A theoretical basis for the Harmonic Balance Method

Johanna D. García-Saldaña, Armengol Gasull

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Abstract

The Harmonic Balance Method provides a heuristic approach for finding truncated Fourier series as an approximation to the periodic solutions of ordinary differential equations. Another natural way for obtaining these types of approximations consists in applying numerical methods. In this paper we recover the pioneering results of Stokes and Urabe that provide a theoretical basis for proving that near these truncated series, whatever is the way they have been obtained, there are actual periodic solutions of the equation. We will restrict our attention to one-dimensional non-autonomous ordinary differential equations, and we apply the obtained results to a concrete example coming from a rigid cubic system. © 2012 Elsevier Inc.
Original languageEnglish
Pages (from-to)67-80
JournalJournal of Differential Equations
Volume254
Issue number1
DOIs
Publication statusPublished - 1 Jan 2013

Keywords

  • Fixed point theorem
  • Fourier series
  • Harmonic Balance Method
  • Hyperbolic limit cycle
  • Planar polynomial system

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    García-Saldaña, J. D., & Gasull, A. (2013). A theoretical basis for the Harmonic Balance Method. Journal of Differential Equations, 254(1), 67-80. https://doi.org/10.1016/j.jde.2012.09.011