The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization
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 <inline-formula&...
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2025-01-01
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| author | Louiza Diabi Adel Ouannas Amel Hioual Giuseppe Grassi Shaher Momani |
| author_facet | Louiza Diabi Adel Ouannas Amel Hioual Giuseppe Grassi Shaher Momani |
| author_sort | Louiza Diabi |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Y</mi></semantics></math></inline-formula>-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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>p</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mn>0</mn></msub></semantics></math></inline-formula> 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. |
| format | Article |
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| institution | Kabale University |
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| spelling | doaj-art-c8fff27d76d445f6820af6c2d582a0352025-01-24T13:39:51ZengMDPI AGMathematics2227-73902025-01-0113223910.3390/math13020239The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and SynchronizationLouiza Diabi0Adel Ouannas1Amel Hioual2Giuseppe Grassi3Shaher Momani4Laboratory of Dynamical Systems and Control, Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, AlgeriaDepartment of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, AlgeriaDepartment of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, AlgeriaDipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, ItalyNonlinear Dynamics Research Center, Ajman University, Ajman 346, United Arab EmiratesThe 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Y</mi></semantics></math></inline-formula>-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 (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>p</mi><mi>E</mi><mi>n</mi></mrow></semantics></math></inline-formula>) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mn>0</mn></msub></semantics></math></inline-formula> 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.https://www.mdpi.com/2227-7390/13/2/239Ueda mapchaoscontrolcommensurate orderincommensurate ordersynchronization |
| spellingShingle | Louiza Diabi Adel Ouannas Amel Hioual Giuseppe Grassi Shaher Momani The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization Mathematics Ueda map chaos control commensurate order incommensurate order synchronization |
| title | The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization |
| title_full | The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization |
| title_fullStr | The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization |
| title_full_unstemmed | The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization |
| title_short | The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization |
| title_sort | discrete ueda system and its fractional order version chaos stabilization and synchronization |
| topic | Ueda map chaos control commensurate order incommensurate order synchronization |
| url | https://www.mdpi.com/2227-7390/13/2/239 |
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