On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating 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 particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

On Approximations of First Integrals for a System of Weakly Nonlinear,Coupled Harmonic Oscillators

In this paper a system of weakly nonlinear, coupled harmonic oscillatorswill be studied. It will be shown that the recently developedperturbation method based on integrating vectors can be used toapproximate first integrals and periodic solutions. To show how thisperturbation method works the method is applied to a system of weaklynonlinear, coupled harmonic oscillators with 1:3 and 3:1 internalresonances. Not only approximations of first integrals will be given,but it will also be shown how, in a rather efficient way, the existenceand stability of time-periodic solutions can be obtained from theseapproximations. In addition some bifurcation diagrams for a set ofvalues of the parameters will be presented.     

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 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 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 ) but also to a strongly nonlinear problem (that is, when ). The model equation as considered in this paper is related to the phenomenon of galloping ofoverhead power transmission lines on which ice has accreted.