标题： Kuramoto 遇见 Koopman：运动常数、对称性和网络图案 显示英文标题

标题： Kuramoto meets Koopman: Constants of motion, symmetries, and network motifs

摘要： Kuramoto模型的部分可积性通常被认为仅限于同质连接的振子或其群组。 然而，对于更一般的图上存在运动常数的确切连接前提条件一直难以捉摸。 利用Koopman生成元的谱性质，我们得出了在任意加权、有向、带符号图上具有异质相位滞后Kuramoto模型中存在不同运动常数的充要条件。 这揭示了一大类支持守恒量的网络基序。 此外，我们识别出生成新运动常数的李对称。 我们的结果为Koopman框架在复杂网络上的非线性动力学提供了严格的理论应用。 显示英文摘要

摘要： The partial integrability of the Kuramoto model is often thought to be restricted to identically connected oscillators or groups thereof. Yet, the exact connectivity prerequisites for having constants of motion on more general graphs have remained elusive. Using spectral properties of the Koopman generator, we derive necessary and sufficient conditions for the existence of distinct constants of motion in the Kuramoto model with heterogeneous phase lags on any weighted, directed, signed graph. This reveals a broad class of network motifs that support conserved quantities. Furthermore, we identify Lie symmetries that generate new constants of motion. Our results provide a rigorous theoretical application of Koopman's framework to nonlinear dynamics on complex networks.