Unfoldings of Singular Hopf Bifurcation

Singular Hopf bifurcation occurs in generic families of vector fields with two slow variables and one fast variable. Prototypes for this bifurcation depend upon several parameters, and the dynamics displayed by these systems is intricate. The core of this paper is a numerical investigation of the bifurcations in one prototypical family. 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 prototype is identified as the primary bifurcation parameter, and the paper presents a catalogue of bifurcation sequences that occur as the primary bifurcation parameter is varied. These results are applied to estimating the parameters for the onset of mixed-mode oscillations in a model of chemical oscillations.