Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator

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Title: Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator
Authors: Beléndez, Augusto | Arribas Garde, Enrique | Beléndez, Tarsicio | Pascual, Carolina | Gimeno, Encarnación | Alvarez, Mariela L.
Research Group/s: Holografía y Procesado Óptico
Center, Department or Service: Universidad de Alicante. Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal | Universidad de Alicante. Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías | Universidad de Castilla-La Mancha. Departamento de Física Aplicada
Keywords: Nonlinears oscillators | Conservative systems | Exact solution | Dynamical systems | Quintic nonlinear oscillator | Jacobian elliptic functions | Symbolic computation
Knowledge Area: Física Aplicada
Date Created: 6-Nov-2016
Issue Date: 2-Feb-2017
Publisher: Hindawi Publishing Corporation
Citation: Advances in Mathematical Physics, Vol. 2017, Article ID 7396063, 14 pages (2017). doi:10.1155/2017/7396063
Abstract: 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.
Sponsor: This work was supported by the “Generalitat Valenciana” of Spain, under Project PROMETEOII/2015/015 and by the Universidad de Alicante, Spain, under Project GITE-09006-UA.
ISSN: 1687-9120 (Print) | 1687-9139 (Online)
DOI: 10.1155/2017/7396063
Language: eng
Type: info:eu-repo/semantics/article
Rights: Copyright © 2017 Augusto Beléndez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Peer Review: si
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