An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

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Title: An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method
Authors: Beléndez, Augusto | Méndez Alcaraz, David Israel | Fernandez-Varo, Helena | Marini, Stephan | Pascual, Inmaculada
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. Departamento de Óptica, Farmacología y Anatomía | Universidad de Alicante. Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías
Keywords: Nonlinear oscillator | Approximate solutions | Duffing-harmonic oscillator | Chebyshev polynomials | Elliptic integrals | Arithmetic-geometric mean
Knowledge Area: Física Aplicada
Date Created: 6-Jun-2009
Issue Date: 15-May-2009
Publisher: Elsevier
Citation: BELÉNDEZ VÁZQUEZ, Augusto, et al. “An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method”. Physics Letters A. Vol. 373, Issue 32 (3 Aug. 2009). ISSN 0375-9601, pp. 2805-2809
Abstract: The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the ‘cubication’ of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this paper predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre’s formula to approximately obtain this mean are used.
Sponsor: This work has been supported by the “Ministerio de Ciencia e Innovación” of Spain, under projects FIS2008-05856-C02-01 and FIS2008-05856-C02-02.
URI: http://hdl.handle.net/10045/11906
ISSN: 0375-9601 (Print) | 1873-2429 (Online)
DOI: 10.1016/j.physleta.2009.05.074
Language: eng
Type: info:eu-repo/semantics/article
Peer Review: si
Publisher version: http://dx.doi.org/10.1016/j.physleta.2009.05.074
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