Homoclinic Orbits of the FitzHugh-Nagumo Equation: The Singular-Limit

TL;DR

This paper uses geometric singular perturbation theory to analyze the bifurcation structure of the FitzHugh-Nagumo equation in the singular limit, providing insights into complex bifurcation curves and their computation.

Contribution

It introduces a geometric singular perturbation approach to understand and compute bifurcation structures in the FitzHugh-Nagumo equation, especially in the singular limit.

Findings

Singular limit analysis accurately captures bifurcation structures.

Geometric methods can compute bifurcation curves inaccessible to continuation.

The singular bifurcation diagram summarizes the bifurcation behavior.

Abstract

The FitzHugh-Nagumo equation has been investigated with a wide array of different methods in the last three decades. Recently a version of the equations with an applied current was analyzed by Champneys, Kirk, Knobloch, Oldeman and Sneyd using numerical continuation methods. They obtained a complicated bifurcation diagram in parameter space featuring a C-shaped curve of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations. We use techniques from multiple time-scale dynamics to understand the structures of this bifurcation diagram based on geometric singular perturbation analysis of the FitzHugh-Nagumo equation. Numerical and analytical techniques show that if the ratio of the time-scales in the FitzHugh-Nagumo equation tends to zero, then our singular limit analysis correctly represents the observed CU-structure. Geometric insight from the analysis can even be used to compute bifurcation curves which are inaccessible via continuation methods. The results of our analysis are summarized in a singular bifurcation diagram.