Nonlinear Dynamics

, Volume 92, Issue 4, pp 2143–2158 | Cite as

Equivalent representation form in the sense of Lyapunov, of nonlinear forced, damped second-order differential equations

  • Alex Elías-Zúñiga
  • Luis Manuel Palacios-Pineda
  • Oscar Martínez-Romero
  • Daniel Olvera-Trejo
Original Paper


This paper focuses on finding in the sense of Lyapunov, the equivalent forced, damped cubic–quintic Duffing equation of nonlinear forced damped, second-order ordinary differential equations that could contain rational or irrational restoring elastic terms. The accuracy obtained from the equivalent expressions of dynamical systems such as the generalized pendulum equation, the power-form elastic term oscillator, oscillatory systems with irrational elastic restoring forces that modeled the motion of a mass attached to two stretched elastic springs, and the motion of the mechanism of dipteran flight motor, is numerically evaluated by computing their corresponding Lyapunov characteristic exponents, the amplitude–frequency, the amplitude–time, phase portraits, Poincare’s maps, and their Kaplan–Yorke dimension plots.


Forced nonlinear oscillators Equivalent restoring forces Power-form elastic term oscillators Lyapunov characteristic exponents Chaos 



This work was funded by the Tecnoló gico de Monterrey, Campus Monterrey through the Research Group in Nanomaterials and Devices Design, and by Conacyt Project Numbers 242269, 255837, 242269, and 255837.


  1. 1.
    Koshlyakov, V.N., Makarov, V.L.: Mechanical systems, equivalent in Lyapunov’s sense to systems not containing non-conservative positional forces. J. Appl. Math. Mech. 71, 10–19 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Zalygina, V.I.: Lyapunov equivalence of systems with unbounded coefficients. J. Math. Sci. 210(2), 210–216 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Sinha, S.C., Srinivasan, P.: A weighted mean square method of linearization in non-linear oscillations. J. Sound Vib. 16, 139–148 (1971)CrossRefMATHGoogle Scholar
  4. 4.
    Yuste, S.B., Sánchez, A.M.: A weighted mean-square method of cubication for non-linear oscillators. J. Sound Vib. 134, 423–433 (1989)CrossRefMATHGoogle Scholar
  5. 5.
    Yuste, S.B.: Cubication of non-linear oscillators using the principle of harmonic balance. Int. J. Nonlinear Mech. 27, 347–356 (1992)CrossRefMATHGoogle Scholar
  6. 6.
    Belendez, A., Alvarez, M.L., Fernandez, E., Pascual, I.: Cubication of conservative nonlinear oscillators. Eur. J. Phys. 30, 973–981 (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Beléndez, A., Méndez, D.I., Fernández, E., Marini, S., Pascual, I.: An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method. Phys. Lett. A 373, 2805–2809 (2009)CrossRefMATHGoogle Scholar
  8. 8.
    Belendez, A., Bernabeu, G., Frances, J., Mendez, D.I., Marini, S.: An accurate closed-form approximate solution for the quintic Duffing oscillator equation. Math. Comput. Model. 52, 637–641 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Elías-Zúñiga, A., Romero-Martínez, O., Córdoba-Díaz, R.K.: Approximate solution for the Duffing-harmonic oscillator by the enhanced cubication method. Math. Probl. Eng. Article ID 618750 (2012)Google Scholar
  10. 10.
    Elías-Zúñiga, A.: Quintication method to obtain approximate analytical solutions of non-linear oscillators. Appl. Math. Comput. 243, 849–855 (2014)MathSciNetMATHGoogle Scholar
  11. 11.
    Parks, P.C.: A. M. Lyapunov’s stability theory—100 years on. IMA J. Math. Control Inf. 9(4), 275–303 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cai, J., Wu, X., Li, Y.P.: An equivalent nonlinearization method for strongly nonlinear oscillations. Mech. Res. Commun. 32, 553–560 (2005)CrossRefMATHGoogle Scholar
  13. 13.
    Trueba, J.L., Rams, J., Sanjuan, M.A.F.: Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators. Int. J. Bifurc. Chaos 10(9), 2257–2267 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Butikov, E.I.: Extraordinary oscillations of an ordinary forced pendulum. Eur. J. Phys. 29, 215–233 (2008)CrossRefMATHGoogle Scholar
  15. 15.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica 16D, 285–317 (1985)MathSciNetMATHGoogle Scholar
  16. 16.
    Sandri, M.: Numerical calculations of Lyapunov exponents. Math. J. 6, 78–84 (1996)Google Scholar
  17. 17.
    Hauptfleisch, H., Gasenzer, T., Meier, K., Nachtmann, O., Schemmel, J.: A computer controlled pendulum with position readout. Am. J. Phys. 78(6), 555–561 (2010)CrossRefGoogle Scholar
  18. 18.
    Taylor, R.L.V.: Attractors: nonstrange to chaotic, Society for Industrial and Applied Mathematics. Undergrad. Res. Online 21(6), 72–80 (2011)CrossRefGoogle Scholar
  19. 19.
    Pilipchuk, V.N.: Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformations. J. Sound Vib. 192(1), 43–64 (1996)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pilipchuk, V.N.: Oscillators with a generalized power-form elastic term. J. Sound Vib. 270, 470–472 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pilipchuk, V.N.: Strongly nonlinear vibrations of damped oscillators with two nonsmooth limits. J. Sound Vib. 302, 398–402 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kovacik, I.: Forced vibrations of oscillators with a purely nonlinear power-form restoring force. J. Sound Vib. 330, 4313–4327 (2011)CrossRefGoogle Scholar
  23. 23.
    Lai, S.K., Xiang, Y.: Application of a generalized Senator Bapat perturbation technique to nonlinear dynamical systems with an irrational restoring force. Comput. Math. Appl. 60, 2078–2086 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Cao, Q., Xiong, Y., Wiercigroch, M.: A novel model of dipteran flight mechanism. Int. J. Dyn. Control 1, 1–11 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Alex Elías-Zúñiga
    • 1
  • Luis Manuel Palacios-Pineda
    • 2
  • Oscar Martínez-Romero
    • 1
  • Daniel Olvera-Trejo
    • 1
  1. 1.Escuela de Ingeniería y CienciasTecnologico de MonterreyMonterreyMéxico
  2. 2.División de Estudios de Posgrado e InvestigaciónTecnológico Nacional de México / Instituto Tecnológico de PachucaPachucaMéxico

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