  We consider a high-$Q$ Duffing oscillator in a weakly non-linear regime with
the driving frequency $\sigma$ varying in time between $\sigma_i$ and
$\sigma_f$ at a characteristic rate $r$. We found that the frequency sweep can
cause controlled transitions between two stable states of the system. Moreover,
these transitions are accomplished via a transient that lingers for a long time
around the third, unstable fixed point of saddle type. We propose a simple
explanation for this phenomenon and find the transient life-time to scale as
$-(\ln{|r-r_c|})/\lambda_r$ where $r_c$ is the critical rate necessary to
induce a transition and $\lambda_r$ is the repulsive eigenvalue of the saddle.
Experimental implications are mentioned.
