Geometry of bifurcation sets: Exploring the parameter space

Abstract

Inspecting a $p$-dimensional parameter space by means of $(p-1)$-dimensional slices, changes can be detected that are only determined by the geometry of the manifolds that compose the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, particularly, the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are discussed in the context of two models of neuron activity: the Hindmarsh-Rose and the FitzHugh-Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter.