Limit cycles of non-smooth oscillators
type: Item
instance of: scholarly article
title: Limit cycles of non-smooth oscillators
zbMATH Open document ID: 1153.34022
published in: Applied Mathematics and Computation
publication date: 2008-06-11T00:00:00Z
review text: The author considers a nonsmooth oscillator of the form \[ \ddot{u}=f(u), \] where \(f:\mathbb R\to\mathbb R\) is a nonlinear, nonsmooth function with \(f(-u)=-f(u)\). The author develops a method to expand both the solution and the frequency of the periodic solution of the above equation in power series of an artificial parameter. Thereto, an artificial parameter-Linstedt-Poincaré method based on the introduction of a linear stiffness term and an artificial parameter \(p\) is applied before a change of the independent variable enables the author to expand the solution and its frequency into a power series of \(p\). By requiring that the solution at each order in the artificial parameter is free from secular terms, algebraic equations for the approximate frequency of oscillations are obtained. The methods does not require \(f\) to be smooth; if \(f\) is not smooth enough, then the derivatives are interpreted in a generalized sense as generalized functions. The method is applied to three examples, namely \(\ddot{u}=-u| u| \), \(\ddot{u}=-\text{sign}(u)\) and \(\ddot{u}=-u^{1/3}\). The first-order approximation is in full accord with that obtained by other methods such as harmonic balance, linearized harmonic balance and fully iterative methods. Furthermore, this method of this paper also allows to obtain second-order corrections.
reviewed by: Peter Giesl
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34A45
zbMATH DE Number: 5286440
zbMATH Keywords: nonlinear oscillations
zbMATH Keywords: nonsmooth oscillators
zbMATH Keywords: artificial parameter
zbMATH Keywords: generalized functions
zbMATH Keywords: Linstedt-Poincaré technique
author: Juan I. Ramos
MaRDI profile type: Publication
full work available at URL: j.amc.2007.10.035
OpenAlex ID: W2007236364
cites work: AN EXPLICIT FORM GENERAL SOLUTION FOR OSCILLATORS WITH A NON-SMOOTH RESTORING FORCE,ẍ+sign(x)f(x)=0
cites work: Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt--Poincaré method
cites work: An analytical approximate technique for a class of strongly non-linear oscillators
cites work: Perturbation methods and non-linear hyperbolic waves
cites work: Solving frontier problems of physics: the decomposition method
cites work: Oscillators with a generalized power-form elastic term
cites work: Oscillations in an \(x^{(2m+2)/(2n+1)}\) potential
cites work: Oscillations in an \(x^{4/3}\) potential
cites work: Analysis of non-linear oscillators having non-polynomial elastic terms
cites work: A classical iteration procedure valid for certain strongly nonlinear oscillators
cites work: Iteration method solutions for conservative and limit-cycle \(x^{1/3}\) force oscillators
cites work: Solutions of nonlinear oscillators with fractional powers by an iteration procedure
cites work: A modified method of equivalent linearization that works even when the non-linearity is not small
cites work: A convolution integral method for certain strongly nonlinear oscillators
cites work: Determination of periodic solution for a \(u^{1/3}\) force by He's modified Lindstedt-Poincaré method
cites work: Generalized harmonic balance/numerical method for determining analytical approximations to the periodic solutions of the \(x^{4/3}\) potential
cites work: Oscillations of non-linear system with restoring force close to \(\mathrm{sign}(x)\)
cites work: Frequencies of oscillators with fractional-power non-linearities
cites work: Solution of a quadratic nonlinear oscillator by the method of harmonic balance
cites work: Nonlinear oscillator with discontinuity by parameter-expansion method
cites work: Q5724171
cites work: Q3244676
cites work: Q5821083
cites work: The theory of generalized functions
cites work: Theories and Applications of Generalized Functions
cites work: Piecewise-linearized methods for oscillators with fractional-power nonlinearities
cites work: Piecewise-linearized methods for oscillators with limit cycles
cites work: Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods
cites work: Q3665013
cites work: Derivatives of the Dirac delta function by explicit construction of sequences
cites work: Nonlinear trigonometric approximation and the Dirac delta function
cites work: Q3671491
DOI: 10.1016/J.AMC.2007.10.035
description: scientific article; zbMATH DE number 5286440