Chaos and Dynamic Behavior of the 4D Hyperchaotic Chen System via Variable-Order Fractional Derivatives

Athar I. Ahmed (), Mohamed Elbadri (), Abeer M. Alotaibi, Manahil A. M. Ashmaig, Mohammed E. Dafaalla and Ilhem Kadri
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Athar I. Ahmed: Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia
Mohamed Elbadri: Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia
Abeer M. Alotaibi: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Manahil A. M. Ashmaig: Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia
Mohammed E. Dafaalla: Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia
Ilhem Kadri: Department of Mathematics, University of Oran 1 Ahmed Ben Bella, Oran 31000, Algeria

Mathematics, 2025, vol. 13, issue 20, 1-24

Abstract: Fractional-order chaotic systems have received increasing attention over the past few years due to their ability to effectively model memory and complexity in nonlinear dynamics. Nonetheless, most of the research conducted so far has been on constant-order formulations, which still have some limitations in terms of adaptability and reality. Thus, to evade these limitations, we present a recently designed four-dimensional hyperchaotic Chen system with variable-order fractional (VOF) derivatives in the Liouville–Caputo sense. In comparison with constant-order systems, the new system possesses excellent performance in numerous aspects. Firstly, with the use of variable-order derivatives, the system becomes more adaptive and flexible, allowing the chaotic dynamics of the system to evolve with changing fractional orders. Secondly, large-scale numerical simulations are conducted, where phase portrait orbits and time series for differences in VOF directly illustrate the effect of the order function on the system’s behavior. Thirdly, qualitative analysis is performed with the help of phase portraits, time series, and Lyapunov exponents to confirm the system’s hyperchaotic behavior and sensitivity to initial and control parameters. Finally, the model developed demonstrates a wide range of dynamic behaviors, which confirms the sufficient efficiency of VOF calculus for modeling complicated nonlinear processes. Numerous analyses indicate that this research not only shows meaningful findings but also provides thoughtful methodologies that might result in subsequent research on fractional-order chaotic systems.

Keywords: fractional calculus; variable-order derivatives; chaos; computational simulation; numerical solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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