The influence of two kinds of time delays on the vibrational resonance of a fractional Mathieu–Duffing oscillator

Abstract

Vibrational resonance is studied in a fractional Mathieu–Duffing oscillator with two types of time delays: fixed and distributed delays. The theoretical expression of the response amplitude is obtained by utilising the method of direct partition of slow and fast motions. Relative errors between the theoretical prediction and the numerical simulation are introduced to verify the validity of analytical approaches. The relative error of the displacement and the relative error of the response amplitude are calculated. Small relative errors show that the theoretical analysis is statistically correct. Therefore, the effects of fractional order, linear stiffness coefficient, low-frequency signal, time delay intensity and damping coefficient on the Mathieu–Duffing oscillator with distributed delay are studied successively. In order to better illustrate the impact of distributed time delay on the model, the case of fixed time delay is analysed and compared, and it can be found that the distributed delay has more significant influence than fixed delay on the system. In addition, the influence of distributed delay on the system is more significant than that of the fixed delay.

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References

  1. 1.

    R Benzi, A Sutera and A Vulpianni, J. Phys. A 14, L453 (1981)

    ADS  Google Scholar 

  2. 2.

    G Litak and M Borowiec, Nonlinear Dyn. 77, 681 (2014)

    Google Scholar 

  3. 3.

    J H Yang, M A F Sanjuán, H G Liu, G Litak and X Li, Commun. Nonlinear Sci. Numer. Simul. 41, 104 (2016)

    ADS  MathSciNet  Google Scholar 

  4. 4.

    M Gitterman, J. Phys. A 34, L355 (2001)

    ADS  MathSciNet  Google Scholar 

  5. 5.

    V N Chizhevsky, E Smeu and G Giacomelli, Phys. Rev. Lett. 91, 220602 (2003)

    ADS  Google Scholar 

  6. 6.

    V N Chizhevsky and G Giacomelli, Phys. Rev. A 71, 011801 (2005)

    ADS  Google Scholar 

  7. 7.

    P S Landa and P V E McClintock, J. Phys. A33, L433 (2000)

    ADS  Google Scholar 

  8. 8.

    B Deng, J Wang, X L Wei, K M Tsang and W L Chan, Chaos 20, 013113 (2010)

    ADS  Google Scholar 

  9. 9.

    Y M Qin, J Wang, C Men, B Deng and X L Wei, Chaos 21, 023133 (2011)

    ADS  Google Scholar 

  10. 10.

    J B Sun, B Deng, C Liu, H T Yu, J Wang, X L Wei and J Zhao, Appl. Mech. Rev. 37, 6311 (2013)

    Google Scholar 

  11. 11.

    J P Baltanás, L Lopez, I I Blechman, P S Landa, A Zaikin, J Kurths and M A F Sanjuán, Phys. Rev. E 67, 066119 (2003)

    ADS  Google Scholar 

  12. 12.

    D L Hu, J H Yang and X B Liu, Commun. Nonlinear Sci. Numer. Simul. 17, 1031 (2012)

    ADS  MathSciNet  Google Scholar 

  13. 13.

    D L Hu, J H Yang and X B Liu, Comput. Biol. Med. 45, 80 (2014)

    Google Scholar 

  14. 14.

    A Daza, A Wagemakers, S Rajasekar and M A F Sanjuán, Commun. Nonlinear Sci. Numer. Simul. 18, 400 (2013)

    Google Scholar 

  15. 15.

    V N Chizhevsky and G G Iacomelli, Phys. Rev. E 70, 062101 (2004)

    ADS  Google Scholar 

  16. 16.

    C G Yao and M Zhan, Phys. Rev. E 81, 061129 (2010)

    ADS  Google Scholar 

  17. 17.

    C J Fang and X B Liu, Chin. Phys. Lett. 29, 050504 (2012)

    ADS  Google Scholar 

  18. 18.

    J H Yang and X B Liu, J. Phys. A 43, 122001 (2010)

    ADS  MathSciNet  Google Scholar 

  19. 19.

    T L M D Mbong, M Siewe and C Tchawoua, Mech. Res. Commun78, 13 (2016)

    Google Scholar 

  20. 20.

    F Yang and K Q Zhu, Theor. Appl. Mech. Lett. 1, 012007 (2011)

    Google Scholar 

  21. 21.

