We show that vortex breakdown appears as a jump bifurcation due to structural instability of swirling flows when their solution locally fails to exist and the flow transits to another stable or metastable state. The flow pattern inside the steady separation zone and, consequently, the vortex breakdown features depend on the flow history. Therefore, we take into account theflow pattern inside the separation zone. The stagnation zone model (without velocity jump) excels the traditional analytic continuation method (leading to a recirculation zone) in that solutions always exist, and, for large enough inflow swirl, exhibit nonuniqueness and folds due to smooth variations of flow parameters, thus predicting the experimentally observed hysteretic jump transitions.©1997 American Institute of Physics.