  In this paper we develop a perturbation method to predict the rate of
occurrence of rare events for singularly perturbed stochastic systems using a
probability density function approach. In contrast to a stochastic normal form
approach, we model rare event occurrences due to large fluctuations
probabilistically and employ a WKB ansatz to approximate their rate of
occurrence. This results in the generation of a two-point boundary value
problem that models the interaction of the state variables and the most likely
noise force required to induce a rare event. The resulting equations of motion
of describing the phenomenon are shown to be singularly perturbed. Vastly
different time scales among the variables are leveraged to reduce the dimension
and predict the dynamics on the slow manifold in a deterministic setting. The
resulting constrained equations of motion may be used to directly compute an
exponent that determines the probability of rare events.
  To verify the theory, a stochastic damped Duffing oscillator with three
equilibrium points (two sinks separated by a saddle) is analyzed. The predicted
switching time between states is computed using the optimal path that resides
in an expanded phase space. We show that the exponential scaling of the
switching rate as a function of system parameters agrees well with numerical
simulations. Moreover, the dynamics of the original system and the reduced
system via center manifolds are shown to agree in an exponentially scaling
sense.
