Instability of periodic traveling wave solutions in a modified FitzHugh–Nagumo model for excitable media

Abstract

We introduce a two-variable system of reaction–diffusion equations for excitable media. We numerically investigate the existence and stability of periodic traveling wave solutions in a two-dimensional parameter plane. Our results based on the method of continuation show a stability change of Eckhaus type. There are two families of periodic traveling waves: fast and slow. The fast family is stable in the case of standard FitzHugh–Nagumo excitable system. However, we observe that the fast family becomes unstable in our model. Consequently, it bifurcates to an oscillating wave. We explain this phenomenon by numerically calculating the essential spectra of the periodic traveling wave solutions. Moreover, we study the stability of the periodic traveling wave solutions for the Aliev–Panfilov excitable system and compare its results with the proposed model. We show that the stability boundary of our model agrees qualitatively with that of the Aliev–Panfilov model, whereas it is different in the FitzHugh–Nagumo model. In addition, our results based on the stability boundary of the models show oscillating periodic traveling waves, except for the FitzHugh–Nagumo model.