We investigate analytically the effect on a period-doubling cascade of slowly sweeping the
bifurcation parameter, by means of asymptotic calculations. First we analyse the behaviour
of the orbits when sweeping through one period-doubling bifurcation. There is still an
effective bifurcation called a dynamic bifurcation, but which is delayed. We compute the
delay. Sweeping in forward and backward directions yields qualitatively different results. The
presence of noise in the system, even of small intensity, turns out to play a key role. We …