Stability of Pulse Solutions for the Discrete FitzHugh–Nagumo System

We show that the fast travelling pulses of the discrete FitzHugh–Nagumo system in the weak-recovery regime are nonlinearly stable. The spectral conditions that need to be verified involve linear operators that are associated to functional differential equations of mixed type. Such equations are ill-posed and do not admit a semi-flow, which precludes the use of standard Evans-function techniques. Instead, we construct the potential eigenfunctions directly by using exponential dichotomies, Fredholm techniques and an infinite-dimensional version of the Exchange Lemma