Invariant Tori and Periodic Orbits in the FitzHugh-Nagumo System

TL;DR

This paper investigates the existence and stability of invariant tori and periodic orbits in the FitzHugh-Nagumo system using averaging theory and numerical methods, revealing new bifurcation phenomena in neural models.

Contribution

It introduces explicit generic conditions for invariant tori bifurcations in the FitzHugh-Nagumo system and analyzes their stability, combining theoretical and numerical approaches.

Findings

Explicit conditions for invariant tori existence

Identification of bifurcation scenarios in neural models

Numerical validation of theoretical results

Abstract

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.