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arXiv · 1712.03212

Homoclinic saddle to saddle-focus transitions in 4D systems

Manu Kalia·Yuri A. Kuznetsov·Hil G. E. Meijer
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Abstract

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is 3-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 3-dimensional stable leading eigenspace. 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 the standard Belyakov case. We also give an example of this bifurcation in the wild case occuring in a perturbed Lorenz-Stenflo 4D ODE model.

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Manu Kalia, Yuri A. Kuznetsov, Hil G. E. Meijer. 2017-12-08. Homoclinic saddle to saddle-focus transitions in 4D systems. https://arxiv.org/abs/1712.03212

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