Unfoldings of singular Hopf bifurcation

TL;DR

This paper analyzes a normal form of singular Hopf bifurcation in systems with two slow and one fast variable, providing bifurcation diagrams, manifold tangencies, and applications to chemical oscillations.

Contribution

It offers a detailed analysis of the bifurcation structure of singular Hopf bifurcation normal forms, including diagrams and parameter catalogs, with applications to chemical oscillation models.

Findings

Bifurcation diagrams of equilibrium points and periodic orbits near singular Hopf bifurcation.

Identification of parameter regions with manifold tangencies linked to mixed-mode oscillations.

Estimation of parameters for the onset of mixed-mode oscillations in chemical models.

Abstract

Singular Hopf bifurcation occurs in generic families of vector-fields with two slow variables and one fast variable. Normal forms for this bifurcation depend upon several parameters, and the dynamics displayed by the normal forms is intricate. This paper analyzes a normal form for this bifurcation. It presents extensive diagrams of bifurcations of equilibrium points and periodic orbits that are close to singular Hopf bifurcation. In addition, parameters are determined where there is a tangency between invariant manifolds that are important in the appearance of mixed-mode oscillations in systems near singular Hopf bifurcation. One parameter of the normal form is identified as the primary bifurcation parameter, and the paper presents a catalog of bifurcation sequences that occur as the primary bifurcation parameter is varied. These results are applied to estimate the parameters for the onset of mixed-mode oscillations in a model of chemical oscillations.