Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit doi:10.3934/dcdss.2009.2.851
John Guckenheimer - Mathematics Department, Cornell University, Ithaca, NY 14853, United States (email) 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 [5] 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.
Keywords: Homoclinic bifurcation, geometric singular perturbation theory, invariant manifolds.
Received: September 2008; Revised: April 2009; Published: September 2009. |
Abstract
Full Text (418.9K)