    F C Meral, T J Royston and R Magin, Commun. Nonlinear Sci. Numer. Simul. 15, 939 (2010)

    ADS  MathSciNet  Google Scholar 

  22. 22.

    Y Q Chen, R T Sun, A H Zhou and N Zaveri, J. Vib. Control 14, 9 (2008)

    Google Scholar 

  23. 23.

    K B Oldham, Adv. Eng. Softw. 41, 9 (2010)

    Google Scholar 

  24. 24.

    F Mainardi, Fractional calculus, in: Fractals and fractional calculus in continuum mechanics (Springer, 1997) pp. 291–348

  25. 25.

    Y A Rossikhin and M V Shitikova, Appl. Mech. Rev. 63, 010801 (2010)

    ADS  Google Scholar 

  26. 26.

    T L M D Mbong, M Siewe and C Tchawoua, Commun. Nonlinear Sci. Numer. Simul. 22, 228 (2015)

    ADS  MathSciNet  Google Scholar 

  27. 27.

    M Li, Math. Prob. Eng. 2010 (2010)

  28. 28.

    O P Agrawal, Nonlinear Dyn. 38, 323 (2004)

    Google Scholar 

  29. 29.

    S Das and I Pan, Fractional-order signal processing: Introductory concepts an applications (Springer Science & Business Media, 2011)

  30. 30.

    J H Yang, Chin. Phys. Lett. 29, 104501 (2012)

    ADS  Google Scholar 

  31. 31.

    J H Yang and H Zhu, Chaos 22, 013112 (2012)

    ADS  MathSciNet  Google Scholar 

  32. 32.

    M Li, Symmetry 10, 40 (2018)

    Google Scholar 

  33. 33.

    N F Pedersen, M R Samuelsen and K Særmark, J. Appl. Phys. 44, 5120 (1973)

    ADS  Google Scholar 

  34. 34.

    V Kaajakari and A Lal, Appl. Phys. Lett. 85, 3923 (2004)

    ADS  Google Scholar 

  35. 35.

    M A Mironov, P A Pyatakov, I I Konopatskaya, G T Clement and N I Vykhodtseva, Acoust. Phys. 55, 567 (2009)

    ADS  Google Scholar 

  36. 36.

    M H El Ouni, N B Kahla and A Preumont, Eng. Struct. 45, 244 (2012)

    Google Scholar 

  37. 37.

    H Plat and I Bucher, J. Sound. Vib. 333, 1408 (2014)

    ADS  Google Scholar 

  38. 38.

    J H Yang, M A F Sanjuán and H G Liu, Eur. Phys. J. B 88, 310 (2015)

    ADS  Google Scholar 

  39. 39.

    Z J Chen and L J Ning, Pramana – J. Phys. 90: 49 (2018)

    ADS  Google Scholar 

  40. 40.

    Z L Yang and L J Ning, Pramana – J. Phys. 92: 89 (2019)

    ADS  Google Scholar 

  41. 41.

    R J Yatawara, R D Neilson and A D S Barr, J. Sound. Vib. 297, 962 (2006)

    ADS  Google Scholar 

  42. 42.

    M Belhaq and S M Sah, Commun. Nonlinear. Sci. Numer. Simul. 13, 1706 (2008)

    ADS  Google Scholar 

  43. 43.

    A Fidlin and J J Thomsen, Int. J. Non-Linear. Mech. 43, 569 (2008)

    ADS  Google Scholar 

  44. 44.

    J J Thomsen, J. Sound. Vib. 311, 1249 (2008)

    ADS  Google Scholar 

  45. 45.

    B Horton, J Sieber, J M T Thompson and M Wiercigroch, Int. J. Non-Linear Mech. 46, 436 (2011)

    ADS  Google Scholar 

  46. 46.

    L Mokni, M Belhaq and F Lakrad, Commun. Nonlinear. Sci. Numer. Simul. 16, 1720 (2011)

    ADS  Google Scholar 

  47. 47.

    R H Huan, W Q Zhu, F Ma and Z H Liu, Shock Vib. 2014, 1 (2014)

    Google Scholar 

  48. 48.

    Y L Song and Y H Peng, Appl. Math. Comput. 181, 1745 (2006)

    MathSciNet  Google Scholar 

  49. 49.

    J Wu, X S Zhan, X H Zhang and H L Gao, Chin. Phys. Lett. 29, 050203 (2012)

    ADS  Google Scholar 

  50. 50.

