A fast Galerkin method to obtain the periodic solutions of a nonlinear oscillator
type: Item
instance of: scholarly article
title: A fast Galerkin method to obtain the periodic solutions of a nonlinear oscillator
zbMATH Open document ID: 0908.65061
DOI: 10.1016/S0096-3003(96)00193-2
author: Juan Peña Miralles
author: Pedro José Jiménez Olivo
author: Damián Ginestar Peiró
author: Gumersindo Verdú Martín
author: José Luis Muñoz-Cobo González
published in: Applied Mathematics and Computation
publication date: 1998-10-08T00:00:00Z
review text: The authors propose an algorithm to determine periodic solutions of nonlinear second-order equations of type \[ x''+ kx'+ (a_1x+ a_2x^2+ a_3x^3)= g(t),\tag{1} \] where \(k\) and \(a_j\) are given constants and \(g(t)\) is a \(T\)-periodic forcing term. Assuming the existence of a periodic solution of (1) with some period \(T'= pT\) and taking the \(m\)th order trigonometrical series approximation to this solution the coefficients of this expansion are determined by a Galerkin type method where the coefficients are approximated iteratively by means of a fast Fourier transform on equally distributed points on a period \([0,T']\). No proofs of the convergence of the proposed iterative schemes and the global errors of the overall process are given. Further, a stability analysis of periodic solutions for equations of this type is also presented. The original equation is linearized around the assumed periodic solution and by Floquet theory some stability conditions on the problem parameters can be derived. Finally, the authors present some numerical results for a Duffing nonlinear oscillator given by the equation. \(x''+ 0.4x'+ x- x^3= b\sin(0.65t)\) that show several families of stable and unstable periodic orbits for different ranges of the \(b\)-parameter.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 65L60
zbMATH DE Number: 1183153
zbMATH Keywords: numerical examples
zbMATH Keywords: algorithm
zbMATH Keywords: periodic solutions
zbMATH Keywords: nonlinear second-order equations
zbMATH Keywords: trigonometrical series approximation
zbMATH Keywords: Galerkin type method
zbMATH Keywords: fast Fourier transform
zbMATH Keywords: stability
zbMATH Keywords: Floquet theory
zbMATH Keywords: Duffing nonlinear oscillator
zbMATH Keywords: periodic orbits
Wikidata QID: Q57732392
reviewed by: Manuel Calvo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1183153