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Chaotic dynamics of the fractionally damped Duffing equation

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  • Sheu, Long-Jye
  • Chen, Hsien-Keng
  • Chen, Juhn-Horng
  • Tam, Lap-Mou

Abstract

Vibration phenomena of the fractionally damped systems have attracted increasing attentions in recent years. In this paper, dynamics of the fractionally damped Duffing equation is examined. The fractionally damped Duffing equation is transformed into a set of fractional integral equations solved by a predictor–corrector method. The effect of fractional order of damping on the dynamic behaviors of the motion is the main subject of the study. In this work, bifurcation of the parameter-dependent system is drawn numerically. The time evolutions of the nonlinear dynamical system responses are also described in phase portraits and the Poincaré map technique. In addition, the occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. Results obtained from this study illustrates that the fractional order of damping has a significant effect on the dynamic behaviors of the motion. The size of the attractor trends to enlarge when fractional order α increases. Regular motions (including period-3 motion) and chaotic motions are examined. Moreover, a period doubling route to chaos is also found. Many period-3 windows are also observed in bifurcation diagram.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:4:p:1459-1468
    DOI: 10.1016/j.chaos.2005.11.066
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    References listed on IDEAS

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    1. Li, Chunguang & Chen, Guanrong, 2004. "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 55-61.
    2. Gao, Xin & Yu, Juebang, 2005. "Chaos in the fractional order periodically forced complex Duffing’s oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1097-1104.
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    Cited by:

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    2. Barba-Franco, J.J. & Gallegos, A. & Jaimes-Reátegui, R. & Pisarchik, A.N., 2022. "Dynamics of a ring of three fractional-order Duffing oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    3. Ali, Irfan & Masood, W. & Rizvi, H. & Alrowaily, Albandari W. & Ismaeel, Sherif M.E. & El-Tantawy, S.A., 2023. "Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    4. 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.
    5. Tam, Lap Mou & Si Tou, Wai Meng, 2008. "Parametric study of the fractional-order Chen–Lee system," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 817-826.
    6. Asiain, Erick & Garrido, Rubén, 2021. "Anti-Chaos control of a servo system using nonlinear model reference adaptive control," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    7. Zhang, Weiwei & Zhou, Shangbo & Li, Hua & Zhu, Hao, 2009. "Chaos in a fractional-order Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1684-1691.
    8. María Pilar Velasco & David Usero & Salvador Jiménez & Luis Vázquez & José Luis Vázquez-Poletti & Mina Mortazavi, 2020. "About Some Possible Implementations of the Fractional Calculus," Mathematics, MDPI, vol. 8(6), pages 1-22, June.

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