On phase-locking of oscillators with delay coupling

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    Abstract

    We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.
    Original languageUndefined
    Title of host publicationProceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)
    EditorsD. Bernardini, G. Rega, F. Romeo
    PublisherEUROMECH
    PagesMS-11
    Number of pages2
    ISBN (Print)978-88-906234-2-4
    Publication statusPublished - Jul 2011

    Publication series

    Name
    PublisherEUROMECH

    Keywords

    • METIS-285212
    • Delay equations
    • IR-79284
    • EWI-21061
    • Stability
    • Numerical bifurcation analysis
    • Synchronization

    Cite this

    Meijer, H. G. E., Kalmar-Nagy, T., & van Gils, S. A. (2011). On phase-locking of oscillators with delay coupling. In D. Bernardini, G. Rega, & F. Romeo (Eds.), Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011) (pp. MS-11). EUROMECH.
    Meijer, Hil Gaétan Ellart ; Kalmar-Nagy, T. ; van Gils, Stephanus A. / On phase-locking of oscillators with delay coupling. Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). editor / D. Bernardini ; G. Rega ; F. Romeo. EUROMECH, 2011. pp. MS-11
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    title = "On phase-locking of oscillators with delay coupling",
    abstract = "We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.",
    keywords = "METIS-285212, Delay equations, IR-79284, EWI-21061, Stability, Numerical bifurcation analysis, Synchronization",
    author = "Meijer, {Hil Ga{\'e}tan Ellart} and T. Kalmar-Nagy and {van Gils}, {Stephanus A.}",
    note = "DOI for the proceedings: 10.3267/ENOC2011Rome",
    year = "2011",
    month = "7",
    language = "Undefined",
    isbn = "978-88-906234-2-4",
    publisher = "EUROMECH",
    pages = "MS--11",
    editor = "D. Bernardini and G. Rega and F. Romeo",
    booktitle = "Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)",

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    Meijer, HGE, Kalmar-Nagy, T & van Gils, SA 2011, On phase-locking of oscillators with delay coupling. in D Bernardini, G Rega & F Romeo (eds), Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). EUROMECH, pp. MS-11.

    On phase-locking of oscillators with delay coupling. / Meijer, Hil Gaétan Ellart; Kalmar-Nagy, T.; van Gils, Stephanus A.

    Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). ed. / D. Bernardini; G. Rega; F. Romeo. EUROMECH, 2011. p. MS-11.

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

    TY - GEN

    T1 - On phase-locking of oscillators with delay coupling

    AU - Meijer, Hil Gaétan Ellart

    AU - Kalmar-Nagy, T.

    AU - van Gils, Stephanus A.

    N1 - DOI for the proceedings: 10.3267/ENOC2011Rome

    PY - 2011/7

    Y1 - 2011/7

    N2 - We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.

    AB - We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.

    KW - METIS-285212

    KW - Delay equations

    KW - IR-79284

    KW - EWI-21061

    KW - Stability

    KW - Numerical bifurcation analysis

    KW - Synchronization

    M3 - Conference contribution

    SN - 978-88-906234-2-4

    SP - MS-11

    BT - Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)

    A2 - Bernardini, D.

    A2 - Rega, G.

    A2 - Romeo, F.

    PB - EUROMECH

    ER -

    Meijer HGE, Kalmar-Nagy T, van Gils SA. On phase-locking of oscillators with delay coupling. In Bernardini D, Rega G, Romeo F, editors, Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). EUROMECH. 2011. p. MS-11