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.


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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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