Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
Volume 2, Issue 4, Pages 851-872Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcdss.2009.2.851
Keywords
Homoclinic bifurcation; geometric singular perturbation theory; invariant manifolds
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Funding
- National Science Foundation
- Department of Energy
- Direct For Computer & Info Scie & Enginr
- Division Of Computer and Network Systems [0832782] Funding Source: National Science Foundation
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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.
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