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.
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References
D.W. Allan. The science of timekeeping. Technical Report 1289, Hewlett Packard, (1997)
E. Doedel, X. Wang, Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations Applied Mathematics Report, California Institute of Technology (1994)
A.K. Poddar, U.L. Rohde, in Crystal Oscillators, Wiley Encyclopedia and Electronics Engineering (2012), pp. 1–38
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.
M. Golubitsky, I.N. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory Vol. II, vol. 69 (Springer, New York, 1988)
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems (Springer, New York, 1990)
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)
G. Sebald, H. Kuwano, D. Guyomar, B. Ducharne, Simulation of a duffing oscillator for broadband piezoelectric energy harvesting. Smart Mater. Struct. 20, 075022 (2011)
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)
B. Van der Pol, On “relaxation-oscillations”. Lond. Edinburgh Dublin Philos. Mag. J. Sci. Ser. 7(2), 978–992 (1926)
P. Holmes, D.R. Rand, Bifurcation of the forced van der pol oscillator. Quart. Appl. Math. 35, 495–509 (1978)
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)
B. van der Pol, J. van der Mark, Frequency demultiplication. Nature 120, 363–364 (1927)
V. Apostolyuk, F. Tay, Dynamics of micromechanical coriolis vibratory gyroscopes. Sensor Lett. 2, 252–259 (2004)
N. Davies. Ring of vibratory gyroscopes with coupling along the drive and sense axes. Master’s thesis, San Diego State University (2011)
A. Shkel. Type i and type ii micromachined vibratory gyroscopes, in Proceedings of IEEE/ION PLANS (San Diego, CA, 2006), pp. 586–593
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)
H. Vu. Ring of Vibratory Gyroscopes with Coupling along the Drive Axis. Ph.D. thesis, San Diego State University (2011)
S.P. Beeby, M.J. Tudor, N.M. White, Energy harvesting vibration sources for microsystems applications. Meas. Sci. Technol. 17, R175–R195 (2006)
B.P. Mann, N.D. Sims, Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound Vib. 319, 515–530 (2009)
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)
M. Krupa, Bifurcations of relative equilibria. SIAM J. Math. Anal. 21(6), 1453–1486 (1990)
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.
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Buono, PL. et al. (2017). Network of Coupled Oscillators for Precision Timing. In: In, V., Longhini, P., Palacios, A. (eds) Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016). ICAND 2016. Lecture Notes in Networks and Systems, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-52621-8_7
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DOI: https://doi.org/10.1007/978-3-319-52621-8_7
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