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A New Nine-Dimensional Chaotic Lorenz System with Quaternion Variables: Complicated Dynamics, Electronic Circuit Design, Anti-Anticipating Synchronization, and Chaotic Masking Communication Application

Author

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  • Emad E. Mahmoud

    (Department of Mathematics, Faculty of Science, Taif University, Taif 888, Saudi Arabia
    Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt)

  • M. Higazy

    (Department of Mathematics, Faculty of Science, Taif University, Taif 888, Saudi Arabia
    Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt)

  • Turkiah M. Al-Harthi

    (Department of Mathematics, Shaqra University, Shaqra 11921, Saudi Arabia)

Abstract

In this paper, a chaotic quaternion autonomous nonlinear structure is introduced and intends to be a contribution. It is the first nonlinear dynamical system with quaternion variables to be studied in the literature. With nine dimensions, the new system is a high-dimensional one. Several vital characteristics and features of this model are investigated, such as its Hamiltonian, symmetry, signal flow graph, dissipation, equilibriums and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams, and chaotic behavior. A circuit implementation is designed to realize the new system, and a scheme is designed to achieve anti-anticipating synchronization (AAS) of two identical chaotic attractors with quaternion variables based on a Lyapunov function and active control. The concept of AAS is yet to be explored in the literature. A simulation experiment is designed and executed to illustrate the effectiveness of the acquired results. After synchronization, numerical outcomes are planned to explain the status variables and errors of these chaotic attractors to prove that AAS is achieved. The secure communication problem is studied based on the obtained events of the AAS of two identical nonlinear Lorenz systems with quaternion variables. AAS connecting the drive and response systems in chaotic systems with quaternion variables is the key to achieving communication. Signal encryption and restoration are simulated numerically.

Suggested Citation

  • Emad E. Mahmoud & M. Higazy & Turkiah M. Al-Harthi, 2019. "A New Nine-Dimensional Chaotic Lorenz System with Quaternion Variables: Complicated Dynamics, Electronic Circuit Design, Anti-Anticipating Synchronization, and Chaotic Masking Communication Applicatio," Mathematics, MDPI, vol. 7(10), pages 1-26, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:877-:d:269349
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    References listed on IDEAS

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    1. Song, Guang-Jing & Wang, Qing-Wen & Yu, Shao-Wen, 2018. "Cramer’s rule for a system of quaternion matrix equations with applications," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 490-499.
    2. Mahmoud, Emad E. & Abo-Dahab, S.M., 2018. "Dynamical properties and complex anti synchronization with applications to secure communications for a novel chaotic complex nonlinear model," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 273-284.
    3. Huang, Chengdai & Cao, Jinde, 2017. "Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 262-275.
    4. Fen, Mehmet Onur, 2017. "Persistence of chaos in coupled Lorenz systems," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 200-205.
    5. Tingwen Huang & David Gao & Chuandong Li & MingQing Xiao, 2012. "Anticipating synchronization through optimal feedback control," Journal of Global Optimization, Springer, vol. 52(2), pages 281-290, February.
    6. Mahmoud, Emad E., 2013. "Modified projective phase synchronization of chaotic complex nonlinear systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 89(C), pages 69-85.
    7. Pavlos, G.P. & Karakatsanis, L.P. & Xenakis, M.N., 2012. "Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma, Part one: Sunspot dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6287-6319.
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    Cited by:

    1. Shu, Jinlong & Wu, Baowei & Xiong, Lianglin, 2022. "Stochastic stability criteria and event-triggered control of delayed Markovian jump quaternion-valued neural networks," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    2. Luigi Fortuna & Arturo Buscarino, 2022. "Analog Circuits," Mathematics, MDPI, vol. 10(24), pages 1-4, December.

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