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
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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",
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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

TY - GEN

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

KW - Resonance tongues

KW - Numerical bifurcation analysis

KW - IR-79062

M3 - Conference contribution

SN - 978-88-906234-2-4

SP - MS03

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