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Systems of oscillatory and excitable media frequently express regular spiral and traveling wave patterns, but bifurcations to complex structures are also common. Spiral wave patterns observed in models of cardiac arrhythmias and chemical oscillations develop alternans and stationary line defects, both of which can be thought of as period-doubling instabilities. Physically, the period-doubled patterns are observed on bounded domains, leading to the question of if and how the domain geometry contributes to the formation of the instability. We explore this question by considering impacts of truncating an infinite to bounded domain, and how properties of single cells can be extended to spatially extended tissues. The first scenario is relevant for determining on what domains the instabilities will be present, and the second is important for understanding if results from in vitro experiments can be applied to in vivo settings.

Bifurcations of bounded spirals may arise from instabilities associated with one of the three distinct regions of the spiral: the core, far-field, and boundary. Here, we introduce a methodology to disentangle the impacts of each region on the instabilities by analyzing spectral properties of spiral waves. We apply our techniques to spirals formed in reaction-diffusion systems and find that the mechanisms driving the period-doubling instabilities are quite different; alternans are driven by the spiral core, whereas line defects appear from boundary effects. Finally, we explore how the spectral properties of reaction-diffusion systems are modified when one species does not diffuse, leading to a rank-deficient diffusion matrix.