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Instabilities (breakup) of spiral waves in two dimensions and their one-dimensional analogs—wave trains triggered by a specific boundary condition—leading to spatiotemporally chaotic dynamics are investigated in a simple activator-inhibitor model. These instabilities always require an absolute instability of the emitted wave trains and coincide with the Eckhaus instability for the excitable case, while for oscillatory conditions the well-known convective variant of the Eckhaus instability is found. The different cases correspond to different spiral breakup phenomenologies.