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Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations

    Serge Bruno Yamgoue Affiliation
    ; Olivier Tiokeng Lekeufack Affiliation
    ; Timoleon Crepin Kofane Affiliation

Abstract

In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach.

Keyword : harmonic balance, linearization, explicit approximations, solitons, hyperbolic solutions

How to Cite
Yamgoue, S. B., Lekeufack, O. T., & Kofane, T. C. (2017). Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations. Mathematical Modelling and Analysis, 22(2), 140-156. https://doi.org/10.3846/13926292.2017.1276983
Published in Issue
Mar 18, 2017
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This work is licensed under a Creative Commons Attribution 4.0 International License.