  We study a noise-induced bifurcation in the vicinity of the threshold by
using a perturbative expansion of the order parameter, called the
Poincar\'e-Lindstedt expansion. Each term of this series becomes divergent in
the long time limit if the power spectrum of the noise does not vanish at zero
frequency. These divergencies have a physical consequence: they modify the
scaling of all the moments of the order parameter near the threshold and lead
to a multifractal behaviour. We derive this anomalous scaling behaviour
analytically by a resummation of the Poincar\'e-Lindstedt series and show that
the usual, deterministic, scalings are recovered when the noise has a low
frequency cut-off. Our analysis reconciles apparently contradictory results
found in the literature.
