IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v172y2020icp321-340.html
   My bibliography  Save this article

Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function

Author

Listed:
  • Eshaghi, Shiva
  • Khoshsiar Ghaziani, Reza
  • Ansari, Alireza

Abstract

This paper deals with the stability and bifurcation of equilibria in a new chaotic fractional-order system in the sense of the Caputo fractional derivative with the chaos entanglement function. We derive conditions under which the system undergoes a Hopf bifurcation and obtain critical parameter value in the Hopf bifurcation. Moreover, the linear feedback control technique is used to control and stabilize the system to equilibrium point in order to eliminate the chaotic vibration. We then design control laws to synchronize two identical chaotic fractional-order systems. Furthermore, by means of numerical simulation, we support the validity of analytical results and reveal more dynamical behaviors consisting chaos, local bifurcation, limit cycles, quasiperiodic and asymptotic stability behaviors. We further emphasize that the order of fractional derivative plays significant roles as the chaos controlling parameter and the Hopf bifurcation parameter.

Suggested Citation

  • Eshaghi, Shiva & Khoshsiar Ghaziani, Reza & Ansari, Alireza, 2020. "Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 321-340.
    • Handle: RePEc:eee:matcom:v:172:y:2020:i:c:p:321-340
    DOI: 10.1016/j.matcom.2019.11.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475419303428
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2019.11.009?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. Ferreira, Bianca Borem & de Paula, Aline Souza & Savi, Marcelo Amorim, 2011. "Chaos control applied to heart rhythm dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 44(8), pages 587-599.
    2. Bernd Blasius & Amit Huppert & Lewi Stone, 1999. "Complex dynamics and phase synchronization in spatially extended ecological systems," Nature, Nature, vol. 399(6734), pages 354-359, May.
    3. Wang, Zhen & Xie, Yingkang & Lu, Junwei & Li, Yuxia, 2019. "Stability and bifurcation of a delayed generalized fractional-order prey–predator model with interspecific competition," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 360-369.
    4. Deshpande, Amey S. & Daftardar-Gejji, Varsha & Sukale, Yogita V., 2017. "On Hopf bifurcation in fractional dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 189-198.
    5. Coronel-Escamilla, A. & Gómez-Aguilar, J.F. & Torres, L. & Escobar-Jiménez, R.F. & Valtierra-Rodríguez, M., 2017. "Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 487(C), pages 1-21.
    6. Zhou, Shangbo & Li, Hua & Zhu, Zhengzhou, 2008. "Chaos control and synchronization in a fractional neuron network system," Chaos, Solitons & Fractals, Elsevier, vol. 36(4), pages 973-984.
    7. Ouannas, Adel & Odibat, Zaid & Hayat, Tasawar, 2017. "Fractional analysis of co-existence of some types of chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 215-223.
    8. Zhang Jiangang & Chu Yandong & Du Wenju & Chang Yingxiang & An Xinlei, 2014. "Hopf Bifurcation Analysis in a New Chaotic System with Chaos Entanglement Function," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-13, July.
    9. Galeone, Luciano & Garrappa, Roberto, 2008. "Fractional Adams–Moulton methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 1358-1367.
    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. Xu, Pan & Fu, Wenlong & Lu, Qipeng & Zhang, Shihai & Wang, Renming & Meng, Jiaxin, 2023. "Stability analysis of hydro-turbine governing system with sloping ceiling tailrace tunnel and upstream surge tank considering nonlinear hydro-turbine characteristics," Renewable Energy, Elsevier, vol. 210(C), pages 556-574.
    2. Chen, Yun & Xu, Yanyi & Lin, Qian & Zhang, Xiyong, 2020. "Model and criteria on the global finite-time synchronization of the chaotic gyrostat systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 515-533.
    3. Zain-Aldeen S. A. Rahman & Basil H. Jasim & Yasir I. A. Al-Yasir & Yim-Fun Hu & Raed A. Abd-Alhameed & Bilal Naji Alhasnawi, 2021. "A New Fractional-Order Chaotic System with Its Analysis, Synchronization, and Circuit Realization for Secure Communication Applications," Mathematics, MDPI, vol. 9(20), pages 1-25, October.

    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. Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2018. "Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 175-185.
    2. Li Wu & Yanjun Yang & Binggeng Xie, 2022. "Modeling Analysis on Coupling Mechanisms of Mountain–Basin Human–Land Systems: Take Yuxi City as an Example," Land, MDPI, vol. 11(7), pages 1-16, July.
    3. Čermák, Jan & Nechvátal, Luděk, 2019. "Stability and chaos in the fractional Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 24-33.
    4. Chen, Yun & Xu, Yanyi & Lin, Qian & Zhang, Xiyong, 2020. "Model and criteria on the global finite-time synchronization of the chaotic gyrostat systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 515-533.
    5. Lounis, Fatima & Boukabou, Abdelkrim & Soukkou, Ammar, 2020. "Implementing high-order chaos control scheme for cardiac conduction model with pathological rhythms," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    6. Ahmadi, Ali Akbar & Majd, Vahid Johari, 2009. "Robust synchronization of a class of uncertain chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1092-1096.
    7. Ge, Zheng-Ming & Chang, Ching-Ming & Chen, Yen-Sheng, 2006. "Anti-control of chaos of single time scale brushless dc motors and chaos synchronization of different order systems," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1298-1315.
    8. Jianguang Zhu & Kai Li & Binbin Hao, 2019. "Image Restoration by Second-Order Total Generalized Variation and Wavelet Frame Regularization," Complexity, Hindawi, vol. 2019, pages 1-16, March.
    9. Wu, Tianyu & Huang, Xia & Chen, Xiangyong & Wang, Jing, 2020. "Sampled-data H∞ exponential synchronization for delayed semi-Markov jump CDNs: A looped-functional approach," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    10. Ge, Zheng-Ming & Chang, Ching-Ming, 2009. "Nonlinear generalized synchronization of chaotic systems by pure error dynamics and elaborate nondiagonal Lyapunov function," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1959-1974.
    11. Wang, Xinhe & Lu, Junwei & Wang, Zhen & Li, Yuxia, 2020. "Dynamics of discrete epidemic models on heterogeneous networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    12. Chen, Hsien-Keng, 2005. "Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lü," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1049-1056.
    13. Hoang, Thang Manh, 2011. "Complex synchronization manifold in coupled time-delayed systems," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 48-57.
    14. Ma, Tingting & Meng, Xinzhu & Hayat, Tasawar & Hobiny, Aatef, 2021. "Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    15. Garrappa, Roberto, 2015. "Trapezoidal methods for fractional differential equations: Theoretical and computational aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 110(C), pages 96-112.
    16. Zhang, Jianlin & Bao, Han & Yu, Xihong & Chen, Bei, 2024. "Heterogeneous coexistence of extremely many attractors in adaptive synapse neuron considering memristive EMI," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    17. Suresh, R. & Senthilkumar, D.V. & Lakshmanan, M. & Kurths, J., 2016. "Emergence of a common generalized synchronization manifold in network motifs of structurally different time-delay systems," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 235-245.
    18. Ghanbari, Behzad, 2021. "On detecting chaos in a prey-predator model with prey’s counter-attack on juvenile predators," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    19. 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).
    20. Balcı, Ercan, 2023. "Predation fear and its carry-over effect in a fractional order prey–predator model with prey refuge," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).

    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:matcom:v:172:y:2020:i:c:p:321-340. 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: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.