupled swarmalators

  • Kevin P. O'Keeffe
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Abstract

Swarmalators are mobile generalizations of phase oscillators. Introduced to model systems in which sync and self-assembly interact, they remain poorly understood theoretically. Unlike the Kuramoto model for coupled oscillators, existing swarmalator models lack a normal-form foundation, and their basic stabilities and bifurcations remain largely unsolved. Here we address both problems. Building on Tanaka's reduction of chemotactic oscillators, we show that the canonical one-dimensional swarmalator model -- previously introduced as an ad hoc toy model -- is recovered in the first-harmonic, zero-lag limit, implying its behavior is generic. We then derive the stability boundaries organizing its four collective states, show they meet at a single cusp, correct a previously published order-parameter formula, and uncover a non-monotonic sync response absent in the Kuramoto model.

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Cite

BibTeXRIS
BibTeX
@misc{okeeffe2026solvable,
  title = {{A solvable normal form for coupled swarmalators}},
  author = {Kevin P. O'Keeffe},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.09810},
  eprint = {2607.09810},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.09810},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A solvable normal form for coupled swarmalators
AU  - Kevin P. O'Keeffe
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.09810
UR  - https://arxiv.org/abs/2607.09810
AB  - Swarmalators are mobile generalizations of phase oscillators. Introduced to model systems in which sync and self-assembly interact, they remain poorly understood theoretically. Unlike the Kuramoto model for coupled oscillators, existing swarmalator models lack a normal-form foundation, and their basic stabilities and bifurcations remain largely unsolved. Here we address both problems. Building on Tanaka's reduction of chemotactic oscillators, we show that the canonical one-dimensional swarmalator model -- previously introduced as an ad hoc toy model -- is recovered in the first-harmonic, zero-lag limit, implying its behavior is generic. We then derive the stability boundaries organizing its four collective states, show they meet at a single cusp, correct a previously published order-parameter formula, and uncover a non-monotonic sync response absent in the Kuramoto model.
ER  -