A study of fixed points and Hopf bifurcation of Fitzhugh-Nagumo model

Phan Van Long Em *

* Corresponding author (pvlem@agu.edu.vn)

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Received: 21 Feb 2017
Revised: 21 Jun 2017
Accepted: 09 Mar 2018
Published: 30 Mar 2018
DOI: 10.22144/ctu.jen.2018.015
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Phan, V. L. E. (2018). A study of fixed points and Hopf bifurcation of Fitzhugh-Nagumo model. CTU Journal of Innovation and Sustainable Development , (08), 112-121. https://doi.org/10.22144/ctu.jen.2018.015
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No. 08 (2018)

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Abstract

In this article, a class of FitzHugh-Nagumo model is studied. First we are finding all necessary conditions for the parameters of system such that we have just one stable fixed point which presents the resting state for this famous model. After that, by using the Hopf’s theorem, we proof analytically the existence of a Hopf bifurcation that is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. With  reasonably generic assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point.
Keywords: FitzHugh-Nagumo model, fixed point, Hopf bifurcation, limit cycle

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