  We generalize stochastic resonance to the non-adiabatic limit by treating the
double-well potential using two quadratic potentials. We use a singular
perturbation method to determine an approximate analytical solution for the
probability density function that asymptotically connects local solutions in
boundary layers near the two minima with those in the region of the maximum
that separates them. The validity of the analytical solution is confirmed
numerically. Free from the constraints of the adiabatic limit, the approach
allows us to predict the escape rate from one stable basin to another for
systems experiencing a more complex periodic forcing.
