On Approximations of First Integrals for a Strongly Nonlinear Forced Oscillator

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 rst integrals. Not only approximations of rst integrals will be given, but it will also be shown how, in a rather ecient way, the existence and stability of time-periodic solutions can be obtained from these approximations. In addition phase portraits, Poincare-return maps, and bifurcation diagrams for a set of values of the parameters will be presented. In particular the strongly nonlinear forced oscillator equation X+X+X3 = _X _X3 + _X cos(2t) will be studied in this paper. It will be shown that the presented perturbation method not only can be applied to a weakly nonlinear oscillator problem (that is, when the parameter = O()) but also to a strongly nonlinear problem (that is, when = O(1)). The model equation as considered in this paper is related to the phenomenon of galloping of overhead power transmission lines on which ice has accreted