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Homoclinic orbits of the FitzHugh-Nagumo equation: the singular-limit. (English) Zbl 1362.34075

Summary: 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 A. R. Champneys et al. [SIAM J. Appl. Dyn. Syst. 6, No. 4, 663–693 (2007; Zbl 1170.34025)] 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.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34E13 Multiple scale methods for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
34C26 Relaxation oscillations for ordinary differential equations
92C20 Neural biology
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations

Citations:

Zbl 1170.34025
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