Cargando…

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 o...

Descripción completa

Detalles Bibliográficos
Autores principales: Willms, Allan R., Kitanov, Petko M., Langford, William F.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society Publishing 2017
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5627120/
https://www.ncbi.nlm.nih.gov/pubmed/28989780
http://dx.doi.org/10.1098/rsos.170777
_version_ 1783268661682765824
author Willms, Allan R.
Kitanov, Petko M.
Langford, William F.
author_facet Willms, Allan R.
Kitanov, Petko M.
Langford, William F.
author_sort Willms, Allan R.
collection PubMed
description 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.
format Online
Article
Text
id pubmed-5627120
institution National Center for Biotechnology Information
language English
publishDate 2017
publisher The Royal Society Publishing
record_format MEDLINE/PubMed
spelling pubmed-56271202017-10-08 Huygens’ clocks revisited Willms, Allan R. Kitanov, Petko M. Langford, William F. R Soc Open Sci Mathematics 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. The Royal Society Publishing 2017-09-06 /pmc/articles/PMC5627120/ /pubmed/28989780 http://dx.doi.org/10.1098/rsos.170777 Text en © 2017 The Authors. http://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Mathematics
Willms, Allan R.
Kitanov, Petko M.
Langford, William F.
Huygens’ clocks revisited
title Huygens’ clocks revisited
title_full Huygens’ clocks revisited
title_fullStr Huygens’ clocks revisited
title_full_unstemmed Huygens’ clocks revisited
title_short Huygens’ clocks revisited
title_sort huygens’ clocks revisited
topic Mathematics
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5627120/
https://www.ncbi.nlm.nih.gov/pubmed/28989780
http://dx.doi.org/10.1098/rsos.170777
work_keys_str_mv AT willmsallanr huygensclocksrevisited
AT kitanovpetkom huygensclocksrevisited
AT langfordwilliamf huygensclocksrevisited