Synchronization of Two Nonidentical Clocks: What Huygens was Able to Observe?

  • Krzysztof Czolczynski
  • Przemysaw Perlikowski
  • Andrzej Stefanski
  • Tomasz Kapitaniak
Part of the Studies in Computational Intelligence book series (SCI, volume 459)


We consider the synchronization of two clocks which are accurate (show the same time) but have pendulums with differentmasses.We show that such clocks hanging on the same beam can show the almost complete (in-phase) and almost antiphase synchronizations. By almost complete and almost antiphase synchronization we defined the periodic motion of the pendulums in which the phase shift between the displacements of the pendulums is respectively close (but not equal) to 0° or 180° . We give evidence that almost antiphase synchronization was the phenomenon observed by Huygens in XVII century.We support our numerical studies by considering the energy balance in the system and showing how the energy is transferred between the pendulums via oscillating beam allowing the pendulums’ synchronization.


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  1. 1.
    Bennet, M., Schatz, M.F., Rockwood, H., Wiesenfeld, K.: Huygens’s clocks. Proc. Roy. Soc. London, A 458, 563–579 (2002)CrossRefGoogle Scholar
  2. 2.
    Blekham, I.I.: Synchronization in Science and Technology. ASME, New York (1988); Birch, T.: The history of The Royal Society of London for improving of natural knowledge, in which the most considerable of those papers communicated to the Society, which have hitherto not been published, are inserted in their proper order, as a supplement to the Philosophical Transactions, vol. 2, pp. 19, 21, 23–24. Johnson, London (1756) (reprint 1968)Google Scholar
  3. 3.
    Britten, F.J.: Britten’s old clocks and watches and their makers; a historical and descriptive account of the different styles of clocks and watches of the past in England and abroad containing a list of nearly fourteen thousand makers. Methuen, London (1973)Google Scholar
  4. 4.
    Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Clustering of Huygens’ Clocks. Prog. Theor. Phys. 122, 1027–1033 (2009a)MATHCrossRefGoogle Scholar
  5. 5.
    Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Clustering and Synchronization of Huygens’ Clocks. Physica A 388, 5013–5023 (2009b)CrossRefGoogle Scholar
  6. 6.
    Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Huygens’ odd sympathy experiment revisited. Int. J. Bifur. Chaos 21 (2011a)Google Scholar
  7. 7.
    Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Why two clocks synchronize: Energy balance of the synchronized clocks. Chaos 21, 023129 (2011b)Google Scholar
  8. 8.
    Dilao, R.: Antiphase and in-phase synchronization of nonlinear oscillators: The Huygens’s clocks system. Chaos 19, 023118 (2009)Google Scholar
  9. 9.
    Fradkov, A.L., Andrievsky, B.: Synchronization and phase relations in the motion of twopendulum system. Int. J. Non-linear Mech. 42, 895 (2007)CrossRefGoogle Scholar
  10. 10.
    Huygens, C.: Letter to de Sluse. In: Oeuveres Completes de Christian Huygens (letters; no. 1333 of 24 February 1665, no. 1335 of 26 February 1665, no. 1345 of 6 March 1665). Societe Hollandaise Des Sciences, Martinus Nijhoff, La Haye (1893)Google Scholar
  11. 11.
    Huygens, C.: Instructions concerning the use of pendulum-watches for finding the longitude at sea. Phil. Trans. R. Soc. Lond. 4, 937 (1669)Google Scholar
  12. 12.
    Golubitsky, M., Stewart, I., Buono, P.L., Collins, J.J.: Symmetry in locomotor central pattern generators and animal gaits. Nature 401, 693–695 (1999)CrossRefGoogle Scholar
  13. 13.
    Kanunnikov, A.Y., Lamper, R.E.: Synchronization of pendulum clocks suspended on an elastic beam. J. Appl. Mech. & Theor. Phys. 44, 748–752 (2003)CrossRefGoogle Scholar
  14. 14.
    Kumon, M., Washizaki, R., Sato, J., Mizumoto, R.K.I., Iwai, Z.: Controlled synchronization of two 1-DOF coupled oscillators. In: Proceedings of the 15th IFAC World Congress, Barcelona (2002)Google Scholar
  15. 15.
    Lepschy, A.M., Mian, G.A., Viaro, U.: Feedback control in ancient water and mechanical clocks. IEEE Trans. Education 35, 3–10 (1993)CrossRefGoogle Scholar
  16. 16.
    Moon, F.C., Stiefel, P.D.: Coexisting chaotic and periodic dynamics in clock escapements. Phil. Trans. R. Soc. A 364, 2539 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Pantaleone, J.: Synchronization of metronomes. Am. J. Phys. 70, 992 (2002)CrossRefGoogle Scholar
  18. 18.
    Pikovsky, A., Roesenblum, M., Kurths, J.: Synchronization: An Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001); Pogromsky, A.Y., Belykh, V.N., Nijmeijer, H.: Controlled synchronization of pendula. In: Proceedings of the 42nd IEEE Conference on Design and Control, Maui, Hawaii, pp. 4381–4385 (2003)Google Scholar
  19. 19.
    Roup, A.V., Bernstein, D.S., Nersesov, S.G., Haddad, W.S., Chellaboina, V.: Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincare maps. Int. J. Control 76, 1685–1698 (2003)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Rawlings, A.L.: The science of clocks and watches. Pitman, New York (1944)Google Scholar
  21. 21.
    Senator, M.: Synchronization of two coupled escapement-driven pendulum clocks. Journal Sound and Vibration 291, 566–603 (2006)CrossRefGoogle Scholar
  22. 22.
    Strogatz, S.H., Stewart, I.: Coupled oscillators and biological synchronization. Scientific American 269(6), 102–109 (1993)CrossRefGoogle Scholar
  23. 23.
    Ulrichs, H., Mann, A., Parlitz, U.: Synchronization and chaotic dynamics of coupled mechanical metronomes. Chaos 19, 043120 (2009)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Krzysztof Czolczynski
    • 1
  • Przemysaw Perlikowski
    • 1
  • Andrzej Stefanski
    • 1
  • Tomasz Kapitaniak
    • 1
  1. 1.Division of DynamicsTechnical University of LodzLodzPoland

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