Multiple Scales via Galerkin Projections: Approximate Asymptotics for Strongly Nonlinear Oscillations

TLDR: In this paper, Galerkin projections are used to obtain approximate realizations of the method of multiple scales and the related method of averaging are commonly used to study slowly modulated oscillations.

Abstract: The method of multiple scales and the related method of averaging are commonly used to study slowly modulated oscillations. If the system of interest is a slightly perturbed harmonic oscillator, then these techniques can be applied easily. If the unperturbed system is strongly nonlinear (though possibly conservative), then these methods can run into difficulties due to the impossibility of carrying out required analytical operations in closed form. In this paper, we abandon the requirement of closed form analytical treatment at all stages. Instead, Galerkin projections are used to obtain approximate realizations of the method of multiple scales. This paper adapts recent work using similar ideas for approximate realizations of the method of averaging. A key contribution of the present work is in the systematic identification and removal of secular terms in the general nonlinear case, a procedure that is more difficult than for the perturbed harmonic oscillator case, and that is unnecessary for averaging. A strength of the present work is that the heuristics (Galerkin) and asymptotics (multiple scales) are kept distinct, leaving room for systematic refinement of the former without compromising the asymptotic features of the latter.