IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v329y2018icp1-4.html
   My bibliography  Save this article

A note on “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli

Author

Listed:
  • Wang, Haijun
  • Li, Xianyi

Abstract

In the recent paper entitled “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli, they proposed the following new four-dimensional (4-D) quadratic autonomous hyper-chaotic system: x1˙=a(x2−x1),x2˙=bx1−x2+ex4−x1x3,x3˙=−cx3+x1x2+x12,x4˙=−dx2, which generates double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Combining theoretical analysis and numerical simulations, they investigated some dynamical properties of that system like Lyapunov exponent spectrum, bifurcation diagram, phase portrait, Hopf bifurcation, etc. In particular, they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled “Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems” [Int. J. Bifurc. Chaos, 21(09) (2011), 2679–2694] by P. Wang et al. However, by means of detailed theoretical analysis, we show that both the conclusion itself and the derivation of its proof in [Appl. Math. Comput. 291 (2016) 323–339] are erroneous. Furthermore, we point out that the method adopted for studying the ultimate bound of that system is not applicable at all. Therefore, the ultimate bound estimation of that system needs further studying in future work.

Suggested Citation

  • Wang, Haijun & Li, Xianyi, 2018. "A note on “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 1-4.
    • Handle: RePEc:eee:apmaco:v:329:y:2018:i:c:p:1-4
    DOI: 10.1016/j.amc.2018.01.027
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300318300481
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2018.01.027?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. Zhiqin Qiao & Xianyi Li, 2014. "Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 20(3), pages 264-283, May.
    2. Zarei, Amin & Tavakoli, Saeed, 2016. "Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 323-339.
    3. Sun, Yeong-Jeu, 2009. "Solution bounds of generalized Lorenz chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 691-696.
    4. Li, Damei & Wu, Xiaoqun & Lu, Jun-an, 2009. "Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1290-1296.
    5. Zhang, Fuchen & Shu, Yonglu & Yang, Hongliang & Li, Xiaowu, 2011. "Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 137-144.
    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. Wang, Haijun & Dong, Guili, 2019. "New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 272-286.

    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. Wang, Haijun & Dong, Guili, 2019. "New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 272-286.
    2. Zhang, Fuchen & Shu, Yonglu & Yang, Hongliang & Li, Xiaowu, 2011. "Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 137-144.
    3. Li, Lijie & Feng, Yu & Liu, Yongjian, 2016. "Dynamics of the stochastic Lorenz-Haken system," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 670-678.
    4. Shu, Yonglu & Xu, Hongxing & Zhao, Yunhong, 2009. "Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2852-2857.
    5. Zhang, Ge & Ma, Jun & Alsaedi, Ahmed & Ahmad, Bashir & Alzahrani, Faris, 2018. "Dynamical behavior and application in Josephson Junction coupled by memristor," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 290-299.
    6. Jian, Jigui & Wu, Kai & Wang, Baoxian, 2020. "Global Mittag-Leffler boundedness and synchronization for fractional-order chaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    7. Zhang, Fuchen & Shu, Yonglu, 2015. "Global dynamics for the simplified Lorenz system model," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 53-60.
    8. Ren, Lei & Lin, Ming-Hung & Abdulwahab, Abdulkareem & Ma, Jun & Saberi-Nik, Hassan, 2023. "Global dynamical analysis of the integer and fractional 4D hyperchaotic Rabinovich system," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    9. Kim, Seong-S. & Choi, Han Ho, 2014. "Adaptive synchronization method for chaotic permanent magnet synchronous motor," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 101(C), pages 31-42.
    10. Gao, Wei & Yan, Li & Saeedi, Mohammadhossein & Saberi Nik, Hassan, 2018. "Ultimate bound estimation set and chaos synchronization for a financial risk system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 154(C), pages 19-33.
    11. Wu, Jiening & Wang, Lidan & Chen, Guanrong & Duan, Shukai, 2016. "A memristive chaotic system with heart-shaped attractors and its implementation," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 20-29.
    12. Bazán Navarro, Ciro Eduardo & Benazic Tomé, Renato Mario, 2024. "Qualitative behavior in a fractional order IS-LM-AS macroeconomic model with stability analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 425-443.
    13. Zarei, Amin & Tavakoli, Saeed, 2016. "Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 323-339.
    14. Yu, Jinwei & Xie, Wei & Zhong, Zhenyu & Wang, Huan, 2022. "Image encryption algorithm based on hyperchaotic system and a new DNA sequence operation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    15. Samia Rezzag & Fuchen Zhang, 2022. "On the Dynamics of New 4D and 6D Hyperchaotic Systems," Mathematics, MDPI, vol. 10(19), pages 1-10, October.

    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:apmaco:v:329:y:2018:i:c:p:1-4. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.