Controlling transitions in a Duffing oscillator by sweeping parameters in time

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 -(ln parallel to r-r(c)parallel to)/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.