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arXiv · 2005.10742

Intrinsic determination of the criticality of a slow-fast Hopf bifurcation

Peter De Maesschalck·Thai Son Doan·Jeroen Wynen
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Abstract

The presence of slow-fast Hopf (or singular Hopf) points in slow-fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In the monograph "Canards from birth to transition", an intrinsic presentation of slow-fast vector fields is initiated, showing hands-on formulas to check for the presence of such singular contact points. We generalize the results in the sense that the criticality of the Hopf bifurcation can be checked with a single formula. We demonstrate the result on a slow-fast system given in non-standard form where slow and fast variables are not separated from each other. The formula is convenient since it does not require any parameterization of the critical curve.

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Peter De Maesschalck, Thai Son Doan, Jeroen Wynen. 2020-05-21. Intrinsic determination of the criticality of a slow-fast Hopf bifurcation. https://doi.org/10.1007/s10884-020-09903-x

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