IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v119y2019icp150-162.html
   My bibliography  Save this article

On fractional–order discrete–time systems: Chaos, stabilization and synchronization

Author

Listed:
  • Khennaoui, Amina-Aicha
  • Ouannas, Adel
  • Bendoukha, Samir
  • Grassi, Giuseppe
  • Lozi, René Pierre
  • Pham, Viet-Thanh

Abstract

In this paper, we propose three systems, namely the fractional Lozi map, the fractional Lorenz map, and the fractional flow map. We study some of the important dynamics of these maps including the bifurcation graphs. In addition, we propose control laws aimed at stabilizing and synchronizing a combination of these three maps. The convergence of the stabilized states and the synchronization errors is established by means of the linearization method. Numerical simulations are also presented to confirm the main findings of the study.

Suggested Citation

  • Khennaoui, Amina-Aicha & Ouannas, Adel & Bendoukha, Samir & Grassi, Giuseppe & Lozi, René Pierre & Pham, Viet-Thanh, 2019. "On fractional–order discrete–time systems: Chaos, stabilization and synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 150-162.
    • Handle: RePEc:eee:chsofr:v:119:y:2019:i:c:p:150-162
    DOI: 10.1016/j.chaos.2018.12.019
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007791831049X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2018.12.019?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. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 285-294.
    2. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "A new derivative with normal distribution kernel: Theory, methods and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 476(C), pages 1-14.
    3. Thabet Abdeljawad & Dumitru Baleanu & Fahd Jarad & Ravi P. Agarwal, 2013. "Fractional Sums and Differences with Binomial Coefficients," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-6, May.
    4. Wu, Guo-Cheng & Baleanu, Dumitru & Xie, He-Ping & Chen, Fu-Lai, 2016. "Chaos synchronization of fractional chaotic maps based on the stability condition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 374-383.
    5. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    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. Danca, Marius-F., 2022. "Fractional order logistic map: Numerical approach," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    2. Dutta, Maitreyee & Roy, Binoy Krishna, 2020. "A new fractional-order system displaying coexisting multiwing attractors; its synchronisation and circuit simulation," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    3. Pakhare, Sumit S. & Bhalekar, Sachin & Gade, Prashant M., 2022. "Synchronization in coupled integer and fractional-order maps," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    4. Ran, Jie & Li, Yu-Qin & Xiong, Yi-Bin, 2022. "On the dynamics of fractional q-deformation chaotic map," Applied Mathematics and Computation, Elsevier, vol. 424(C).
    5. Mayada Abualhomos & Abderrahmane Abbes & Gharib Mousa Gharib & Abdallah Shihadeh & Maha S. Al Soudi & Ahmed Atallah Alsaraireh & Adel Ouannas, 2023. "Bifurcation, Hidden Chaos, Entropy and Control in Hénon-Based Fractional Memristor Map with Commensurate and Incommensurate Orders," Mathematics, MDPI, vol. 11(19), pages 1-19, October.
    6. Xin, Baogui & Peng, Wei & Kwon, Yekyung, 2020. "A discrete fractional-order Cournot duopoly game," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).
    7. Wang, Lingyu & Sun, Kehui & Peng, Yuexi & He, Shaobo, 2020. "Chaos and complexity in a fractional-order higher-dimensional multicavity chaotic map," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Roman Ivanovich Parovik, 2022. "Studies of the Fractional Selkov Dynamical System for Describing the Self-Oscillatory Regime of Microseisms," Mathematics, MDPI, vol. 10(22), pages 1-14, November.
    9. Li, Yuqing & He, Xing & Zhang, Wei, 2020. "The fractional difference form of sine chaotification model," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    10. Liu, Xianggang & Ma, Li, 2020. "Chaotic vibration, bifurcation, stabilization and synchronization control for fractional discrete-time systems," Applied Mathematics and Computation, Elsevier, vol. 385(C).
    11. Ouannas, Adel & Khennaoui, Amina-Aicha & Odibat, Zaid & Pham, Viet-Thanh & Grassi, Giuseppe, 2019. "On the dynamics, control and synchronization of fractional-order Ikeda map," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 108-115.

    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. Singh, C.S. & Singh, Harendra & Singh, Somveer & Kumar, Devendra, 2019. "An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1440-1448.
    2. Yavuz, Mehmet & Bonyah, Ebenezer, 2019. "New approaches to the fractional dynamics of schistosomiasis disease model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 373-393.
    3. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    4. Ma, Chaoqun & Ma, Zonggang & Xiao, Shisong, 2019. "A closed-form pricing formula for vulnerable European options under stochastic yield spreads and interest rates," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 59-68.
    5. Owolabi, Kolade M., 2019. "Mathematical modelling and analysis of love dynamics: A fractional approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 849-865.
    6. Al-khedhairi, A. & Elsadany, A.A. & Elsonbaty, A., 2019. "Modelling immune systems based on Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 25-39.
    7. Kumar, Sachin & Pandey, Prashant, 2020. "A Legendre spectral finite difference method for the solution of non-linear space-time fractional Burger’s–Huxley and reaction-diffusion equation with Atangana–Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    8. Owolabi, Kolade M. & Pindza, Edson, 2019. "Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 146-157.
    9. Ahmad, Zubair & Ali, Farhad & Khan, Naveed & Khan, Ilyas, 2021. "Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    10. Mishra, Jyoti, 2019. "Modified Chua chaotic attractor with differential operators with non-singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 64-72.
    11. Kubeka, Amos S. & Doungmo Goufo, Emile F. & Khumalo, Melusi, 2018. "On the quasi-normal modes of a Schwarzschild white hole for the lower angular momentum and perturbation by non-local fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 348-357.
    12. Saqib, Muhammad & Khan, Ilyas & Shafie, Sharidan, 2018. "Application of Atangana–Baleanu fractional derivative to MHD channel flow of CMC-based-CNT's nanofluid through a porous medium," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 79-85.
    13. Gatabazi, P. & Mba, J.C. & Pindza, E. & Labuschagne, C., 2019. "Grey Lotka–Volterra models with application to cryptocurrencies adoption," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 47-57.
    14. Shah, Syed Azhar Ali & Khan, Muhammad Altaf & Farooq, Muhammad & Ullah, Saif & Alzahrani, Ebraheem O., 2020. "A fractional order model for Hepatitis B virus with treatment via Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    15. Pang, Denghao & Jiang, Wei & Liu, Song & Jun, Du, 2019. "Stability analysis for a single degree of freedom fractional oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 498-506.
    16. Dubey, Ved Prakash & Dubey, Sarvesh & Kumar, Devendra & Singh, Jagdev, 2021. "A computational study of fractional model of atmospheric dynamics of carbon dioxide gas," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    17. Qureshi, Sania & Bonyah, Ebenezer & Shaikh, Asif Ali, 2019. "Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    18. Gatabazi, P. & Mba, J.C. & Pindza, E., 2019. "Modeling cryptocurrencies transaction counts using variable-order Fractional Grey Lotka-Volterra dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 283-290.
    19. Abu Arqub, Omar & Maayah, Banan, 2019. "Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 163-170.
    20. Ouannas, Adel & Khennaoui, Amina-Aicha & Odibat, Zaid & Pham, Viet-Thanh & Grassi, Giuseppe, 2019. "On the dynamics, control and synchronization of fractional-order Ikeda map," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 108-115.

    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:119:y:2019:i:c:p:150-162. 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.