A study of resonance tongues near a Chenciner bifurcation using MatcontM

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    Abstract

    MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol'd tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [Baesens&Mackay,2007] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves.
    Original languageUndefined
    Title of host publicationProceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)
    EditorsD. Bernardini, G. Rega, F. Romeo
    PublisherEUROMECH
    PagesMS03
    Number of pages5
    ISBN (Print)978-88-906234-2-4
    Publication statusPublished - Jul 2011

    Publication series

    Name
    PublisherEuromech

    Keywords

    • METIS-285214
    • MatcontM
    • EWI-21063
    • Resonance tongues
    • Numerical bifurcation analysis
    • IR-79062

    Cite this

    Govaerts, W., Kouznetsov, I. A., Meijer, H. G. E., & Neirynck, N. (2011). A study of resonance tongues near a Chenciner bifurcation using MatcontM. In D. Bernardini, G. Rega, & F. Romeo (Eds.), Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011) (pp. MS03). EUROMECH.
    Govaerts, W. ; Kouznetsov, Iouri Aleksandrovitsj ; Meijer, Hil Gaétan Ellart ; Neirynck, N. / A study of resonance tongues near a Chenciner bifurcation using MatcontM. Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). editor / D. Bernardini ; G. Rega ; F. Romeo. EUROMECH, 2011. pp. MS03
    @inproceedings{96982835787a4951b1495eda2fd917a6,
    title = "A study of resonance tongues near a Chenciner bifurcation using MatcontM",
    abstract = "MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol'd tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [Baesens&Mackay,2007] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves.",
    keywords = "METIS-285214, MatcontM, EWI-21063, Resonance tongues, Numerical bifurcation analysis, IR-79062",
    author = "W. Govaerts and Kouznetsov, {Iouri Aleksandrovitsj} and Meijer, {Hil Ga{\'e}tan Ellart} and N. Neirynck",
    note = "DOI for the proceedings: 10.3267/ENOC2011Rome",
    year = "2011",
    month = "7",
    language = "Undefined",
    isbn = "978-88-906234-2-4",
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    Govaerts, W, Kouznetsov, IA, Meijer, HGE & Neirynck, N 2011, A study of resonance tongues near a Chenciner bifurcation using MatcontM. in D Bernardini, G Rega & F Romeo (eds), Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). EUROMECH, pp. MS03.

    A study of resonance tongues near a Chenciner bifurcation using MatcontM. / Govaerts, W.; Kouznetsov, Iouri Aleksandrovitsj; Meijer, Hil Gaétan Ellart; Neirynck, N.

    Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). ed. / D. Bernardini; G. Rega; F. Romeo. EUROMECH, 2011. p. MS03.

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

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    T1 - A study of resonance tongues near a Chenciner bifurcation using MatcontM

    AU - Govaerts, W.

    AU - Kouznetsov, Iouri Aleksandrovitsj

    AU - Meijer, Hil Gaétan Ellart

    AU - Neirynck, N.

    N1 - DOI for the proceedings: 10.3267/ENOC2011Rome

    PY - 2011/7

    Y1 - 2011/7

    N2 - MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol'd tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [Baesens&Mackay,2007] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves.

    AB - MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol'd tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [Baesens&Mackay,2007] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves.

    KW - METIS-285214

    KW - MatcontM

    KW - EWI-21063

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    KW - Numerical bifurcation analysis

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    SN - 978-88-906234-2-4

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

    A2 - Bernardini, D.

    A2 - Rega, G.

    A2 - Romeo, F.

    PB - EUROMECH

    ER -

    Govaerts W, Kouznetsov IA, Meijer HGE, Neirynck N. A study of resonance tongues near a Chenciner bifurcation using MatcontM. In Bernardini D, Rega G, Romeo F, editors, Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011). EUROMECH. 2011. p. MS03