Abstract
The solutions of a class of nonlinear second-order differential equations with a cubic term in the dependent variable being related to Duffing oscillators are obtained by means of the factorization technique. The Lagrangian, the Hamiltonian and the constant of motion are also found through a correspondence with an autonomous system. A physical example is worked out in this frame.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Holmes, P., Rand, D.: Phase portraits and bifurcations of the non-linear oscillator: \(\ddot{x}+(\alpha+\gamma x^{2})\dot{x}+\beta x+ \delta x^{3}=0\). Int. J. Non-linear Mech. 15, 449–458 (1980)
Holmes, P., Whitley, D.: On the attracting set for Duffing’s equation. Physica D 7, 111–123 (1983)
Ueda, Y.: Survey of regular and chaotic phenomena in the forced Duffing oscillator. Chaos Solitons Fractals 1, 199–231 (1991)
Baltanás, J.P., Trueba, J.L., Sanjuán, M.A.F.: Energy dissipation in nonlinearly damped Duffing oscillator. Physica D 159, 22–34 (2001)
Borowiec, M., Litak, G., Syta, A.: Vibration of the Duffing oscillator: effect of fractional damping. Shock Vib. 14, 29–36 (2007)
Duchesne, B., Fischer, C.W., Gray, C.G., Jeffrey, K.R.: Chaos in the motion of an inverted pendulum: an undergraduate laboratory experiment. Am. J. Phys. 59, 987–992 (1991)
Jones, B.K., Trefan, G.: The Duffing oscillator: a precise electronic analog chaos demonstrator for undergraduate laboratory. Am. J. Phys. 69, 464–469 (2001)
Zaitsev, S., Almog, R., Shtempluck, O., Buks, E.: Nonlinear damping in nanomechanical beam oscillator. arXiv:cond-mat/0503130
Belhaq, M., Lakrad, F.: Prediction of homoclinic bifurcation: the elliptic averaging method. Chaos Solitons Fractals 11, 2251–2258 (2000)
Yan, Z.: A sinh-Gordon equation expansion method to construct doubly periodic solutions for nonlinear differential equations. Chaos Solitons Fractals 16, 291–297 (2003)
Cveticanin, L.: Homotopy-perturbation method for pure nonlinear differential equation. Chaos Solitons Fractals 30, 1221–1230 (2006)
Estévez, P.G., Kuru, Ş., Negro, J., Nieto, L.M.: Travelling wave solutions of two-dimensional Korteweg-de Vries-Burgers and Kadomtsev-Petviashvili equations. J. Phys. A, Math. Gen. 39, 11441–11452 (2006)
Cornejo-Pérez, O., Negro, J., Nieto, L.M., Rosu, H.C.: Travelling-wave solutions for Korteweg-de Vries-Burgers equations through factorization. Found. Phys. 36, 1587–1599 (2006)
Estévez, P.G., Kuru, Ş., Negro, J., Nieto, L.M.: Travelling wave solutions of the generalized Benjamin-Bona-Mohany equation. Chaos Solitons Fractals 40, 2031–2040 (2009)
Cornejo-Pérez, O.: Traveling wave solutions for some factorized nonlinear PDEs. J. Phys. A, Math. Theor. 42, 035204 (2009)
Estévez, P.G., Kuru, Ş., Negro, J., Nieto, L.M.: Factorization of a class of almost linear second-order differential equations. J. Phys. A, Math. Theor. 40, 9819–9824 (2007)
Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)
Clarkson, P.A.: Painlevé equations nonlinear special functions. J. Comput. Appl. Math. 153, 127–140 (2003)
Gettys, W.E., Ray, J.R., Breitenberge, E.: Bohlin’s and other integrals for the damped harmonic oscillator. Am. J. Phys. 49, 162–164 (1981)
Byrd, P.F., Friedman, D.: Handbook of elliptic integrals for engineers and physicists. Springer, Berlin (1954)
Whittaker, E.T., Watson, G.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Estévez, P.G., Kuru, Ş., Negro, J. et al. Solutions of a Class of Duffing Oscillators with Variable Coefficients. Int J Theor Phys 50, 2046–2056 (2011). https://doi.org/10.1007/s10773-010-0560-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-010-0560-6