We model an atomic Bose-Einstein condensate (BEC) near an instability, looking for universal features. Instabilities are often associated with bifurcations where the classical field theory provided here by the Gross-Pitaevskii equation predicts that two or more solutions appear or disappear. Simple examples of such a situation can be realized in a BEC in a double well potential or in a BEC rotating in a ring trap. We analyze this problem using both Bogoliubov theory and exact diagonalization. The former describes elementary excitations which display complex frequencies near the bifurcation. We make connections to the description of bifurcations using catastrophe theory but modified to include field quantization.