The method of multiple scales and the perturbation-incremental method for autonomous non-linear oscillators

Abstract

In this report, a method of multiple scales is presented for the analysis of weakly non-linear oscillators in the form 2 2 2 0 , d u du u F u dt dt +ω =ε  , where , du F u dt is arbitrary non-linear function of its arguments and 0 <ε ≪1. Maple, a computer tool, is used for the analysis. Then a perturbation-incremental method is presented for the analysis of strongly non-linear oscillators in the form ɺxɺ+ g ( x) =λ f ( x, xɺ ) xɺ , where g ( x) and f ( x, xɺ ) are arbitrary non-linear functions of their arguments and λ > 0 . The perturbation-incremental method is an extension of the classical perturbation method to the case where λ is not necessarily small. The method incorporates salient features from both the perturbation method and the incremental method. Matlab is used for the perturbation-incremental method. Limit cycles of an oscillator can be calculated to any desired degree of accuracy. The stability of a limit cycle is also considered.