FitzHugh-Nagumo equation: bifurcations, slow-fast system and dynamics near infinity

TL;DR

This paper provides a detailed qualitative analysis of bifurcations, phase portraits, and dynamics near infinity in the FitzHugh-Nagumo system, including explicit bifurcation descriptions and connections to slow-fast dynamics.

Contribution

It offers a comprehensive bifurcation analysis of the three-parameter FitzHugh-Nagumo system, including explicit descriptions of singularities and their unfoldings, and studies dynamics near infinity.

Findings

Identification of a double-zero bifurcation with Z2-symmetry

Explicit bifurcation and transition curves including pitchfork, Hopf, and homoclinic bifurcations

Analysis of dynamics near infinity in the FitzHugh-Nagumo system

Abstract

We focus on the qualitative analysis of the phase portraits arising in the three-parameter FitzHugh-Nagumo system and its compactified form. The investigation is split into three parameter-dependent cases. In one of these cases, the system displays a double-zero bifurcation with Z2-symmetry, a singularity of codimension two. For this case, we provide explicit descriptions of the bifurcation and transition curves unfolding the singularity, including pitchfork, Hopf, Belyakov, and double homoclinic bifurcations. Furthermore, we present the corresponding bifurcation diagrams. We bridge this analysis with the theory on the framework of slow-fast family and to the presence of canards. We complete our study with an analysis of the dynamics near infinity for the family under consideration. This study complements the work summarized in Georgescu, Rocsoreanu, and Giurgiteanu, Global Bifurcations in the FitzHugh-Nagumo Model, Trends in Mathematics: Bifurcations, Symmetry and Patterns (2003).