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Bifurcation, Hidden Chaos, Entropy and Control in Hénon-Based Fractional Memristor Map with Commensurate and Incommensurate Orders

Author

Listed:
  • Mayada Abualhomos

    (Applied Science Research Center (ASRC), Applied Science Private University, Amman 11942, Jordan)

  • Abderrahmane Abbes

    (Laboratory of Mathematics, Dynamics and Modelization, Badji Mokhtar-Annaba University, Annaba 23000, Algeria)

  • Gharib Mousa Gharib

    (Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan)

  • Abdallah Shihadeh

    (Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 330127, Zarqa 13133, Jordan)

  • Maha S. Al Soudi

    (Department of Basic Scientific Sciences, Applied Science Private University, Amman 11931, Jordan)

  • Ahmed Atallah Alsaraireh

    (Department of Computer Information Systems, The University of Jordan, Amman 11942, Jordan)

  • Adel Ouannas

    (Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria)

Abstract

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting bifurcation diagrams, and applying the sample entropy test to assess the complexity and validate the chaotic characteristics. However, since the proposed fractional map has no fixed points, the outcomes reveal that the map can exhibit a wide range of hidden dynamical behaviors. This phenomenon significantly augments the complexity of the fractal structure inherent to the chaotic attractors. Moreover, we introduce nonlinear controllers designed for stabilizing and synchronizing the proposed fractional Hénon-based memristor map. The research emphasizes the system’s sensitivity to fractional-order parameters, resulting in the emergence of distinct dynamic patterns. The memristor-based chaotic map exhibits rich and intricate behavior, making it a captivating and significant area of investigation.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4166-:d:1253692
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    References listed on IDEAS

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    1. Lai, Qiang & Lai, Cong & Zhang, Hui & Li, Chunbiao, 2022. "Hidden coexisting hyperchaos of new memristive neuron model and its application in image encryption," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    2. Peng, Yuexi & Liu, Jun & He, Shaobo & Sun, Kehui, 2023. "Discrete fracmemristor-based chaotic map by Grunwald–Letnikov difference and its circuit implementation," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    3. 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.
    4. Liu, Xinkang & Sun, Kehui & Wang, Huihai & He, Shaobo, 2023. "A class of novel discrete memristive chaotic map," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    5. Lin, Hairong & Wang, Chunhua & Sun, Jingru & Zhang, Xin & Sun, Yichuang & Iu, Herbert H.C., 2023. "Memristor-coupled asymmetric neural networks: Bionic modeling, chaotic dynamics analysis and encryption application," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
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