Slow Passage through Multiple Parametric Resonance Tongues

This work concerns linear parametrically excited systems that involve multiple resonances. The property of such systems is that if the parameters are fixed and lie inside a resonance tongue, the motion becomes unbounded as time goes to infinity. In this work we consider what happens when the parameters are not fixed, but rather are constrained to vary slowly in time, passing into and out of the resonance tongues. One might expect that during the time in which the motion lies inside a tongue the solution grows, and that the slower the passage through the tongue the more time is spent inside the tongue, and the larger the resulting growth. We show that this is not always the case. In particular we investigate the effect of initial conditions and relative forcing amplitudes on the growth or amplification of the solution. We address the problem of how to choose these parameters so as to minimize growth (i.e., to de-amplify the solution) after passage through multiple tongues.