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Analysis on limit cycle of fractional-order van der Pol oscillator

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  • Shen, Yongjun
  • Yang, Shaopu
  • Sui, Chuanyi

Abstract

In this paper the approximately analytical solution of van der Pol (VDP) oscillator with two kinds of fractional-order derivatives is obtained based on averaging method. Two equivalent system parameters, i.e. equivalent damping coefficient and equivalent stiffness coefficient, are defined, which could characterize the effects of the fractional parameters on the limit cycle in fractional-order VDP oscillator. The same points and differences between the traditional integer-order and fractional-order VDP oscillator are analyzed and summarized in detail. The differences are focused on the convergence speed and frequency characteristic of the limit cycle in VDP oscillator. The comparison between the analytical and numerical solution verifies the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the fractional parameters on the convergence speed and frequency characteristic of the limit cycle in fractional-order VDP oscillator are illustrated based on some typical system parameters.

Suggested Citation

  • Shen, Yongjun & Yang, Shaopu & Sui, Chuanyi, 2014. "Analysis on limit cycle of fractional-order van der Pol oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 94-102.
    • Handle: RePEc:eee:chsofr:v:67:y:2014:i:c:p:94-102
    DOI: 10.1016/j.chaos.2014.07.001
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    References listed on IDEAS

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    1. Chen, Juhn-Horng & Chen, Wei-Ching, 2008. "Chaotic dynamics of the fractionally damped van der Pol equation," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 188-198.
    2. Sheu, Long-Jye & Chen, Hsien-Keng & Chen, Juhn-Horng & Tam, Lap-Mou, 2007. "Chaotic dynamics of the fractionally damped Duffing equation," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1459-1468.
    3. Yang, Shaopu & Shen, Yongjun, 2009. "Recent advances in dynamics and control of hysteretic nonlinear systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1808-1822.
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    Cited by:

    1. Niu, Jiangchuan & Liu, Ruyu & Shen, Yongjun & Yang, Shaopu, 2019. "Stability and bifurcation analysis of single-degree-of-freedom linear vibro-impact system with fractional-order derivative," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 14-23.
    2. Amar Debbouche & Juan J. Nieto & Delfim F. M. Torres, 2017. "Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 174(1), pages 7-31, July.
    3. Azhdari, Meysam & Binazadeh, Tahereh, 2022. "A novel adaptive SMC strategy for sustained oscillations in nonlinear sandwich systems based on stable limit cycle approach," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    4. Li, Zengshan & Chen, Diyi & Ma, Mengmeng & Zhang, Xinguang & Wu, Yonghong, 2017. "Feigenbaum's constants in reverse bifurcation of fractional-order Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 116-123.
    5. Xu, Beibei & Chen, Diyi & Zhang, Hao & Wang, Feifei, 2015. "Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 50-61.
    6. Guo, Feng & Wang, Xue-yuan & Qin, Ming-wei & Luo, Xiang-dong & Wang, Jian-wei, 2021. "Resonance phenomenon for a nonlinear system with fractional derivative subject to multiplicative and additive noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 562(C).
    7. Dai, Hongzhe & Zheng, Zhibao & Wang, Wei, 2017. "On generalized fractional vibration equation," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 48-51.
    8. Hakimi, A.R. & Azhdari, M. & Binazadeh, T., 2021. "Limit cycle oscillator in nonlinear systems with multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    9. 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.

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