A Perturbation Method for Truly Nonlinear Oscillator Differential Equations
by
Ronald E. Mickens
Clark Atlanta University, Box 172 - Physics Department, Atlanta, GA 30314, USA

The standard perturbation methods are based on the assumption that when a small parameter, appearing in the differential equation, is set equal to zero, the resulting equation is that for the linear harmonic oscillator [1]. Recently, we have constructed a new method to effectively deal with the case where no linear harmonic oscillator limiting situation exists. Our procedure combines a linearization of the ëlastic" force term of the nonlinear differential equation and then the application of the method of first-order averaging [2]. The critical part of this procedure is the selection of the proper technique for the linearization of the nonlinear elastic force function. We do this by utilizing the properties of the Fourier expansion of this term along with results from lowest order harmonic balance [2]. The value of this procedure for calculating analytical approximations to the periodic solutions of such "truly" nonlinear oscillator differential equations will be illustrated by applying it to four such model equations.

This work was supported in part by research grants from DOE and the MBRS-SCORE Program at Clark Atlanta University.

1. J. A. Murdock, Perturbations: Theory and Methods (Wiley-Interscience, New York, 1991). 2. R. E. Mickens, Oscillations in Planar Dynamic Systems (World Scientific, Singapore, 1996).

Date received: January 5, 2003