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On the dynamical investigation and synchronization of variable-order fractional neural networks: the Hopfield-like neural network model

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Abstract

Since the variable-order fractional systems show more complex characteristics and more degrees of freedom due to time-varying fractional derivatives, we introduce a variable-order fractional Hopfield-like neural network in this paper. First, the properties and dynamical behavior of the system are studied. The variable-order derivative’s effects on the system’s behavior are investigated through the Lyapunov exponents and bifurcation diagram; an emerging Feigenbaum tree of period-4 bubble is observed, which appears with the creation and annihilation of periodic orbits. A general basin of attraction for the fractional-order neural network is presented, demonstrating that its dynamical behaviors are extremely sensitive to initial conditions resulting in different periodic orbits and chaotic attractors’ coexistence. After that, an adaptive control scheme is proposed for the variable-order fractional system. Through Lyapunov theorem and Barbalat’s Lemma, the system’s convergence and stability under the proposed control scheme are proven. The main advantages of the proposed controller are its guaranteed stability, robustness against uncertainties, and simplicity. Finally, the synchronization results are presented. Numerical simulations show the excellent performance of the proposed controller for the variable-order fractional Hopfield-like neural network.

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Acknowledgements

The research was supported by the Taif University Researchers Supporting Project number (TURSP-2020/77), Taif University, Taif, Saudi Arabia.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Manitoba, Winnipeg, R3T 5V6, Canada

    Hadi Jahanshahi

  2. Facultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, Av. Universidad S/N, Cd. Universitaria, C.P. 66455, San Nicolás de los Garza, NL, Mexico

    Ernesto Zambrano-Serrano

  3. Department of Banking and Finance, FEMA, University of Malta, Msida MSD, 2080, Malta

    Stelios Bekiros

  4. Department of Economics, European University Institute, Via delle Fontanelle, 18, Florence, I-50014, Italy

    Stelios Bekiros

  5. School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, 430074, China

    Zhouchao Wei

  6. Zhejiang Institute, China University of Geosciences, Hangzhou, 311305, Zhejiang, China

    Zhouchao Wei

  7. Laboratory of Nonlinear Systems, Circuits and Complexity, Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloníki, Greece

    Christos Volos

  8. Division of Graduate Studies and Research, Tijuana Institute of Technology, Tijuana, Mexico

    Oscar Castillo

  9. Department of Mechanical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia

    Ayman A. Aly

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Jahanshahi, H., Zambrano-Serrano, E., Bekiros, S. et al. On the dynamical investigation and synchronization of variable-order fractional neural networks: the Hopfield-like neural network model. Eur. Phys. J. Spec. Top. 231, 1757–1769 (2022). https://doi.org/10.1140/epjs/s11734-022-00450-8

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