Almost-global convergence to twisted states in discrete Huygens rings
Abstract
We study the discrete Huygens phase map on the ring Cn, the simplest oscillator network in which the coupling can be frustrated. Reciprocal coupling favours phase opposition between neighbours, while the edge phase differences must sum to zero modulo 2π around the cycle; for odd n these two requirements are incompatible. The parity of n determines the absolute minima of the potential. When 4| n, the critical set also contains circles of critical points, all of which meet at two completely degenerate configurations. After quotienting by the common phase, the map is a gradient step for an intrinsic potential on an (n−1)-torus, and the cycle spectrum gives an explicit interval of interaction strengths on which every orbit converges to a critical point. Our main result is that, on this whole interval and for every n≥3, almost every orbit converges to a uniform twisted minimum, including at interaction strengths for which the map is not invertible. When 4| n, the non-minimal points of the critical circles are treated by a centre-stable manifold argument. At the two degenerate intersections, a local blow-up reduces the problem to the cubic leading term of the potential and its restriction to the unit sphere. Orbits approaching along directions where this restriction is positive form a null set, because the cubic is harmonic in the intrinsic metric. No orbit approaches along its zero level, where the critical points are non-degenerate saddles on invariant rays. Hence the degenerate equilibria also have basins of zero measure.
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BibTeX
@misc{buescu2026almostglobal,
title = {{Almost-global convergence to twisted states in discrete Huygens rings}},
author = {Jorge Buescu and Henrique M. Oliveira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19953},
eprint = {2610.19953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19953},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Almost-global convergence to twisted states in discrete Huygens rings
AU - Jorge Buescu
AU - Henrique M. Oliveira
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19953
UR - https://arxiv.org/abs/2610.19953
AB - We study the discrete Huygens phase map on the ring C_n, the simplest oscillator network in which the coupling can be frustrated. Reciprocal coupling favours phase opposition between neighbours, while the edge phase differences must sum to zero modulo 2π around the cycle; for odd n these two requirements are incompatible. The parity of n determines the absolute minima of the potential. When 4| n, the critical set also contains circles of critical points, all of which meet at two completely degenerate configurations. After quotienting by the common phase, the map is a gradient step for an intrinsic potential on an (n−1)-torus, and the cycle spectrum gives an explicit interval of interaction strengths on which every orbit converges to a critical point. Our main result is that, on this whole interval and for every n≥3, almost every orbit converges to a uniform twisted minimum, including at interaction strengths for which the map is not invertible. When 4| n, the non-minimal points of the critical circles are treated by a centre-stable manifold argument. At the two degenerate intersections, a local blow-up reduces the problem to the cubic leading term of the potential and its restriction to the unit sphere. Orbits approaching along directions where this restriction is positive form a null set, because the cubic is harmonic in the intrinsic metric. No orbit approaches along its zero level, where the critical points are non-degenerate saddles on invariant rays. Hence the degenerate equilibria also have basins of zero measure.
ER -