The Kuramoto model serves as a paradigm to study the phenomenon of spontaneous collective synchronization. We study here a nontrivial generalization of the Kuramoto model by including an interaction that breaks explicitly the rotational symmetry of the model. In an inertial frame (e.g., the laboratory frame), the Kuramoto model does not allow for a stationary state, that is, a state with time-independent value of the so-called Kuramoto (complex) synchronization order parameter $z\equiv r{e}^{i\psi }$. Note that a time-independent $z$ implies $r$ and $\psi$ are both time independent, with the latter fact corresponding to a state in which $\psi$ rotates at zero frequency (no rotation). In this backdrop, we ask: Does the introduction of the symmetry-breaking term suffice to allow for the existence of a stationary state in the laboratory frame? Compared to the original model, we reveal a rather rich phase diagram of the resulting model, with the existence of both stationary and standing wave phases. While in the former the synchronization order parameter $r$ has a long-time value that is time independent, one has in the latter an oscillatory behavior of the order parameter as a function of time that nevertheless yields a nonzero and time-independent time average. Our results are based on numerical integration of the dynamical equations as well as an exact analysis of the dynamics by invoking the so-called Ott-Antonsen ansatz that allows to derive a reduced set of time-evolution equations for the order parameter.

中文翻译：

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在存在其他破坏旋转对称性的相互作用的情况下，仓本模型。

仓本模型是研究自发集​​体同步现象的范例。我们在这里研究了Kuramoto模型的非平凡归纳，方法是包括一个明确打破模型旋转对称性的相互作用。在惯性坐标系（例如实验室坐标系）中，仓本模型不允许稳态，即状态与时间无关的所谓仓本（复杂）同步阶数参数的状态$ž\equiv \mathrm{[R}{Ë}^{\mathrm{一世}\psi }$。请注意，与时间无关$ž$ 暗示 $\mathrm{[R}$ 和 $\psi$ 都是时间独立的，后一个事实对应于一个状态 $\psi$以零频率旋转（不旋转）。在这种背景下，我们问：引入对称破缺术语是否足以使实验室框架中存在稳态？与原始模型相比，我们揭示了所得模型的相当丰富的相位图，同时存在驻波和驻波两个相位。而在前者中，同步顺序参数$\mathrm{[R}$具有一个与时间无关的长时间值，后者在时间上具有作为时间函数的阶数参数的振荡行为，但仍会产生非零且与时间无关的时间平均值。我们的结果基于动力学方程的数值积分以及对动力学的精确分析，方法是调用所谓的Ott-Antonsen ansatz，该动力学允许导出阶数参数的一组简化的时间演化方程。