# Determining the source of period-doubling instabilities in spiral waves

## TL;DR

This paper introduces a spectral analysis methodology to identify whether period-doubling instabilities in spiral waves originate from the core, boundary effects, or far-field regions, with applications to cardiac and chemical models.

## Contribution

It provides a new approach to distinguish the sources of instabilities in spiral waves by analyzing spectral properties, revealing different mechanisms for alternans and line defects.

## Key findings

- Alternans are driven by the spiral core.
- Line defects originate from boundary effects.
- Eigenfunction shapes result from interactions with continuous spectra.

## 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.