Annals of Mathematical Sciences and Applications

Volume 6 (2021)

Number 1

A homotopy perturbation method for a class of truly nonlinear oscillators

Pages: 3 – 23



So-Hsiang Chou (Dept. of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio, U.S.A.)

C. Attanayake (Department of Mathematics, Miami University, Middletown, Ohio, U.S.A.)

C. Thapa (Department of Mathematical Sciences, Delaware State University, Dover, Del., U.S.A.)


We apply a homotopy perturbation method to a class of nonlinear oscillators with a restoring force proportional to the odd powers $p$ of the displacement. To derive the amplitude-frequency relations indexed by $p$, a Lindstedt–Poincaré procedure is applied. Unlike in the case when $p$ is a fixed numerical value, for general $p$, deriving formulas via traditional symbolic manipulation will result in plethora of hypergeometric and trigonometric expressions, and the technique of killing the secular terms becomes unmanageable. In this paper, we propose a functional analytic framework so that these difficulties can be overcome. We introduce a Volterra integral representation of the displacement, and the annihilation of the secular terms is replaced by enforcing an orthogonal solvability condition. All computations are in terms of inner product operations. We demonstrate by numerical examples that the first two or three approximates are sufficiently accurate for this class of truly nonlinear oscillators.


homotopy, perturbation method, nonlinear oscillators, Lindstedt–Poincaré method, amplitude-frequency relation

2010 Mathematics Subject Classification

Primary 33F05. Secondary 34C15, 34E10.

Received 4 October 2020

Accepted 26 November 2020

Published 6 October 2021