The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization

Louiza Diabi, Adel Ouannas, Amel Hioual, Giuseppe Grassi () and Shaher Momani
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Louiza Diabi: Laboratory of Dynamical Systems and Control, Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
Adel Ouannas: Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
Amel Hioual: Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
Giuseppe Grassi: Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy
Shaher Momani: Nonlinear Dynamics Research Center, Ajman University, Ajman 346, United Arab Emirates

Mathematics, 2025, vol. 13, issue 2, 1-24

Abstract: The Ueda oscillator is one of the most popular and studied nonlinear oscillators. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on Y -th Caputo fractional difference and thoroughly investigates its chaotic dynamics via commensurate and incommensurate orders. Applying numerical methods like maximum Lyapunov exponent spectra, bifurcation plots, and phase plane. We demonstrate the emergence of chaotic attractors influenced by fractional orders and system parameters. Advanced complexity measures, including approximation entropy ( A p E n ) and C 0 complexity, are applied to validate and measure the nonlinear and chaotic nature of the system; the results indicate a high level of complexity. Furthermore, we propose a control scheme to stabilize and synchronize the introduced Ueda map, ensuring the convergence of trajectories to desired states. MATLAB R2024a simulations are employed to confirm the theoretical findings, highlighting the robustness of our results and paving the way for future works.

Keywords: Ueda map; chaos; control; commensurate order; incommensurate order; synchronization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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