IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v99y2017icp116-123.html
   My bibliography  Save this article

Feigenbaum's constants in reverse bifurcation of fractional-order Rössler system

Author

Listed:
  • Li, Zengshan
  • Chen, Diyi
  • Ma, Mengmeng
  • Zhang, Xinguang
  • Wu, Yonghong

Abstract

This paper demonstrates the existence of Feigenbaum's constants in reverse bifurcation for fractional-order Rössler system. First, the numerical algorithm of fractional-order Rössler system is presented. Then, the definition of Feigenbaum's constants in reverse bifurcation is provided. Third, in order to observe the effect of fractional-order to Feigenbaum's constants in reverse bifurcation, a series of bifurcation diagrams are computed. The Feigenbaum's constants in reverse bifurcation are measured and the error percentage in fractional-order Rössler system is presented. The simulation results show that Feigenbaum's constants exist in reverse bifurcation for fractional-order Rössler system. Especially, the Feigenbaum's constants still exist in the periodic windows. A summary on previous others’ works about Feigenbaum's constants is proposed. This paper draw a conclusion that the constants are universal in both period-doubling bifurcation and reverse bifurcation for both integer and fractional-order system.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:99:y:2017:i:c:p:116-123
    DOI: 10.1016/j.chaos.2017.03.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077917300668
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2017.03.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Letellier, Christophe & Bennoud, Mounia & Martel, Gilles, 2007. "Intermittency and period-doubling cascade on tori in a bimode laser model," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 782-794.
    3. Yu, Yongguang & Li, Han-Xiong & Wang, Sha & Yu, Junzhi, 2009. "Dynamic analysis of a fractional-order Lorenz chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1181-1189.
    4. 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.
    5. 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.
    6. Goldfain, Ervin, 2006. "Feigenbaum scaling, Cantorian space–time and the hierarchical structure of standard model parameters," Chaos, Solitons & Fractals, Elsevier, vol. 30(2), pages 324-331.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Li, Peiluan & Gao, Rong & Xu, Changjin & Ahmad, Shabir & Li, Ying & Akgül, Ali, 2023. "Bifurcation behavior and PDγ control mechanism of a fractional delayed genetic regulatory model," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ge, Zheng-Ming & Yi, Chang-Xian, 2007. "Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 42-61.
    2. Hanshuo Qiu & Xiangzi Zhang & Huaixiao Yue & Jizhao Liu, 2023. "A Novel Eighth-Order Hyperchaotic System and Its Application in Image Encryption," Mathematics, MDPI, vol. 11(19), pages 1-29, September.
    3. 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).
    4. Zheng, Yongai & Ji, Zhilin, 2016. "Predictive control of fractional-order chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 307-313.
    5. Laarem, Guessas, 2021. "A new 4-D hyper chaotic system generated from the 3-D Rösslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it’s fractional order model and chaos sy," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    6. Petráš, Ivo, 2008. "A note on the fractional-order Chua’s system," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 140-147.
    7. Leng, Xiangxin & Gu, Shuangquan & Peng, Qiqi & Du, Baoxiang, 2021. "Study on a four-dimensional fractional-order system with dissipative and conservative properties," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    8. Silva-Juárez, Alejandro & Tlelo-Cuautle, Esteban & de la Fraga, Luis Gerardo & Li, Rui, 2021. "Optimization of the Kaplan-Yorke dimension in fractional-order chaotic oscillators by metaheuristics," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    9. Junmei Guo & Chunrui Ma & Xinheng Wang & Fangfang Zhang & Michaël Antonie van Wyk & Lei Kou, 2021. "A New Synchronization Method for Time-Delay Fractional Complex Chaotic System and Its Application," Mathematics, MDPI, vol. 9(24), pages 1-20, December.
    10. Kamal, F.M. & Elsonbaty, A. & Elsaid, A., 2021. "A novel fractional nonautonomous chaotic circuit model and its application to image encryption," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    11. Deepika, Deepika & Kaur, Sandeep & Narayan, Shiv, 2018. "Uncertainty and disturbance estimator based robust synchronization for a class of uncertain fractional chaotic system via fractional order sliding mode control," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 196-203.
    12. Ravi P. Agarwal & Snezhana Hristova & Donal O’Regan, 2023. "Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory," Mathematics, MDPI, vol. 11(18), pages 1-23, September.
    13. Andrade, Dana I. & Specchia, Stefania & Fuziki, Maria E.K. & Oliveira, Jessica R.P. & Tusset, Angelo M. & Lenzi, Giane G., 2024. "Dynamic analysis and SDRE control applied in a mutating autocatalyst with chaotic behavior," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    14. Runlong Peng & Cuimei Jiang & Rongwei Guo, 2021. "Partial Anti-Synchronization of the Fractional-Order Chaotic Systems through Dynamic Feedback Control," Mathematics, MDPI, vol. 9(7), pages 1-13, March.
    15. Ahmad, Shabir & Ullah, Aman & Akgül, Ali, 2021. "Investigating the complex behaviour of multi-scroll chaotic system with Caputo fractal-fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    16. 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.
    17. Chiou, Juing-Shian & Cheng, Chun-Ming, 2009. "Stabilization analysis of the switched discrete-time systems using Lyapunov stability theorem and genetic algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 751-759.
    18. Yildirim, Gokce & Tanyildizi, Erkan, 2023. "An innovative approach based on optimization for the determination of initial conditions of continuous-time chaotic system as a random number generator," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    19. 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.
    20. Gilardi-Velázquez, H.E. & Echenausía-Monroy, J.L. & Jaimes-Reátegui, R. & García-López, J.H. & Campos, Eric & Huerta-Cuellar, G., 2022. "Deterministic coherence resonance analysis of coupled chaotic oscillators: fractional approach," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:99:y:2017:i:c:p:116-123. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.