Stability of periodic traveling waves in the Aliev–Panfilov reaction–diffusion system

• We study the stability of periodic traveling wave solutions (PTWs) for the Aliev–Panfilov model of cardiac excitation.
• Our calculations of essential spectra show a stability change of Eckhaus type in the PTWs, when a model parameter is varied.
• A stability boundary between stable and unstable PTWs is calculated.
• We show that the stable wave bifurcates to an oscillating pattern, based on the stability boundary.
• The far-field spiral breakup is found numerically based on the instability of the PTWs.

We study the two-component Aliev–Panfilov reaction–diffusion system of cardiac excitation. It is known that the model exhibits spiral wave instability in two-dimensional spatial domains. In order to describe the spiral wave instability, it is important to understand periodic traveling wave instability resulting from the model. We determine the existence and stability of periodic traveling waves in the model. In addition, we calculate the stability boundary between stable and unstable periodic traveling waves in a two-dimensional parameter plane. It is observed that the periodic traveling waves express instability by a stability change of Eckhaus type. As a result, a stable wave bifurcates to an oscillating periodic traveling wave. We describe these phenomena by calculating the essential spectra of the waves. Furthermore, we study the stability of the waves as a function of the gaps between two nullclines. In two dimensions, we determine the spiral wave instability based on the stability boundary of the periodic traveling waves.