Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis
This paper investigates the behavior and stability of the fractional-order Gray–Scott model, with a specific focus on achieving finite-time stability and synchronization. It introduces essential concepts, including the Gamma function, the Riemann–Liouville fractional-order integral operator, the Cap...
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| Format: | Article |
| Language: | English |
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KeAi Communications Co., Ltd.
2025-12-01
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| Series: | International Journal of Cognitive Computing in Engineering |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666307425000142 |
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| author | Shaher Momani Iqbal M. Batiha Issam Bendib Adel Ouannas Amel Hioual Dalah Mohamed |
| author_facet | Shaher Momani Iqbal M. Batiha Issam Bendib Adel Ouannas Amel Hioual Dalah Mohamed |
| author_sort | Shaher Momani |
| collection | DOAJ |
| description | This paper investigates the behavior and stability of the fractional-order Gray–Scott model, with a specific focus on achieving finite-time stability and synchronization. It introduces essential concepts, including the Gamma function, the Riemann–Liouville fractional-order integral operator, the Caputo fractional derivative, and the Mittag-Leffler function, to establish a foundational framework for subsequent analysis. Equilibrium points are defined, distinguishing between initial and finite-time equilibria, and the conditions for finite-time stability, including settling time, are precisely outlined. Stability results for this model are presented through theorems with detailed proofs, elucidating the roles of Lyapunov functions, class functions, and other system parameters. Furthermore, the paper explores finite-time synchronization schemes in master–slave systems, providing a mathematical framework for understanding and achieving synchronization within a finite time frame. This framework illuminates synchronization dynamics and their practical implications for controlling complex systems. Additionally, numerical examples illustrate finite-time stability and synchronization within the Gray–Scott reaction–diffusion model. |
| format | Article |
| id | doaj-art-3cf5b9d493c3426f8451d3f18b35e6c8 |
| institution | OA Journals |
| issn | 2666-3074 |
| language | English |
| publishDate | 2025-12-01 |
| publisher | KeAi Communications Co., Ltd. |
| record_format | Article |
| series | International Journal of Cognitive Computing in Engineering |
| spelling | doaj-art-3cf5b9d493c3426f8451d3f18b35e6c82025-08-20T02:15:28ZengKeAi Communications Co., Ltd.International Journal of Cognitive Computing in Engineering2666-30742025-12-01638039010.1016/j.ijcce.2025.02.004Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysisShaher Momani0Iqbal M. Batiha1Issam Bendib2Adel Ouannas3Amel Hioual4Dalah Mohamed5Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan; Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab EmiratesNonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates; Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan; Corresponding author at: Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan.Applied Mathematics & Modeling Laboratory, Department of Mathematics, Faculty of Exact Sciences, University of Brothers Mentouri, Constantine 25000, AlgeriaDepartment of Mathematics and Computer Science, University of Oum El Bouaghi, Oum El, Bouaghi 04000, AlgeriaDepartment of Mathematics and Computer Science, University of Oum El Bouaghi, Oum El, Bouaghi 04000, AlgeriaApplied Mathematics & Modeling Laboratory, Department of Mathematics, Faculty of Exact Sciences, University of Brothers Mentouri, Constantine 25000, AlgeriaThis paper investigates the behavior and stability of the fractional-order Gray–Scott model, with a specific focus on achieving finite-time stability and synchronization. It introduces essential concepts, including the Gamma function, the Riemann–Liouville fractional-order integral operator, the Caputo fractional derivative, and the Mittag-Leffler function, to establish a foundational framework for subsequent analysis. Equilibrium points are defined, distinguishing between initial and finite-time equilibria, and the conditions for finite-time stability, including settling time, are precisely outlined. Stability results for this model are presented through theorems with detailed proofs, elucidating the roles of Lyapunov functions, class functions, and other system parameters. Furthermore, the paper explores finite-time synchronization schemes in master–slave systems, providing a mathematical framework for understanding and achieving synchronization within a finite time frame. This framework illuminates synchronization dynamics and their practical implications for controlling complex systems. Additionally, numerical examples illustrate finite-time stability and synchronization within the Gray–Scott reaction–diffusion model.http://www.sciencedirect.com/science/article/pii/S2666307425000142Fractional-order Gray–Scott modelFinite-time stabilityFinite-time synchronizationLyapunov functions |
| spellingShingle | Shaher Momani Iqbal M. Batiha Issam Bendib Adel Ouannas Amel Hioual Dalah Mohamed Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis International Journal of Cognitive Computing in Engineering Fractional-order Gray–Scott model Finite-time stability Finite-time synchronization Lyapunov functions |
| title | Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis |
| title_full | Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis |
| title_fullStr | Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis |
| title_full_unstemmed | Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis |
| title_short | Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis |
| title_sort | examining finite time behaviors in the fractional gray scott model stability synchronization and simulation analysis |
| topic | Fractional-order Gray–Scott model Finite-time stability Finite-time synchronization Lyapunov functions |
| url | http://www.sciencedirect.com/science/article/pii/S2666307425000142 |
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