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On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

Authors:	Waluya S B van Horssen W T

Institution:	(1) Department of Applied Mathematical Analysis, Faculty of Information Technology and Systems, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands

Abstract:	In this paper a strongly nonlinear forced oscillator will be studied. It will be shown that the recently developed perturbation method based on integrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how, in a rather efficient way, the existence and stability oftime-periodic solutions can be obtained from these approximations. In additionphase portraits, Poincaré-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particularthe strongly nonlinear forced oscillator equation $\ddot X + X + {\lambda }X^3 = \varepsilon \left( {{\delta }\dot X - \beta \dot X^3 + \gamma \dot X\cos \left( {2t} \right)} \right)$ will be studied in this paper. It will be shown that the presentedperturbation method not onlycan be applied to a weakly nonlinear oscillator problem (that is, when the parameter ${\lambda } = {\mathcal{O}}\left( \varepsilon \right)$ ) but also to a strongly nonlinear problem (that is, when ${\lambda } = {\mathcal{O}}\left( 1 \right)$ ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.

Keywords:	integrating factor integrating vector first integral perturbation method asymptotic approximation of first integral periodic solution strongly nonlinear forced oscillator
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