A theoretical basis for the Harmonic Balance Method

Johanna D García-Saldaña (1) , Armengol Gasull (1)

1 UAB - Universitat Autònoma de Barcelona = Autonomous University of Barcelona = Universidad Autónoma de Barcelona

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.

Mots clés: Harmonic Balance Method; Planar polynomial system; Hyperbolic limit cycle; Fourier series; Fixed point theorem

Domaines: Mathématiques [math]

Johanna D García-Saldaña, Armengol Gasull. A theoretical basis for the Harmonic Balance Method. Journal of Differential Equations, 2013, 254 (1), pp.67-80. ⟨10.1016/j.jde.2012.09.011⟩. ⟨hal-01635166⟩