Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System

The FitzHugh-Nagumo system is a \(4\)-parameter family of \(3\)D vector field used for modeling neural excitation and nerve impulse propagation. The origin represents a Hopf-zero equilibrium in the FitzHugh-Nagumo system for two classes of parameters. In this paper, we employ recent techniques in av...

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Published in:arXiv.org 2024-08
Main Authors: Cândido, Murilo R, Novaes, Douglas D, Sadri, Nasrin
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Bifurcation theory
Differential equations
Fields (mathematics)
Invariants
Orbits
Parameters
Toruses
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description The FitzHugh-Nagumo system is a \(4\)-parameter family of \(3\)D vector field used for modeling neural excitation and nerve impulse propagation. The origin represents a Hopf-zero equilibrium in the FitzHugh-Nagumo system for two classes of parameters. In this paper, we employ recent techniques in averaging theory to investigate, besides periodic solutions, the bifurcation of invariant tori within the aforementioned families. We provide explicit generic conditions for the existence of these tori and analyze their stability properties. Furthermore, we employ the backward differentiation formula to solve the stiff differential equations and provide numerical simulations for each of the mentioned results.
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subjects Bifurcation theory
Differential equations
Fields (mathematics)
Invariants
Orbits
Parameters
Toruses
title Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System
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