https://doi.org/10.1140/epjs/s11734-022-00450-8
Regular Article
On the dynamical investigation and synchronization of variable-order fractional neural networks: the Hopfield-like neural network model
1
Department of Mechanical Engineering, University of Manitoba, R3T 5V6, Winnipeg, Canada
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
3
Department of Banking and Finance, FEMA, University of Malta, 2080, Msida MSD, Malta
4
Department of Economics, European University Institute, Via delle Fontanelle, 18, I-50014, Florence, Italy
5
School of Mathematics and Physics, China University of Geosciences (Wuhan), 430074, Wuhan, China
6
Zhejiang Institute, China University of Geosciences, 311305, Hangzhou, Zhejiang, China
7
Laboratory of Nonlinear Systems, Circuits and Complexity, Department of Physics, Aristotle University of Thessaloniki, 54124, Thessaloníki, Greece
8
Division of Graduate Studies and Research, Tijuana Institute of Technology, Tijuana, Mexico
9
Department of Mechanical Engineering, College of Engineering, Taif University, P.O. Box 11099, 21944, Taif, Saudi Arabia
Received:
14
February
2021
Accepted:
13
January
2022
Published online:
27
January
2022
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.
© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2022
