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

    Keywords

    • homoclinic bifurcations
    • numerical bifurcation analysis

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