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Synchronization of asymmetrically coupled systems

  • J. Pena RamirezEmail author
  • I. Ruiz Ramos
  • J. Alvarez
Original Paper
  • 54 Downloads

Abstract

In this paper, the dynamics of two mechanical oscillators interacting through an asymmetric coupling is considered. In the uncontrolled system, the asymmetry in the coupling prevents the oscillators from reaching synchronization. Therefore, a state feedback controller is designed and applied to the coupling system in order to remove the asymmetry. The resulting closed-loop dynamics is analyzed by using the Poincaré method, and analytic conditions for the existence of stable synchronous solutions are derived. It is demonstrated that two synchronous solutions coexist in the closed-loop system, namely in-phase and anti-phase synchronization. Additionally, analytic expressions for the amplitude, frequency, and phase of these solutions—which are not determined by a reference signal, but rather by the intrinsic properties of the coupled systems—are provided and a characteristic equation for determining the stability of the synchronous solution is given. Finally, the obtained theoretical results are illustrated by numerical simulations, including a numerical study on the robustness of the synchronous solution.

Keywords

Synchronization Asymmetric coupling Poincaré method of perturbation 

Notes

Acknowledgements

This research has been partially supported by Mexican Council for Science and Technology, CONACYT, under Grant ‘Catedra CONACYT No. 888’.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Strogatz, S.: Sync: The Emerging Science of Spontaneous Order. Hyperon, New York (2003)Google Scholar
  2. 2.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  3. 3.
    Rodriguez-Angeles, A., Nijmeijer, H.: Mutual synchronization of robots via estimated state feedback: a cooperative approach. IEEE Trans. Control Syst. Technol. 12, 542–554 (2004)CrossRefGoogle Scholar
  4. 4.
    Rohden, M., Sorge, A., Timme, M., Witthaut, D.: Self-organized synchronization in decentralized power grids. Phys. Rev. Lett. 109, 064101 (2012)CrossRefGoogle Scholar
  5. 5.
    Kocarev, L., Parlitz, U.: General approach for chaotic synchronization with applications to communication. Phys. Rev. Lett. 74, 5028 (1995)CrossRefGoogle Scholar
  6. 6.
    Blekhman, I.I., Fradkov, A.L., Nijmeijer, H., Pogromsky, Y.A.: On self-synchronization and controlled synchronization. Syst. Control Lett. 31, 299–305 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Huygens, C.: Oeuvres complètes de Christiaan Huygens. In: Nijhoff, M. (ed.) Correspondance, vol. 5, pp. 1664–1665. La Societe Hollandaise des Sciences, The Hague (1893)Google Scholar
  8. 8.
    Belykh, V.N., Pankratova, E.V.: Shilnikov chaos in oscillators with Huygens coupling. Int. J. Bifurc. Chaos 24, 1440007 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pena Ramirez, J., Olvera, L.A., Nijmeijer, H., Alvarez, J.: The sympathy of two pendulum clocks: beyond Huygens’ observations. Sci. Rep. 6, 23580 (2016)CrossRefGoogle Scholar
  10. 10.
    Willms, A.R., Kitanov, P.M., Langford, W.F.: Huygens’ clocks revisited. R. Soc. Open Sci. 4, 1–33 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kapitaniak, M., Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Synchronization of clocks. Phys. Rep. 517, 1–70 (2012)CrossRefGoogle Scholar
  12. 12.
    Czolczynski, K., Perlikowski, P., Stefanski, A., Kapitaniak, T.: Huygens’ odd sympathy experiment revisited. Int. J. Bifurc. Chaos 21, 2047–2056 (2011)CrossRefGoogle Scholar
  13. 13.
    Wiesenfeld, K., Borrero-Echeverry, D.: Huygens (and others) revisited. Chaos Interdiscip. J. Nonlinear Sci. 21, 047515 (2011)CrossRefGoogle Scholar
  14. 14.
