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...
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| Format: | Article |
| Language: | English |
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The Royal Society
2017-01-01
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| Series: | Royal Society Open Science |
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| Online Access: | https://royalsocietypublishing.org/doi/pdf/10.1098/rsos.170777 |
| _version_ | 1818980198785220608 |
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| author | Allan R. Willms Petko M. Kitanov William F. Langford |
| author_facet | Allan R. Willms Petko M. Kitanov William F. Langford |
| author_sort | Allan R. Willms |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-20T17:11:37Z |
| format | Article |
| id | doaj.art-648836d3c76947a6887a0af9e27a4b3a |
| institution | Directory Open Access Journal |
| issn | 2054-5703 |
| language | English |
| last_indexed | 2024-12-20T17:11:37Z |
| publishDate | 2017-01-01 |
| publisher | The Royal Society |
| record_format | Article |
| series | Royal Society Open Science |
| spelling | doaj.art-648836d3c76947a6887a0af9e27a4b3a2022-12-21T19:32:07ZengThe Royal SocietyRoyal Society Open Science2054-57032017-01-014910.1098/rsos.170777170777Huygens’ clocks revisitedAllan R. WillmsPetko M. KitanovWilliam F. LangfordIn 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.https://royalsocietypublishing.org/doi/pdf/10.1098/rsos.170777huygens' clocksdouble hopf bifurcationequivariant bifurcation theorysynchronizationnonlinear resonancecoupled identical oscillators |
| spellingShingle | Allan R. Willms Petko M. Kitanov William F. Langford Huygens’ clocks revisited Royal Society Open Science huygens' clocks double hopf bifurcation equivariant bifurcation theory synchronization nonlinear resonance coupled identical oscillators |
| 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 | huygens' clocks double hopf bifurcation equivariant bifurcation theory synchronization nonlinear resonance coupled identical oscillators |
| url | https://royalsocietypublishing.org/doi/pdf/10.1098/rsos.170777 |
| work_keys_str_mv | AT allanrwillms huygensclocksrevisited AT petkomkitanov huygensclocksrevisited AT williamflangford huygensclocksrevisited |