Homoclinic saddle to saddle-focus transitions in 4D systems

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    Abstract

    A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model.
    Original languageEnglish
    Pages (from-to)2024-2054
    Number of pages31
    JournalNonlinearity
    Volume32
    Issue number6
    DOIs
    Publication statusPublished - Jun 2019

    Fingerprint

    Homoclinic
    saddles
    Saddle
    eigenvalues
    Eigenvalue
    Bifurcation (mathematics)
    Ordinary differential equations
    Bifurcation
    Eigenspace
    Bifurcation Point
    Bifurcation Curve
    Complex conjugate
    Codimension
    Ordinary differential equation
    differential equations
    Distinct
    Three-dimensional
    curves
    Standards

    Keywords

    • homoclinic bifurcations
    • numerical bifurcation analysis

    Cite this

    @article{aa0bcfc1c11245f49fdd5455ba7de596,
    title = "Homoclinic saddle to saddle-focus transitions in 4D systems",
    abstract = "A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model.",
    keywords = "homoclinic bifurcations, numerical bifurcation analysis",
    author = "Manu Kalia and Kouznetsov, {Iouri Aleksandrovitsj} and Meijer, {Hil Ga{\'e}tan Ellart}",
    year = "2019",
    month = "6",
    doi = "10.1088/1361-6544/ab0041",
    language = "English",
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    journal = "Nonlinearity",
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    }

    Homoclinic saddle to saddle-focus transitions in 4D systems. / Kalia, Manu ; Kouznetsov, Iouri Aleksandrovitsj; Meijer, Hil Gaétan Ellart.

    In: Nonlinearity, Vol. 32, No. 6, 06.2019, p. 2024-2054.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - Homoclinic saddle to saddle-focus transitions in 4D systems

    AU - Kalia, Manu

    AU - Kouznetsov, Iouri Aleksandrovitsj

    AU - Meijer, Hil Gaétan Ellart

    PY - 2019/6

    Y1 - 2019/6

    N2 - A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model.

    AB - A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model.

    KW - homoclinic bifurcations

    KW - numerical bifurcation analysis

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    U2 - 10.1088/1361-6544/ab0041

    DO - 10.1088/1361-6544/ab0041

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    EP - 2054

    JO - Nonlinearity

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    SN - 0951-7715

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    ER -