A posteriori estimates of the error of the method of harmonic balance in the problem of periodic motions of autonomous systems (Q1063160)

Dynamic systems of the form \(y=w(p)x\), \(x=f(y)\) are considered, where w, f are smooth real valued functions, \(x=x(t)\), \(y=y(t)\) and \(p=d/dt\). Periodic solutions x,y of a given period T are sought by means of projections on sin \(k\omega\) t, cos \(k\omega\) t, \(\omega =2\pi /T\), \(k=1,2,...,n=\max k\), i.e. essentially by the Ritz-Galerkin method. The latter is considered as an extension of the method of harmonic balance \((n=1)\). The error of the partial sum approximation is estimated and the projection method justified by means of an auxiliary integral equation involving a particular Green function [already introduced by the author in ''Oscillations of nonlinear systems'' (Russian), Nauka, Moscow (1969; Zbl 0204.461)]. The validity conditions are rather stringent.