Abstract
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.
Highlights
Christiaan Huygens was a manufacturer of accurate pendulum clocks in the seventeenth century
The paradox is resolved by the Equivariant Hopf Bifurcation Theorem, where it is shown that the anti-phase oscillations have a spatio-temporal symmetry, which consists of the above permutation symmetry in space combined with a half-period phase shift in time
The analysis presented here provides a detailed description of the behaviour of a system with two weakly coupled identical oscillators near the onset of oscillations from the rest state due to a double Hopf bifurcation
Summary
Christiaan Huygens was a manufacturer of accurate pendulum clocks in the seventeenth century. One of the most complete studies of Huygens’ clocks to date is by Bennett et al [3], who used mathematical modelling and numerical computation as well as physical experiments to explore the synchronization of pendulum clocks. Many new studies of Huygens’ clocks have appeared since the publication of Bennett et al [3] Most of these studies report anti-phase synchronization, as observed by Huygens himself. Previous authors have modelled Huygens’ experiment with systems of differential equations that implicitly preserve this symmetry property, but they have not explicitly made use of the symmetry in their analysis. It is explained why Huygens saw only anti-phase solutions. In-phase solutions and other behaviour reported by other authors is explained by mapping their physical systems into the same normal form. New types of behaviour and solutions are studied, such as a period-doubling sequence leading from a synchronized breather to a chaotic breather
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