  We derive the exact bifurcation diagram of the Duffing oscillator with
parametric noise thanks to the analytical study of the associated Lyapunov
exponent. When the fixed point is unstable for the underlying deterministic
dynamics, we show that the system undergoes a noise-induced reentrant
transition in a given range of parameters. The fixed point is stabilised when
the amplitude of the noise belongs to a well-defined interval. Noisy
oscillations are found outside that range, i.e., for both weaker and stronger
noise.
