2020/08/29 by Takashi Teramoto, Teramoto, Takashi, Peter van Heijster +1
Computer Science · Mathematics · Medicine · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2008.12942
openalex publication_date 2020/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.