doi:10.3934/dcdss.2009.2.851          Full text: (418.9K)    

John Guckenheimer - Mathematics Department, Cornell University, Ithaca, NY 14853, United States (email)

Christian Kuehn - Center for Applied Mathematics, 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.
Mathematics Subject Classification: Primary: 34E13, 34E15, 37G20, 37D10; Secondary: 34C26, 37D45.

Received:   September   2008;   Revised:   April  2009;   Published:   September  2009.

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