    A Mesbahi, M Haeri, M Nazari and E A Butcher, Int. J. Control 88, 622 (2015)

    Google Scholar 

  51. 51.

    R Caponetto, G Dongola, L Fortuna and I Petrás̆, (World Scientific, 2010)

  52. 52.

    I Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their appliactions (Elsevier, 1998) Vol. 198

  53. 53.

    C A Monje, Y Q Chen, B M Vinagre, D Y Xue and V Feliu Batlle, Fractional-order systems and controls: Fundamentals and applications (Springer Science & Business Media, 2010)

  54. 54.

    J M Cushing, J. Math. Biol. 4(3), 257 (1977)

    MathSciNet  Google Scholar 

  55. 55.

    I I Blekhman, Vibrational mechanics: Nonlinear dynamic effects, general approach, applications (World Scientific, 2000)

  56. 56.

    J H Yang, Diffraction and resonance in fractional-order systems (Science Press, Beijing, 2017)

    Google Scholar 

Download references

Acknowledgements

This work was partially funded by the National Natural Science Foundation of China under Grant No. 11202120 and the Fundamental Research Funds for the Central Universities under No. GK201901008.

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Correspondence to Lijuan Ning.

Appendix A. An appendix section

Appendix A. An appendix section

In light of \({\mathrm{eq}.}\) (5) and \(x=X+\Psi \),

$$\begin{aligned} Z\left( {t,x\left( t \right) } \right)= & {} \int _{ - \infty }^t {\sigma {\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }} \left( {x\left( \tau \right) } \right) \,\mathrm {d}\tau } \nonumber \\= & {} \int _{ - \infty }^t {\sigma {\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }}\left( {X\left( \tau \right) + \Psi \left( \tau \right) } \right) \,\mathrm {d}\tau } \nonumber \\= & {} Z\left( {t,X\left( t \right) } \right) + Z\left( {t,\Psi \left( t \right) } \right) . \end{aligned}$$
(A.1)

For \(\Psi (t) = {A_H}\cos \left( {\Omega t + {\Phi _H}} \right) \), then

$$\begin{aligned} Z\left( {t,\Psi \left( t \right) } \right)= & {} \int _{ - \infty }^t {\sigma {\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }}\Psi \left( \tau \right) \,\mathrm {d}\tau }\nonumber \\= & {} \sigma {A_H}\int _{ - \infty }^t {{\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }}\cos ( {\Omega \tau + {\Phi _H}} ) \,\mathrm {d}\tau } \nonumber \\= & {} \sigma {A_H}{\mathrm {e}^{ - \sigma t}} \int _{ - \infty }^t {{\mathrm {e}^{\sigma \tau }} \cos ( {\Omega \tau + {\Phi _H}} )\,\mathrm {d}\tau }.\nonumber \\ \end{aligned}$$
(A.2)

According to the divisional integration method, the integral value in (A.2) is

$$\begin{aligned}&\int _{ - \infty }^t {{\mathrm {e}^{\sigma \tau }} \cos ( {\Omega \tau + {\Phi _H}} ) \,\mathrm {d}\tau } \nonumber \\&\quad = \frac{{{\mathrm {e}^{\sigma t}}}}{{\sqrt{{\Omega ^2} +{\sigma ^2}} }}\sin \left( {\Omega t + {\Phi _H} +\arctan \frac{\sigma }{\Omega }} \right) . \end{aligned}$$

We have

$$\begin{aligned} \mathrm{Z}\left( {t,\Psi \left( t \right) } \right) =\frac{{\sigma {A_H}}}{{\sqrt{{\Omega ^2} +{\sigma ^2}} }}\sin ( {\Omega t + {\Phi _H} + \theta }) \end{aligned}$$
(A.3)

where

$$\begin{aligned} \theta = \arctan \frac{\sigma }{\Omega }. \end{aligned}$$

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Ning, L., Guo, W. The influence of two kinds of time delays on the vibrational resonance of a fractional Mathieu–Duffing oscillator. Pramana - J Phys 94, 40 (2020). https://doi.org/10.1007/s12043-019-1905-1

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Keywords

  • Fractional Mathieu–Duffing oscillator
  • distributed delay
  • fixed time delay
  • relative error

PACS Nos

  • 12.60.Jv
  • 12.10.Dm
  • 98.80.Cq
  • 11.30.Hv