Slow Passage Through Resonance and Resonance Tongues

Abstract

In Chap. 11 the autoresonance phenomenon due to a slow variation of frequency of the driving force or the slow variation of a parameter of nonlinear systems is described. Two other interesting effects that can be realizable in the case of slowly varying a parameter are the slow passage through resonance and resonance tongues. For example, consider a linear or a nonlinear system driven by a periodic external force with a fixed frequency ω _f. Assume that the amplitude of oscillation is maximum at ω _f = ω _f,max with a value A _max. Suppose ω is allowed to vary slowly with time and write $\omega _{\mathrm{f}}(t) =\omega +\epsilon t$, ε ≪ 1. In a typical case when ω is fixed at a value smaller than ω _f,max then the amplitude of oscillation varies with time, reaches a maximum value at a certain time t′ such that $\omega _{\mathrm{f}}(t) =\omega +\epsilon t' <\omega _{\mathrm{f,max}}$ and then decreases. The new resonance frequency ω _f(t) is lower than the frequency ω _f,max. This is termed as slow passage through resonance and is a transient phenomenon. Such a passing through has been found to occur in systems exhibiting parametric resonance also.