Determining the Source of Period-Doubling Instabilities in Spiral Waves

Abstract

Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, which can both be thought of as period-doubling instabilities. These instabilities are observed on bounded domains and may be caused by the spiral core, far-field asymptotics, or boundary conditions. Here, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves and boundary sinks on bounded domains with appropriate boundary conditions. We apply our techniques to spirals formed in reaction-diffusion systems to investigate how and why alternans and line defects develop. Our results indicate that the mechanisms driving these instabilities are quite different; alternans are driven by the spiral core, whereas line defects appear from boundary effects. Moreover, we find that the shape of the alternans eigenfunction is due to the interaction of a point eigenvalue with curves of continuous spectra.