Symmetry-based selection rules for higher-order interactions in coupled oscillators

Year: 2026

Type: preprint

Abstract: Pairwise interactions among general nonlinear oscillators can be reduced, via phase reduction, to a Kuramoto-type phase coupling $\sin(- θ_j+θ_k )$. For higher-order interactions, multiple phase couplings exist -- such as $\sin(-2θ_j+θ_k+θ_l )$ and $\sin(-θ_j+2θ_k-θ_l)$. Since different nonpairwise coupling functions produce qualitatively different dynamics, it is important to understand which phase couplings should be included in coupled phase oscillator models. In this Letter, we establish selection rules for higher-order phase coupling functions. These selection rules, which can be applied without the need of explicit phase reduction, are solely based on the symmetry of the isolated oscillator velocity field and the $n$-body interaction functions. As phase reduction established the mechanistic basis for the Kuramoto model, our results provide a theoretical link between physical systems and higher-order phase models.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2606.04904

Authors: Léon, Iván, Muolo, Riccardo, Zhang, Yuanzhao, Lucas, Maxime

Institutions: Universidad de Cantabria, Systems Biology Institute, RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences, Santa Fe Institute, University of Namur, UCLouvain

Topic: Nonlinear Dynamics and Pattern Formation

Subfield: Computer Networks and Communications

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Santa Fe Institute, Fonds De La Recherche Scientifique - FNRS, Japan Society for the Promotion of Science, European Regional Development Fund, Agencia Estatal de Investigación

OpenAlex ID: W7163545166