Using the main results of the Kuramoto theory of globally coupled phase oscillators combined with methods from probability and generalized function theory in a geometric analysis, we extend Kuramoto's results and obtain a mathematical description of the instantaneous frequency (phase-velocity) distribution. Our result is validated against numerical simulations, and we illustrate it in cases in which the natural frequencies have normal and Beta distributions. In both cases, we vary the coupling strength and compare systematically the distribution of time-averaged frequencies (a known result of Kuramoto theory) to that of instantaneous frequencies, focusing on their qualitative differences near the synchronized frequency and in their tails. For a class of natural frequency distributions with power-law tails, which includes the Cauchy-Lorentz distribution, we analyze the tails of the instantaneous frequency distribution by means of an asymptotic formula obtained from a power-series expansion.

中文翻译：

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仓本模型中的瞬时频率

利用Kuramoto全局耦合相位振荡器理论的主要结果，结合概率分析和广义函数理论的方法，在几何分析中扩展了Kuramoto的结果，并获得了瞬时频率（相速度）分布的数学描述。我们的结果通过数值模拟进行了验证，并且在自然频率具有正态分布和Beta分布的情况下进行了说明。在这两种情况下，我们都改变了耦合强度，并系统地比较了时均频率（仓本理论的已知结果）与瞬时频率的分布，着眼于它们在同步频率及其尾部的定性差异。对于一类具有幂律尾部的固有频率分布，