Localized Patterns in a Three-Component FitzHugh–Nagumo Model Revisited Via an Action Functional

Abstract

In this manuscript, we combine geometrical singular perturbation techniques and an action functional to revisit—and further study—the existence and stability of stationary localized structures in a singularly perturbed three-component FitzHugh–Nagumo model. In particular, the action functional replaces the Melnikov integral approach used in Doelman et al. (J Dyn Differ Equ 21:73–115, 2009) to explicitly derive existence conditions for stationary localized structures. In addition, the action functional also gives critical information on the stability of these stationary localized structures, thus circumventing a tedious Evans function computation. This highlights the strength of the action functional approach.