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arXiv · 2008.12942

Traveling pulse solutions in a three-component FitzHugh-Nagumo Model

Takashi Teramoto·Peter van Heijster
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Abstract

We use geometric singular perturbation techniques combined with an action functional approach to study traveling pulse solutions in a three-component FitzHugh--Nagumo model. First, we derive the profile of traveling $1$-pulse solutions with undetermined width and propagating speed. Next, we compute the associated action functional for this profile from which we derive the conditions for existence and a saddle-node bifurcation as the zeros of the action functional and its derivatives. We obtain the same conditions by using a different analytical approach that exploits the singular limit of the problem. We also apply this methodology of the action functional to the problem for traveling $2$-pulse solutions and derive the explicit conditions for existence and a saddle-node bifurcation. From these we deduce a necessary condition for the existence of traveling $2$-pulse solutions. We end this article with a discussion related to Hopf bifurcations near the saddle-node bifurcation.

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Takashi Teramoto, Peter van Heijster. 2021-02-26. Traveling pulse solutions in a three-component FitzHugh-Nagumo Model. https://arxiv.org/abs/2008.12942

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