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

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Abstract

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.

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References

  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)

    Article  MathSciNet  Google Scholar 

  2. Zalygina, V.I.: Lyapunov equivalence of systems with unbounded coefficients. J. Math. Sci. 210(2), 210–216 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Sinha, S.C., Srinivasan, P.: A weighted mean square method of linearization in non-linear oscillations. J. Sound Vib. 16, 139–148 (1971)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  5. Yuste, S.B.: Cubication of non-linear oscillators using the principle of harmonic balance. Int. J. Nonlinear Mech. 27, 347–356 (1992)

    Article  MATH  Google Scholar 

  6. Belendez, A., Alvarez, M.L., Fernandez, E., Pascual, I.: Cubication of conservative nonlinear oscillators. Eur. J. Phys. 30, 973–981 (2009)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  10. Elías-Zúñiga, A.: Quintication method to obtain approximate analytical solutions of non-linear oscillators. Appl. Math. Comput. 243, 849–855 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Parks, P.C.: A. M. Lyapunov’s stability theory—100 years on. IMA J. Math. Control Inf. 9(4), 275–303 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cai, J., Wu, X., Li, Y.P.: An equivalent nonlinearization method for strongly nonlinear oscillations. Mech. Res. Commun. 32, 553–560 (2005)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Butikov, E.I.: Extraordinary oscillations of an ordinary forced pendulum. Eur. J. Phys. 29, 215–233 (2008)

    Article  MATH  Google Scholar 

  15. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica 16D, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Sandri, M.: Numerical calculations of Lyapunov exponents. Math. J. 6, 78–84 (1996)

    Google Scholar 

  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)

    Article  Google Scholar 

  18. Taylor, R.L.V.: Attractors: nonstrange to chaotic, Society for Industrial and Applied Mathematics. Undergrad. Res. Online 21(6), 72–80 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pilipchuk, V.N.: Oscillators with a generalized power-form elastic term. J. Sound Vib. 270, 470–472 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pilipchuk, V.N.: Strongly nonlinear vibrations of damped oscillators with two nonsmooth limits. J. Sound Vib. 302, 398–402 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kovacik, I.: Forced vibrations of oscillators with a purely nonlinear power-form restoring force. J. Sound Vib. 330, 4313–4327 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cao, Q., Xiong, Y., Wiercigroch, M.: A novel model of dipteran flight mechanism. Int. J. Dyn. Control 1, 1–11 (2013)

    Article  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Escuela de Ingeniería y Ciencias, Tecnologico de Monterrey, Ave. Eugenio Garza Sada 2501, 64849, Monterrey, NL, México

    Alex Elías-Zúñiga, Oscar Martínez-Romero & Daniel Olvera-Trejo

  2. División de Estudios de Posgrado e Investigación, Tecnológico Nacional de México / Instituto Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, 42080, Pachuca, HGO, México

    Luis Manuel Palacios-Pineda

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  1. Alex Elías-Zúñiga
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  2. Luis Manuel Palacios-Pineda
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  3. Oscar Martínez-Romero
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  4. Daniel Olvera-Trejo
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Correspondence to Alex Elías-Zúñiga.

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Dedicated to Octavio Herrera Giammattei on the occasion of his 80th birthday.

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Elías-Zúñiga, A., Palacios-Pineda, L.M., Martínez-Romero, O. et al. Equivalent representation form in the sense of Lyapunov, of nonlinear forced, damped second-order differential equations. Nonlinear Dyn 92, 2143–2158 (2018). https://doi.org/10.1007/s11071-018-4186-1

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  • DOI: https://doi.org/10.1007/s11071-018-4186-1

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