Data from: Huygens' clocks revisited

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.

1665年，惠更斯（Huygens）观察到，两台通过重梁弱耦合的完全相同的摆钟，可快速实现同步，拥有一致的周期与振幅，但两摆的摆动方向完全相反。这一现象如今被称为反相同步（anti-phase synchronization）。本文针对包括惠更斯摆钟在内的一大类耦合全同振子的行为展开分析，所采用的方法为等变分支理论（equivariant bifurcation theory）。文中推导了此类系统的等变标准型（equivariant normal form），并对所有可能存在的解进行了表征。本文明确给出了物理系统参数到标准型参数的转换关系，并将其分别应用于惠更斯摆钟的物理参数，以及近期相关研究中的实验参数。研究表明，惠更斯的物理系统仅能呈现反相运动，这解释了为何惠更斯仅观测到该类现象。与之形成对比的是，部分近期研究者在各自的实验系统中观测到了同相（in-phase）或其他更为复杂的运动形式。本文阐明了这类系统具备哪些物理特性，使得其他类型的稳定解得以存在。本分析不仅在统一框架下解释了此前已观测到的各类解，还引入了其他作者尚未分类的行为，例如同步环面呼吸子（synchronized toroidal breather）与混沌环面呼吸子（chaotic toroidal breather）。