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

TL;DR

This paper introduces an intrinsic, formula-based method to determine the criticality of slow-fast Hopf bifurcations in planar systems, simplifying analysis without normal form transformation or parameterization.

Contribution

It generalizes previous approaches by providing a single, easy-to-use formula for assessing Hopf bifurcation criticality in complex slow-fast systems.

Findings

The formula accurately determines bifurcation criticality.

Applicable to non-standard slow-fast systems.

Simplifies bifurcation analysis without normal form or parameterization.

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.