Low-Dimensional Reduction of Mean-Field Coupled Dynamical Systems with Units Described by Two Real Variables: Classification and Applications
Abstract
Low-dimensional reduction is crucial for understanding network systems with many units. In a recent advance, Martens and Dharmavaram developed a theory of exact low-dimensional reduction for a general class of mean-field coupled dynamical systems characterized by a mean-field Lie-Scheffers structure. In this paper, we classify mean-field coupled dynamical systems with a mean-field Lie-Scheffers structure, where each unit is described by two real variables. Our classification is based on González-López, Kamran, and Olver's local classification of Lie systems on the plane near generic points and yields 28 types of mean-field coupled dynamical systems. For mean-field coupled systems with units described by two real variables, we first present a general theory on classification and low-dimensional reduction based on the mean-field Lie-Scheffers structure, and then discuss applications to populations of coupled oscillators. We show that low-dimensional reduction is possible for a broad class of populations of limit-cycle oscillators, extending the previously reported reducible class of populations of Stuart-Landau oscillators to include many new models.
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BibTeX
@misc{tokunaga2026lowdimensional,
title = {{Low-Dimensional Reduction of Mean-Field Coupled Dynamical Systems with Units Described by Two Real Variables: Classification and Applications}},
author = {Kai Tokunaga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28715},
eprint = {2610.28715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28715},
}
RIS
TY - UNPB
TI - Low-Dimensional Reduction of Mean-Field Coupled Dynamical Systems with Units Described by Two Real Variables: Classification and Applications
AU - Kai Tokunaga
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28715
UR - https://arxiv.org/abs/2610.28715
AB - Low-dimensional reduction is crucial for understanding network systems with many units. In a recent advance, Martens and Dharmavaram developed a theory of exact low-dimensional reduction for a general class of mean-field coupled dynamical systems characterized by a mean-field Lie-Scheffers structure. In this paper, we classify mean-field coupled dynamical systems with a mean-field Lie-Scheffers structure, where each unit is described by two real variables. Our classification is based on González-López, Kamran, and Olver's local classification of Lie systems on the plane near generic points and yields 28 types of mean-field coupled dynamical systems. For mean-field coupled systems with units described by two real variables, we first present a general theory on classification and low-dimensional reduction based on the mean-field Lie-Scheffers structure, and then discuss applications to populations of coupled oscillators. We show that low-dimensional reduction is possible for a broad class of populations of limit-cycle oscillators, extending the previously reported reducible class of populations of Stuart-Landau oscillators to include many new models.
ER -