发表机构
Chung-Ang University; Sungkyunkwan University; Korea Institute for Advanced Study（中央大学; 成均馆大学; 韩国高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究探讨高维Kuramoto模型中的序参量选择，发现球面上平衡态为唯一稳定候选，而圆上则因有限N效应产生两簇锁定态，形成连续与量子化选择机制。
AI 中文摘要
我们研究了一个高维Kuramoto模型，该模型包含吸引性的两两耦合和排斥性的高阶效应。尽管两两相互作用有利于同步，但高阶相互作用阻止了完全同步，并选择了与平衡态相对应的中间相干水平。对于维度至少为二的单位球面，我们提供了平衡态的显式吸引域。我们还对非零均值平衡态进行了分类，并证明所有非平衡态都是线性不稳定的。这表明在高维中，平衡态是唯一自然的稳定候选态。相比之下，在圆上，相同的模型简化为一个高阶Kuramoto型模型，该模型展现出一种定性不同的选择机制，该机制从根本上依赖于维度。在这种情况下，有限$N$时的群体不平衡阻止了精确的静止平衡，而是产生了共同的角漂移，从而形成相位锁定态。我们为两簇锁定态构建了吸引域，其中相位分为两组，并确定了相应的有限$N$序参量选择值。这种现象被称为连续与量子化的序参量选择。我们进一步通过根选择机制证明了为什么在大多数模拟中通常观察到这种两簇锁定态。最后，我们证明具有大群体不平衡的两簇锁定态是线性不稳定的，这解释了为什么只有某些锁定态在动态上是稳健的。
英文摘要
We study a high-dimensional Kuramoto model with attractive pairwise coupling and a repulsive higher-order effect. Although the pairwise interaction favors synchronization, the higher-order interaction prevents complete synchronization and selects an intermediate level of coherence corresponding to the balanced equilibria. For the unit sphere of dimension at least two, we provide an explicit basin of attraction for the balanced equilibria. We also classify nonzero-mean equilibria and show that all non-balanced equilibria are linearly unstable. This indicates that balanced states are the only natural stable candidates in the higher dimensions. In contrast, on the circle, the same model reduces to a higher-order Kuramoto-type model which exhibits a qualitatively different selection mechanism that depends fundamentally on the dimension. In this case, population imbalance at finite $N$ prevents exact stationary balance and instead generates a common angular drift producing phase-locked states. We construct basins of attraction for two-cluster locked states, in which the phases split into two groups, and identify the corresponding finite-$N$ selected values of the order parameters. This phenomenon is called continuous-versus-quantized order-parameter selection. We further demonstrate, through a root-selection mechanism, why such two-cluster locked states are typically observed in most simulations. Finally, we show that two-cluster locked states with a large population imbalance are linearly unstable, which explains why only certain locked states are dynamically robust.
