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Bifurcation Analysis, Synchronization and FPGA Implementation of a New 3-D Jerk System with a Stable Equilibrium

Author

Listed:
  • Sundarapandian Vaidyanathan

    (Centre for Control Systems, Vel Tech University, 400 Feet Outer Ring Road, Avadi, Chennai 600 062, Tamil Nadu, India)

  • Ahmad Taher Azar

    (College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
    Faculty of Computers and Artificial Intelligence, Benha University, Benha 13511, Egypt
    Automated Systems and Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh 11586, Saudi Arabia)

  • Ibrahim A. Hameed

    (Department of ICT and Natural Sciences, Norwegian University of Science and Technology, Larsgardsvegen, 2, 6009 Alesund, Norway)

  • Khaled Benkouider

    (Non Destructive Testing Laboratory, Automatic Department, Jijel University, BP 98, Jijel 18000, Algeria)

  • Esteban Tlelo-Cuautle

    (Electronics Department, Instituto Nacional de Astrofisica, Optica y Electronica (INAOE), Puebla 72840, Mexico)

  • Brisbane Ovilla-Martinez

    (Computer Science Department, CINVESTAV—The Center for Research and Advanced Studies of the National Polytechnic Institute, Av. IPN 2508, Mexico City 07360, Mexico)

  • Chang-Hua Lien

    (Department of Marine Engineering, National Kaohsiung University of Science and Technology, Kaohsiung 81157, Taiwan)

  • Aceng Sambas

    (Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Gong Badak 21300, Terengganu, Malaysia
    Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Tasikmalaya 46196, Indonesia)

Abstract

This research paper addresses the modelling of a new 3-D chaotic jerk system with a stable equilibrium. Such chaotic systems are known to exhibit hidden attractors. After the modelling of the new jerk system, a detailed bifurcation analysis has been performed for the new chaotic jerk system with a stable equilibrium. It is shown that the new jerk system has multistability with coexisting attractors. Next, we apply backstepping control for the synchronization design of a pair of new jerk systems with a stable equilibrium taken as the master-slave chaotic systems. Lyapunov stability theory is used to establish the synchronization results for the new jerk system with a stable equilibrium. Finally, we show that the FPGA design of the new jerk system with a stable equilibrium can be implemented using the FPGA Zybo Z7-20 development board. The design of the new jerk system consists of multipliers, adders and subtractors. It is observed that the experimental attractors are in good agreement with simulation results.

Suggested Citation

  • Sundarapandian Vaidyanathan & Ahmad Taher Azar & Ibrahim A. Hameed & Khaled Benkouider & Esteban Tlelo-Cuautle & Brisbane Ovilla-Martinez & Chang-Hua Lien & Aceng Sambas, 2023. "Bifurcation Analysis, Synchronization and FPGA Implementation of a New 3-D Jerk System with a Stable Equilibrium," Mathematics, MDPI, vol. 11(12), pages 1-22, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2623-:d:1166759
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    References listed on IDEAS

    as
    1. Vijayakumar M.D. & Anitha Karthikeyan & Jozef Zivcak & Ondrej Krejcar & Hamidreza Namazi, 2021. "Dynamical Behavior of a New Chaotic System with One Stable Equilibrium," Mathematics, MDPI, vol. 9(24), pages 1-9, December.
    2. Zhang, Sen & Zeng, Yicheng, 2019. "A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 25-40.
    3. P. Babu Dhivakaran & A. Vinodkumar & S. Vijay & S. Lakshmanan & J. Alzabut & R. A. El-Nabulsi & W. Anukool, 2022. "Bipartite Synchronization of Fractional-Order Memristor-Based Coupled Delayed Neural Networks with Pinning Control," Mathematics, MDPI, vol. 10(19), pages 1-13, October.
    4. Lai, Qiang & Chen, Zhijie, 2023. "Grid-scroll memristive chaotic system with application to image encryption," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    5. Shahna, K.U., 2023. "Novel chaos based cryptosystem using four-dimensional hyper chaotic map with efficient permutation and substitution techniques," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
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