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December  2009, 2(4): 851-872. doi: 10.3934/dcdss.2009.2.851

Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit

John Guckenheimer 1, and  Christian Kuehn 2,

1. 

Mathematics Department, Cornell University, Ithaca, NY 14853, United States
2. 

Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, United States

Received  September 2008 Revised  April 2009 Published  September 2009

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: geometric singular perturbation theory, invariant manifolds., Homoclinic bifurcation.
Mathematics Subject Classification: Primary: 34E13, 34E15, 37G20, 37D10; Secondary: 34C26, 37D4.
Citation: John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851
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