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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...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
The Royal Society Publishing
2017
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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 |
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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 |