Period–doubling bifurcations of a fairly general unimodal map yn+1= ƒ(yn, xn) are considered with linearly varying parameter xn+1= xn± ε (ε≪ 1). The sweeping delays the apparent bifurcations. There are structurally different types of trajectories sweeping forwards (+ε) and sweeping backwards (−ε). A matched asymptotic approach is used to analyse the system. Following Baesens, adiabatic manifolds are obtained for the period−1 and period−2 regions. An outer expansion that is singular at the bifurcation is matched to an inner expansion valid in an O(ε½) boundary layer. The inner expansion brings out the normal form, in this case a pitchfork bifurcation. Local stability of the manifolds is examined. Typical trajectories sweeping down can again be described by similar outer and inner expansions. Sweeping up, however, typical trajectories remain at least O(ε) close to the unstable period−1 manifold before a rapid transition to period 2 (a discrete canard). An inner–inner expansion is needed to describe how perturbations or noise are critical in determining where the rapid transition occurs. Trajectories sweeping up are thus described by a uniformly valid triple–deck matched asymptotic expansion. A specific example of a quadratic map is used to compare this expansion with numerical simulations. The example shows explicitly the effect of both sweep rate and initial value or noise on determining where the rapid transition occurs.