Passage Through Resonance for a One-Dimensional Oscillator with Slowly Varying Frequency

Abstract

This paper concerns a model problem illustrating the techniques needed to handle passage through resonance for oscillatory systems with slowly varying frequencies. The model consists of a linear oscillator with a slowly varying frequency which at some time coincides with the constant frequency of the forcing function. It is shown that the solution can be constructed by matching the two asymptotic expansions which one obtains for oscillations near and away from resonance. As each of these asymptotic expansions depends simultaneously on two time scales, this example combines use of the two principal techniques of singular perturbations. The results show that the amplitude increases, then decreases as the natural frequency passes through the resonant value. Extension of the techniques to systems of differential equations is also indicated.