  In various models and systems involving the escape of periodically forced
particle from the potential well, a common pattern is observed. Namely, the
minimal forcing amplitude required for the escape exhibits sharp minimum for
the excitation frequency below the natural frequency of small oscillations in
the well. The paper explains this regularity by exploring the transient escape
dynamics in simple benchmark potential wells. In the truncated parabolic well,
in absence of the damping the minimal forcing amplitude obviously tends to zero
for the natural excitation frequency. Addition of weak symmetric softening
nonlinearity to the truncated parabolic well leads to the nonzero forcing
minimum below the natural frequency. We explicitly compute this shift in the
principal approximation by considering the slow-flow dynamics in conditions of
the principal 1:1 resonance. Essentially nonlinear model, analyzed with the
help of transformation to action-angle variables, demonstrates very similar
qualitative features of the transient escape dynamics.
