Abstract
The method of multiple scales and the related method of averaging are commonly used tostudy slowly modulated oscillations. If the system of interest is a slightlyperturbed harmonic oscillator, then these techniques can be applied easily. If the unperturbed system is strongly nonlinear (though possiblyconservative), then these methods can run into difficulties due to the impossibilityof carrying out required analytical operations in closed form.
In this paper, we abandon the requirement of closed form analyticaltreatment at all stages. Instead, Galerkin projections are used toobtain approximate realizations of the method of multiple scales. Thispaper adapts recent work using similar ideas for approximaterealizations of the method of averaging. A key contribution of thepresent work is in the systematic identification and removal of secularterms in the general nonlinear case, a procedure that is more difficultthan for the perturbed harmonic oscillator case, and that is unnecessaryfor 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 formerwithout compromising the asymptotic features of the latter.
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Das, S.L., Chatterjee, A. Multiple Scales via Galerkin Projections: Approximate Asymptotics for Strongly Nonlinear Oscillations. Nonlinear Dynamics 32, 161–186 (2003). https://doi.org/10.1023/A:1024447407071
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DOI: https://doi.org/10.1023/A:1024447407071