  Semiclassical transition probabilities characterize transfer of energy
between "hard" and "soft" modes in various physical systems. We establish the
boundary problem for singular euclidean solutions used to calculate such
probabilities. Solutions are found numerically for a system of two interacting
quartic oscillators. In the double-well case, we find numerical evidence that
certain regular {\em minkowskian} trajectories have approximate stopping points
or, equivalently, are approximately periodic. This property leads to estimates
of tunneling excitation probabilities in that system and suggests that similar
estimates may be possible in other systems with tunneling.
