### Abstract

Copyright © 2014 John Wiley & Sons, Ltd. We characterize the values of the parameters for which a zero-Hopf equilibrium point takes place at the singular points, namely, O (the origin), P+, and P- in the FitzHugh-Nagumo system. We find two two-parameter families of the FitzHugh-Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero-Hopf equilibrium point O. We prove that there exist three two-parameter families of the FitzHugh-Nagumo system for which the equilibrium point at P+ and at P- is a zero-Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P+ and P-.

Original language | English |
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Pages (from-to) | 4289-4299 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 38 |

Issue number | 17 |

DOIs | |

Publication status | Published - 30 Nov 2015 |

### Keywords

- averaging theory
- FitzHugh-Nagumo system
- periodic orbit
- zero-Hopf bifurcation

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## Cite this

Euzébio, R. D., Llibre, J., & Vidal, C. (2015). Zero-Hopf bifurcation in the FitzHugh-Nagumo system.

*Mathematical Methods in the Applied Sciences*,*38*(17), 4289-4299. https://doi.org/10.1002/mma.3365