Planar target patterns are radially symmetric time-periodic structures that connect a core region with a spatially periodic traveling wave in the far field. These patterns arise in a number of different applications, including chemical reaction patterns. We are interested in understanding the robustness of these patterns (eg do these patterns select the wave number of the asymptotic wave train or do they come in one-parameter families) and their stability with respect to small perturbations. Existence and robustness of small target patterns was previously studied near degenerate oscillatory instabilities. Here, we study the large-amplitude case and use a combination of spatial dynamical systems and Fredholm techniques to prove that target patterns uniquely select the asymptotic wave number provided the asymptotic wave trains have positive group velocity and are spectrally stable.
