Philipp Meerkamp

This dissertation concerns singular Hopf bifurcation in slow-fast vector fields with one fast and two slow variables. Singular Hopf bifurcation occurs generically in this setting. Singular Hopf bifurcation is important in the understanding of mixed-mode oscillations in systems modeling electrical activity in neural cells and chemical oscillations. Three approaches are taken in this study of singular Hopf bifurcation.

First, we analyze a normal form for singular Hopf bifurcation. Normal forms for this bifurcation depend upon several parameters, and the dynamics displayed by the normal forms is intricate. Extensive bifurcation diagrams of equilibrium points and periodic orbits that are close to singular Hopf bifurcation are presented. In addition, parameters are determined where there is a tangency between invariant manifolds that can mark the onset of mixed-mode oscillations in systems near singular Hopf bifurcation. One parameter of the normal form is identified as the primary bifurcation parameter, and we present a catalog of bifurcation sequences that occur as the primary bifurcation parameter is varied. These results are used to estimate the parameters for the onset of mixed-mode oscillations in a model of chemical oscillations.

Second, we present an in-depth study singular Hopf bifurcation and mixed-mode oscillations in a system modeling electric activity in a pituitary lactotroph cell. We transform the lactotroph model into singular Hopf normal form and analyze in detail to which extent the normal form approximates the lactotroph model. It is shown that mixed-mode oscillations in the lactotroph model are over large parameter ranges due to singular Hopf bifurcation.

Third, we describe a numerical method to rigorously compute enclosures of slow manifolds. Locally partitioning parameter space into regions with distinct behaviors, slow manifolds are central objects in the study of vector fields with multiple time scales as well as of mixed-mode oscillations. Important parts of the analysis of the singular Hopf normal form also involve slow manifolds. Computing slow manifolds poses two main challenges. First, slow manifolds are defined asymptotically in terms of the parameter ε separating the time scales and are thus non-unique fuzzy objects for ε > 0. We meet this challenge by defining computable slow manifolds, which can be computed for ε > 0. Second, the separation of time scales poses problems for many numerical schemes. The method to rigorously compute enclosures of slow manifolds described here becomes more efficient as ε → 0.