  Bifurcations in dynamical systems are often studied experimentally and
numerically using a slow parameter sweep. Focusing on the cases of
period-doubling and pitchfork bifurcations in maps, we show that the adiabatic
approximation always breaks down sufficiently close to the bifurcation, so that
the upsweep and downsweep dynamics diverge from one another, disobeying
standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and
downsweep trajectories for sufficiently slow sweep rates, revealing that the
slow trajectories depend essentially on a structural asymmetry parameter, whose
effect is negligible for the stationary dynamics. We obtain explicit asymptotic
expressions for the universal trajectories, and use them to calculate the area
of the hysteresis loop enclosed between the upsweep and downsweep trajectories
as a function of the asymmetry parameter and the sweep rate.
