Controlling transitions in a Duffing oscillator by sweeping parameters in time

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.