    Dilao, R.: Antiphase and in-phase synchronization of nonlinear oscillators: the Huygens’s clocks system. Chaos Interdiscip. J. Nonlinear Sci. 19, 023118 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Xin, X., Muraoka, Y., Hara, S.: Analysis of synchronization of n metronomes on a cart via describing function method: new results beyond two metronomes. In: Proceedings of the 2016 American Control Conference (ACC), July 6–8, Boston, MA, pp. 6604–6609 (2016)Google Scholar
  16. 16.
    Strogatz, S., Abrams, D.M., McRobie, A., Eckhardt, B., Ott, E.: Crowd synchrony on the Millennium Bridge. Nature 438, 43–44 (2005)CrossRefGoogle Scholar
  17. 17.
    Belykh, I., Jeter, R., Belykh, V.: Foot force models of crowd dynamics on a wobbly bridge. Sci. Adv. 3, 1–12 (2017)CrossRefGoogle Scholar
  18. 18.
    Teoh, C.S., Davis, L.E.: A coupled pendula system as an analogy to coupled transmission lines. IEEE Trans. Educ. 39, 548–557 (1996)CrossRefGoogle Scholar
  19. 19.
    Liu, Z., Tian, Y., Zhou, C.: Controlled anti-phase synchronization of passive gait. In: Proceedings of the 2006 IEEE International Conference on Robotics and Biomimetics, Kunming, China (2016)Google Scholar
  20. 20.
    Czoczyski, K., Perlikowski, P., Stefaski, A., Kapitaniak, T.: Why two clocks synchronize: energy balance of the synchronized clocks. Chaos Interdiscip. J. Nonlinear Sci. 21, 023129 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pena Ramirez, J., Nijmeijer, H.: The Poincaré method: a powerful tool for analyzing synchronization of coupled oscillators. Indagationes Mathematicae 48, 1–20 (2016)zbMATHGoogle Scholar
  22. 22.
    Blekhman, I.I.: Synchronization in Science and Technology. ASME Press, New York (1988)Google Scholar
  23. 23.
    Bittanti, S., Colaneri, P.: Periodic Systems: Filtering and Control. Springer, London (2009)zbMATHGoogle Scholar
  24. 24.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)zbMATHGoogle Scholar
  25. 25.
    Gritli, H., Khraief, N., Chemori, A., Belghith, S.: Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC. Nonlinear Dyn. 89, 2195–2226 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jovanovic, V., Koshkin, S.: Synchronization of Huygens’ clocks and the Poincaré method. J. Sound Vibr. 331, 2887–2900 (2012)CrossRefGoogle Scholar
  27. 27.
    Marquez, S., Alvarez, M., Plaza, J.A., Villanueva, L.G., Dominguez, C., Lechuga, L.M.: Asymmetrically coupled resonators for mass sensing. Appl. Phys. Lett. 11, 113101 (2017)CrossRefGoogle Scholar
  28. 28.
    Noh, J.D.: Assymetrically coupled directed percolation systems. Phys. Rev. Lett. 94, 145702 (2005)CrossRefGoogle Scholar
  29. 29.
    Hirata Salazar, G.: Design and implementation of control algorithms for attenuation of seismic vibrations in a class of structures. Ph.D. Thesis, CICESE (2016)Google Scholar
  30. 30.
    van der Pol, B.: On relaxation-oscillations. Philos. Mag. Ser. 7 2, 978–992 (1926)CrossRefGoogle Scholar
  31. 31.
    Jovanovic, V., Koshkin, S.: Synchronization of Huygens’ clocks and the Poincaré method. J. Sound Vibr. 331, 2887–2900 (2012)CrossRefGoogle Scholar
  32. 32.
    Ruiz Ramos, I., Pena Ramirez, J., Alvarez, J.: Synchronous behavior in asymmetrically coupled pendulums. In: Proceedings of the 2017 International Symposium on Nonlinear Theory and its Applications, December 4–7, Cancun, Mexico (2017)Google Scholar
  33. 33.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76, 924–941 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Choukchou-Braham, A., Cherki, B., Djemai, M., Busawon, K.: Analysis and Control of Underactuated Mechanical Systems. Springer, Basel (2014)CrossRefGoogle Scholar
  35. 35.
    Pena Ramirez, J., Aihara, K., Fey, R.H.B., Nijmeijer, H.: Further understanding of Huygens’ coupled clocks: the effect of stiffness. Phys. D Nonlinear Phenom. 270, 11–19 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.CONACYT-CICESE, Center for Scientific Research and Higher Education at EnsenadaEnsenadaMexico
  2. 2.Tijuana University of TechnologyTijuanaMexico
  3. 3.Center for Scientific Research and Higher Education at EnsenadaEnsenadaMexico

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