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Network of Coupled Oscillators for Precision Timing

  • Pietro-Luciano BuonoEmail author
  • Bernard Chan
  • Jocirei Ferreira
  • Patrick Longhini
  • Antonio Palacios
  • Steven Reeves
  • Visarath In
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 6)

Abstract

Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. It was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation. Satellite-based global positioning and navigation systems such as the GPS use the same principle. However, even the most sophisticated satellite navigation equipment cannot operate in every environment. In response to this need, we present a computational and analytical study of a network based model of a high-precision, inexpensive, Coupled Oscillator System and Timing device. Preliminary results from computer simulations seem to indicate that timing errors decrease as 1 / N when N crystals are coupled as oppose to \(1{/}\sqrt{N}\) for an uncoupled assemble. This manuscript is aimed, however, at providing a complete classification of the various patterns of collective behavior that are created, mainly, through symmetry-breaking bifurcations. The results should provide guidelines for follow-up simulations, design and fabrication tasks.

Keywords

Periodic Solution Irreducible Representation Hopf Bifurcation Hopf Bifurcation Point Crystal Oscillator 
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.

Notes

Acknowledgements

Visarath In, Antonio Palacios, and Pietro-Luciano Buono are conducting (as part of on-going work) all the theoretical calculations on the generic nonlinear system, as well as on specific applications. Antonio Palacios was supported by ASEE ONR Summer Faculty.

References

  1. 1.
    D.W. Allan. The science of timekeeping. Technical Report 1289, Hewlett Packard, (1997)Google Scholar
  2. 2.
    E. Doedel, X. Wang, Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations Applied Mathematics Report, California Institute of Technology (1994)Google Scholar
  3. 3.
    A.K. Poddar, U.L. Rohde, in Crystal Oscillators, Wiley Encyclopedia and Electronics Engineering (2012), pp. 1–38Google Scholar
  4. 4.
    J. Wang, R. Wu, J. Du, T. Ma, D. Huang, W. Yan. The nonlinear thickness-shear ovibrations of quartz crystal plates under a strong electric field, in IEEE International Ultrasonics Symposium Proceedings, vol. 10.1109 (IEEE, 2011), pp. 320–323.Google Scholar
  5. 5.
    M. Golubitsky, I.N. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory Vol. II, vol. 69 (Springer, New York, 1988)Google Scholar
  6. 6.
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems (Springer, New York, 1990)CrossRefzbMATHGoogle Scholar
  7. 7.
    V. In, A. Palacios, A. Bulsara, P. Longhini, A. Kho, J. Neff, S. Baglio, B. Ando, Complex behavior in driven unidirectionally coupled overdamped duffing elements. Phys. Rev. E, 73(6):066121 (2006)Google Scholar
  8. 8.
    G. Sebald, H. Kuwano, D. Guyomar, B. Ducharne, Simulation of a duffing oscillator for broadband piezoelectric energy harvesting. Smart Mater. Struct. 20, 075022 (2011)CrossRefGoogle Scholar
  9. 9.
    E.V. Appleton, B. van der Pol, On a type of oscillation-hysteresis in a simple triode generator. Lond. Edinburgh Dublin Philos. Mag. J. Sci. Ser. 6(43), 177–193 (1922)CrossRefGoogle Scholar
  10. 10.
    B. Van der Pol, On “relaxation-oscillations”. Lond. Edinburgh Dublin Philos. Mag. J. Sci. Ser. 7(2), 978–992 (1926)Google Scholar
  11. 11.
    P. Holmes, D.R. Rand, Bifurcation of the forced van der pol oscillator. Quart. Appl. Math. 35, 495–509 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    B. van der Pol, Forced oscillations in a circuit with non-linear resistance (reception with reactive triode). Lond. Edinburgh Dublin Philos. Mag. J. Sci. Ser. 7(3), 65–80 (1927)Google Scholar
  13. 13.
    B. van der Pol, J. van der Mark, Frequency demultiplication. Nature 120, 363–364 (1927)CrossRefGoogle Scholar
  14. 14.
    V. Apostolyuk, F. Tay, Dynamics of micromechanical coriolis vibratory gyroscopes. Sensor Lett. 2, 252–259 (2004)CrossRefGoogle Scholar
  15. 15.
    N. Davies. Ring of vibratory gyroscopes with coupling along the drive and sense axes. Master’s thesis, San Diego State University (2011)Google Scholar
  16. 16.
    A. Shkel. Type i and type ii micromachined vibratory gyroscopes, in Proceedings of IEEE/ION PLANS (San Diego, CA, 2006), pp. 586–593Google Scholar
  17. 17.
    H. Vu, A. Palacios, V. In, P. Longhini, J. Neff, Two-time scale analysis of a ring of coupled vibratory gyroscopes. Phys. Rev. E. 81, 031108 (2010)CrossRefGoogle Scholar
  18. 18.
    H. Vu. Ring of Vibratory Gyroscopes with Coupling along the Drive Axis. Ph.D. thesis, San Diego State University (2011)Google Scholar
  19. 19.
    S.P. Beeby, M.J. Tudor, N.M. White, Energy harvesting vibration sources for microsystems applications. Meas. Sci. Technol. 17, R175–R195 (2006)CrossRefGoogle Scholar
  20. 20.
    B.P. Mann, N.D. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319, 515–530 (2009)CrossRefGoogle Scholar
  21. 21.
    A. Matus-Vargas, H.G. Gonzalez-Hernandez, B. Chan, A. Palacios, P.-L. Buono, V. In, S. Naik, A. Phipps, P. Longhini, Dynamics, bifurcations and normal forms in arrays of magnetostrictive energy harvesters with all-to-all coupling. Int. J. Bifurc. Chaos 25(2), 1550026 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    M. Krupa, Bifurcations of relative equilibria. SIAM J. Math. Anal. 21(6), 1453–1486 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Pietro-Luciano Buono
    • 1
    Email author
  • Bernard Chan
    • 2
  • Jocirei Ferreira
    • 3
  • Patrick Longhini
    • 4
  • Antonio Palacios
    • 2
  • Steven Reeves
    • 2
  • Visarath In
    • 4
  1. 1.Faculty of ScienceUniversity of Ontario Institute of TechnologyOshawaCanada
  2. 2.Nonlinear Dynamical Systems Group, Department of MathematicsSan Diego State UniversitySan DiegoUSA
  3. 3.Institute of Exact and Earth Science-CUA, Federal University of Mato GrossoCuiabáBrazil
  4. 4.Space and Naval Warfare Systems Center, Code 2363San DiegoUSA

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