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An analytical criterion for jump phenomena in fractional Duffing oscillators

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  • Liu, Q.X.
  • Liu, J.K.
  • Chen, Y.M.

Abstract

This paper presents an analytical study on a general kind of fractional Duffing oscillators subjected to harmonic excitations. The Caputo-type fractional derivatives are transformed into improper double integrals by employing a memory-free principle. The integrals and the cubic stiffness are further handled by equivalent linearization. An equivalent linear equation is then deduced, based on which amplitude–frequency responses can be obtained analytically. According to the attained amplitude–frequency curve, we present an analytical criterion for jump phenomena of the oscillating amplitude due to varying excitation frequency. The analytical results are validated by numerical examples.

Suggested Citation

  • Liu, Q.X. & Liu, J.K. & Chen, Y.M., 2017. "An analytical criterion for jump phenomena in fractional Duffing oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 216-219.
    • Handle: RePEc:eee:chsofr:v:98:y:2017:i:c:p:216-219
    DOI: 10.1016/j.chaos.2017.03.030
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    References listed on IDEAS

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    1. Liu, Q.X. & Liu, J.K. & Chen, Y.M., 2015. "Non-diminishing relative error of the predictor–corrector algorithm for certain fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 117(C), pages 10-19.
    2. Balescu, R., 2007. "V-Langevin equations, continuous time random walks and fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 62-80.
    3. Tofighi, Ali, 2003. "The intrinsic damping of the fractional oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 329(1), pages 29-34.
    4. Yang, Yongge & Xu, Wei & Gu, Xudong & Sun, Yahui, 2015. "Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 190-204.
    5. Achar, B.N.Narahari & Hanneken, J.W. & Enck, T. & Clarke, T., 2001. "Dynamics of the fractional oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 297(3), pages 361-367.
    6. Agrawal, S.K. & Srivastava, M. & Das, S., 2012. "Synchronization of fractional order chaotic systems using active control method," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 737-752.
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    Cited by:

    1. Zhu, Jue & Yuan, Wei-bin & Li, Long-yuan, 2021. "Cross-sectional flattening-induced nonlinear damped vibration of elastic tubes subjected to transverse loads," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. Valentine Kim & Roman Parovik, 2020. "Mathematical Model of Fractional Duffing Oscillator with Variable Memory," Mathematics, MDPI, vol. 8(11), pages 1-14, November.

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