Computational and experimental works reveal that the coupling of similar crystal oscillators leads to a variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns, which have the potential for developing precision timing devices through phase drift reduction. Among all observed patterns, the standard traveling wave, in which consecutive crystals oscillate out of phase by [Formula: see text], where [Formula: see text] is the network size, leads to optimal phase drift error that scales down as [Formula: see text] as opposed to [Formula: see text] for an uncoupled ensemble. In this manuscript, we provide an analytical proof of the scaling laws, for uncoupled and coupled symmetric networks, and show that [Formula: see text] is the fundamental limit of phase-error reduction that one can obtain with a symmetric network of nonlinear oscillators of any type, not just crystals.

中文翻译：

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用于精密定时的耦合非线性振荡器的相位漂移比例律

计算和实验工作表明，相似晶体振荡器的耦合导致了多种集体模式，主要是各种形式的离散旋转波和同步模式，它们具有通过减少相位漂移来开发精密计时装置的潜力。在所有观察到的模式中，标准行波，其中连续晶体按 [公式：见文本] 异相振荡，其中 [公式：见文本] 是网络大小，导致最佳相位漂移误差按比例缩小为 [公式: see text] 而不是 [Formula: see text] 的解耦合奏。在这份手稿中，我们为非耦合和耦合对称网络提供了比例定律的分析证明，并表明 [公式：