Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Variational Analysis and Aerospace Engineering

Part of the book series: Springer Optimization and Its Applications ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University PressCambridge, 2001.

    MATH  Google Scholar 

  2. M. Bennett, M.F. Schatz, H. Rockwood, K. Weisenfeld, Huygensís clocks, Proc. R. Soc. Lond. A, 458 (2002) 563–579.

    Article  MATH  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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. 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. J. Pantaleone, Synchronization of metronomes, Am. J. Phys., 70 (2002) 992–1000.

    Article  Google Scholar 

  8. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.

    MATH  Google Scholar 

  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. S.H. Strogatz, I. Stewart, Coupled oscillators and biological synchronization, Scient. Am., 269, n 6 (1993) 68–75.

    Google Scholar 

  11. A.A. Andronov, A.A. Witt, S.E. Khaikin, Theory of Oscillators, Pergamon, Oxford, 1966.

    MATH  Google Scholar 

  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. R. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982.

    MATH  Google Scholar 

  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 

Download references

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).

Author information

Authors and Affiliations

  1. NonLinear Dynamics Group, Instituto Superior Técnico Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Rui Dilão

Authors
  1. Rui Dilão
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui Dilão .

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag New York

About this paper

Cite this paper

Dilão, R. (2009). On the Problem of Synchronization of Identical Dynamical Systems: The Huygens’s Clocks. In: Variational Analysis and Aerospace Engineering. Springer Optimization and Its Applications, vol 33. Springer, New York, NY. https://doi.org/10.1007/978-0-387-95857-6_10

Download citation

Publish with us

Policies and ethics

Search

Navigation