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This paper introduces a numerical method for computing the spectrum ofadjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusionsystems in arbitrary geometries. The method is illustrated by computing over ahundred eigenfunctions associated with an unstable time-periodic single-spiralsolution of the Karma model on a square domain. We show that all leadingadjoint eigenfunctions are exponentially localized in the vicinity of thespiral tip, although the marginal modes (response functions) demonstrate thestrongest localization. We also discuss the implications of the localizationfor the dynamics and control of unstable spiral waves. In particular, theinteraction with no-flux boundaries leads to a drift of spiral waves which canbe understood with the help of the response functions.

Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation