Slow Passage through Multiple Parametric Resonance Tongues

Abstract

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.

Fractional Mathieu equation

“…This work represents a first step in developing a complete theory of fractional parametric excitation. Related work that lies ahead could include the effects of phenomena that have been applied to non-fractional Mathieu equations, such as nonlinearity [19], quasiperiodic forcing [30], delay [13], partial differential equations [18] and slow passage through resonance [3].…”

Approximations for crossing two nearby spin resonances

“…It has been shown by Refs. [10,11] and others that, for a system which crosses a parametric resonance, a decent approximation can be achieved in a piecewise fashion. The oscillatory parts of the kernel contribute only inside the region of the so-called parametric-resonance tongue.…”

“…So for our case, an approximate solution can be developed following Refs. [10,11], by using the multiscales method. We expand around the parametric resonance θ ¼ θ r and rewrite Ω 2 ðθ þ θ r Þ:…”

“…Considering for now that C þr is larger, we can follow Refs. [10,11]. By defining new coordinates η ¼ ϵ 1 θ and t ¼ θ, our differential equation becomes (keeping only first-order ϵ 1 terms)…”

“…However, following Refs. [10,11], who considered a similar system with linear terms and multiple frequency terms, we can break up the Ω 2 into regions where the polynomial terms dominate and those where the parametric resonance dominates and then attempt a piecewise approximation that captures the important dynamics of this system.…”