A theoretical basis for the Harmonic Balance Method

TL;DR

This paper establishes a theoretical foundation for the Harmonic Balance method, showing that approximate Fourier series solutions correspond to actual periodic solutions in certain differential equations.

Contribution

It recovers and extends classical results to justify the Harmonic Balance method's approximations for one-dimensional non-autonomous ODEs.

Findings

Theoretical proof linking truncated Fourier series to true solutions.

Application of results to specific planar autonomous systems.

Validation of the Harmonic Balance method's accuracy.

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 type 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 results obtained to a couple of concrete examples coming from planar autonomous systems.