Rare but stable: hidden states in the 1D swarmalator model
Abstract
The one-dimensional swarmalator model is usually understood through three macroscopic states: synchrony, the phase wave, and asynchrony. We show that the identical model also contains a much larger set of hidden stable equilibria - families of q-twisted and m-clustered states - which random initial conditions almost never reach, because many are embedded in a degenerate neutral manifold of vanishing global basin. They are nonetheless locally robust: under Gaussian perturbations of per-coordinate amplitude ε their survival radii shrink only algebraically, as 1/q and 1/m, in sharp contrast to the Gaussian basins and winding ceiling of twisted states on a nearest-neighbour Kuramoto ring, both removed by mean-field coupling. A block-circulant Jacobian fixes the stability in closed form, and far-kick experiments indicate the basins are compact cores rather than tentacled sets. Several patterns reported in extended swarmalator models thus already exist, stable or neutrally stable, in the minimal one.
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BibTeX
@misc{djeudjo2026rare,
title = {{Rare but stable: hidden states in the 1D swarmalator model}},
author = {Rommel Tchinda Djeudjo and Kevin P. O'Keeffe and Timoteo Carletti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33929},
eprint = {2609.33929},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33929},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rare but stable: hidden states in the 1D swarmalator model
AU - Rommel Tchinda Djeudjo
AU - Kevin P. O'Keeffe
AU - Timoteo Carletti
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33929
UR - https://arxiv.org/abs/2609.33929
AB - The one-dimensional swarmalator model is usually understood through three macroscopic states: synchrony, the phase wave, and asynchrony. We show that the identical model also contains a much larger set of hidden stable equilibria - families of q-twisted and m-clustered states - which random initial conditions almost never reach, because many are embedded in a degenerate neutral manifold of vanishing global basin. They are nonetheless locally robust: under Gaussian perturbations of per-coordinate amplitude ε their survival radii shrink only algebraically, as 1/q and 1/m, in sharp contrast to the Gaussian basins and winding ceiling of twisted states on a nearest-neighbour Kuramoto ring, both removed by mean-field coupling. A block-circulant Jacobian fixes the stability in closed form, and far-kick experiments indicate the basins are compact cores rather than tentacled sets. Several patterns reported in extended swarmalator models thus already exist, stable or neutrally stable, in the minimal one.
ER -