We consider a high-$Q$ Duffing oscillator in a weakly nonlinear 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 lifetime to scale as $-\left(\ln \mid r-r_c\mid \right)∕{\lambda }_r$, 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.

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