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On the Problem of Synchronization of Identical Dynamical Systems: The Huygens’s Clocks

  • Rui Dilão
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)

Abstract

In 1665, Christiaan Huygens reported the observation of the synchronization of two pendulum clocks hanged on the wall of his workshop. After synchronization, the clocks swung exactly in the same frequency and 180° out of phase—anti-phase synchronization. Here, we propose and analyze a new interaction mechanism between oscillators leading to exact anti-phase and in-phase synchronization of pendulum clocks, and we determine a sufficient condition for the existence of an exact anti-phase synchronous state. We show that exact anti-phase and in-phase synchronous states can coexist in phase space, and the periods of the synchronous states are different from the eigenperiods of the individual oscillators. We analyze the robustness of the system when the parameters of the individual pendulum clocks are varied, and we show numerically that exact anti-phase and in-phase synchronous states exist in systems of coupled oscillators with different parameters.

Keywords

Phase Space Nonlinear Oscillator Couple Oscillator Attachment Point Positive Real Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgments

I would like to thank the support of the Ettore Majorana Center for Scientific Culture and the hospitality of the organizers of the conference “Variational Analysis and Aerospace Engineering,” dedicated to Prof. Angelo Miele on his 85th birthday. This work has been partially supported by a Fundação para a Ciência e a Tecnologia (FCT) pluriannual funding grant to the NonLinear Dynamics Group (GDNL).

References

  1. 1.
    A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University PressCambridge, 2001.MATHGoogle Scholar
  2. 2.
    M. Bennett, M.F. Schatz, H. Rockwood, K. Weisenfeld, Huygensís clocks, Proc. R. Soc. Lond. A, 458 (2002) 563–579.MATHCrossRefGoogle Scholar
  3. 3.
    M. Kumon, R. Washizaki, J. Sato, R.K.I. Mizumoto, Z. Iwai, Controlled synchronization of two 1-DOF coupled oscillators, in: Proc. of the 15th Triennial World Congress of IFAC, Barcelona, 2002.Google Scholar
  4. 4.
    A.L. Fradkov, B. Andrievsky, Synchronization and phase relations in the motion of two-pendulum system, Int. J. Non-Linear Mechanics, 42 (2007) 895–901.CrossRefGoogle Scholar
  5. 5.
    B. Andrievsky, A. Fradkov, S. Gavrilov, V. Konoplev, Modeling and Synchronization of the Mechatronic Vibrational Stand, in Proc. 2nd Intern. Conf. “Physics and Control”, IEEE, St. Petersburg, 2005, pp.165–168.Google Scholar
  6. 6.
    I.I. Blekhman, Yu.A. Bortsov, A.A. Burmistrov, A.L. Fradkov, S.V. Gavrilov, O.A. Kononov, B.P. Lavrov, V.M. Shestakov, P.V. Sokolov, O.P. Tomchina, Computer-controlled vibrational setup for education and research, in Proc. 14th IFAC World Congress, vol. M, 1999, pp. 193–197.Google Scholar
  7. 7.
    J. Pantaleone, Synchronization of metronomes, Am. J. Phys., 70 (2002) 992–1000.CrossRefGoogle Scholar
  8. 8.
    Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.MATHGoogle Scholar
  9. 9.
    D. J. Kortweg, Les Horloges Sympathiques de Huygens, Archives Neerlandaises, Series II , Tome XI, pp. 273–295, Martinus Nijhoff, The Hague, 1906.Google Scholar
  10. 10.
    S.H. Strogatz, I. Stewart, Coupled oscillators and biological synchronization, Scient. Am., 269, n 6 (1993) 68–75.Google Scholar
  11. 11.
    A.A. Andronov, A.A. Witt, S.E. Khaikin, Theory of Oscillators, Pergamon, Oxford, 1966.MATHGoogle Scholar
  12. 12.
    R. Dilão, Anti-phase and in-phase synchronization of nonlinear oscillators: The Huygens’s clocks system, Chaos 19,023118 (2009), DOI: 10.1063/1.3139177.Google Scholar
  13. 13.
    R. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982.MATHGoogle Scholar
  14. 14.
    A. Hurwitz, On the Conditions under which an Equation has only Roots with Negative Real Parts, Mathematische Annalen, 46, (1895) 273–284. Reprinted in “Selected Papers on Mathematical Trends in Control Theory”, R. Bellman, R. Kalaba (Ed.), Dover, New York, 1964.Google Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.NonLinear Dynamics GroupInstituto Superior Técnico Av. Rovisco PaisLisbonPortugal

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