ELSEVIER Physica D 118(1998)221-249 PHYSICA Reduction of secular error in approximations to harmonic oscillator systems with nonlinear perturbations Peter B. Kahn a, Yair Zarmi b a Physics Department State UniversiO, ofNew York, Stony Brook, NY 11794, USA b Centerj[w Energy & Environmental Physics, Jacob Blaustein Institute for Desert Research, Ben-Gurion Universi O' of the Negev, Sede Boqer Campus 84990, lsrael Received 17 September 1997; received in revised lbrm 3 December 1997; accepted 8 January 1998 Communicated by A.M. Albano Abstract The freedom inherent in perturbative expansions is exploited within the framework of the method of normal forms, to demonstrate that the secular errors, evolving in approximations to solutions for harmonic oscillator systems with small nonlinear perturbations, can be reduced significantly by the minimal normalfi~rm (MNF) choice of the tree resonant terms in the expansion. The MNF choice, previously developed for systems with one degree of freedom, is extended to higher- dimensional ones by applying it to the phase component of the normal form equations. The results are shown for a sequence of examples of growing complexity, starting with single oscillators and ending with three coupled oscillators. © 1998 Elsevier Science B.V. PACS: 03.40-t: 02.90+p Kevwords: Nonlinear oscillations Normal forms; Secular error 1. Introduction The present knowledge concerning the estimation of, or control over, errors in perturbative approximations to solutions of dynamical systems that are subjected to small nonlinear interactions is rather limited. This is the case even for Hamiltonian systems, where motion is bounded due to energy conservation. Our knowledge is extensive only in the case of a single harmonic oscillator that is perturbed by a small nonlinearity. For the latter, it can be shown in expansion methods that account for the effect of the nonlinear perturbation on the frequency (the methods of averaging [1,2], multiple timescales [3], and normal forms [4-9]) that the zero-order term constitutes an O(e) approximation for time spans of O(1/~), e(lEI << 1) being the strength of the perturbation. Extension to higher orders is possible (see, e.g., [2]). The error in approximations to the full solution, x(t), arises from two sources: (1) the error in the amplitude induces a bounded error; (2) The error in the phase, which grows linearly 0167-2789/98/$19.00 Copyright © 1998 Elsevier Science B.V. All rights reserved P11S0167-2789(98)00007-4