Closed-form exact solutions for the unforced quintic nonlinear oscillator
type: Item
instance of: scholarly article
title: Closed-form exact solutions for the unforced quintic nonlinear oscillator
zbMATH Open document ID: 1401.34002
DOI: 10.1155/2017/7396063
author: A. Beléndez
author: E. Arribas
author: T. Beléndez
author: C. Pascual
author: Encarnación Gimeno
author: M. L. Alvarez
published in: Advances in Mathematical Physics
publication date: 2018-10-23T00:00:00Z
review text: Summary: Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at \(x = 0\) are also considered.
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 6962665
zbMATH Keywords: closed-form exact solutions
zbMATH Keywords: periodic motion
zbMATH Keywords: complete elliptic integral of the first kind
zbMATH Keywords: periods
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2017/7396063
OpenAlex ID: W2585949672
Wikidata QID: Q59142466
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description: scientific article; zbMATH DE number 6962665