标题： 拓扑Kuramoto模型在细胞复形上的相位锁定和多稳定性 显示英文标题

标题： Phase locking and multistability in the topological Kuramoto model on cell complexes

摘要： 拓扑 Kuramoto 模型通过包含高阶相互作用，扩展了经典的同步模型，其振子动力学定义在单纯复形或细胞复形中的任意维度的单元上。 在本文中，我们展示了拓扑 Kuramoto 模型中的多稳态，并开发了拓扑非线性 Kirchhoff 条件算法，以识别任意细胞复形上的所有相位锁定状态。 该算法基于将 Kirchhoff 定律推广到任意维度的细胞复形以及单元之间的非线性相互作用。 通过将此框架应用于环、正则多面体和单形，作为更大网络的最小代表性模块，我们推导出了显式界限（基于绕数约束），这些界限决定了共存稳定状态的数量。 我们发现了多稳态的结构级联，这些级联既继承自较低维度也继承自较高维度，并证明了细胞复形可以产生比相同维度的单纯复形更丰富的多稳态模式。 此外，我们发现细胞复形中的多稳态模式似乎由边界单元的数量决定，这暗示了一种可能的普遍模式。 显示英文摘要

摘要： The topological Kuramoto model generalizes classical synchronization models by including higher-order interactions, with oscillator dynamics defined on cells of arbitrary dimension within simplicial or cell complexes. In this article, we demonstrate multistability in the topological Kuramoto model and develop the topological nonlinear Kirchhoff conditions algorithm to identify all phase-locked states on arbitrary cell complexes. The algorithm is based on a generalization of Kirchhoff's laws to cell complexes of arbitrary dimension and nonlinear interactions between cells. By applying this framework to rings, Platonic solids, and simplexes, as minimal representative motifs of larger networks, we derive explicit bounds (based on winding number constraints) that determine the number of coexisting stable states. We uncover structural cascades of multistability, inherited from both lower and higher dimensions and demonstrate that cell complexes can generate richer multistability patterns than simplicial complexes of the same dimension. Moreover, we find that multistability patterns in cell complexes appear to be determined by the number of boundary cells, hinting a possible universal pattern.