In modern times, satellite-based global positioning and navigation systems, such as the GPS, include precise time-keeping devices, e.g. atomic clocks, which are crucial for navigation and for a wide range of economic and industrial applications. However, precise timing might not be available when the environment renders satellite equipment inoperable. In response to this critical need, we have been carrying out, over the past three years, theory and preliminary experiments [Buono et al., 2018a; Buono et al., 2018b; Palacios et al., 2020], towards developing a novel and inexpensive precision timing device that can function independently of GPS availability. The fundamental idea is to exploit collective behavior generated by networks of coupled nonlinear oscillators. Common sense may suggest that synchronized oscillations may lead to higher accuracy. Previous works show, however, that it is not synchronization but rather, traveling wave patterns, in which consecutive oscillators are out of phase by a constant amount, that can better reduce the negative effects of noise and material imperfections which cause phase drift. In this work we advance the state-of-art in the network-based concept by studying, mainly computationally, collective behavior in networks of Colpitts oscillators. These type of oscillators are chosen because they offer a wide range of advantages (such as the ability to tune up the oscillations over a broad frequency range). The results highlight the regions of parameter space, including coupling strength, where traveling wave patterns have the largest basins of attraction and the ability to reduce phase drift by a 1/N scaling law, where N is the number of oscillators in the network. The results should also provide guidelines for follow-up design and fabrication tasks of a network-based technology for precision timing.

中文翻译：

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耦合 Colpitts 振荡器网络中的相位漂移

在现代，基于卫星的全球定位和导航系统，例如 GPS，包括精确的计时设备，例如原子钟，这对于导航以及广泛的经济和工业应用至关重要。但是，当环境使卫星设备无法运行时，可能无法获得精确的时间。针对这一关键需求，我们在过去三年中一直在进行理论和初步实验 [Buono等。, 2018a; 博诺等。, 2018b; 帕拉西奥斯等。, 2020]，致力于开发一种新颖且廉价的精密计时设备，该设备可以独立于 GPS 可用性运行。基本思想是利用耦合非线性振荡器网络产生的集体行为。常识可能表明同步振荡可能会导致更高的精度。然而，以前的工作表明，不是同步而是行波模式，其中连续的振荡器异相恒定量，可以更好地减少噪声和材料缺陷的负面影响，从而导致相位漂移。在这项工作中，我们通过主要通过计算研究 Colpitts 振荡器网络中的集体行为来推进基于网络的概念的最新技术。选择这些类型的振荡器是因为它们具有广泛的优势（例如能够在较宽的频率范围内调整振荡）。结果突出了参数空间的区域，包括耦合强度，其中行波模式具有最大的吸引力盆地，并且能够通过 a1/ñ比例律，其中ñ是网络中振荡器的数量。结果还应为基于网络的精确定时技术的后续设计和制造任务提供指导。