  Transitions between multiple stable states of nonlinear systems are
ubiquitous in physics, chemistry, and beyond. Two types of behaviors are
usually seen as mutually exclusive: unpredictable noise-induced transitions and
predictable bifurcations of the underlying vector field. Here, we report a new
situation, corresponding to a fluctuating system approaching a bifurcation,
where both effects collaborate. We show that the problem can be reduced to a
single control parameter governing the competition between deterministic and
stochastic effects. Two asymptotic regimes are identified: when the control
parameter is small (e.g. small noise), deviations from the deterministic case
are well described by the Freidlin-Wentzell theory. In particular, escapes over
the potential barrier are very rare events. When the parameter is large (e.g.
large noise), such events become typical. Unlike pure noise-induced
transitions, the distribution of the escape time is peaked around a value which
is asymptotically predicted by an adiabatic approximation. We show that the two
regimes are characterized by qualitatively different reacting trajectories,
with algebraic and exponential divergence, respectively.
