Different orders jerk models: Dynamics, synchronization and their application in image encryption

Tarek M. Abed-Elhameed; Mohamed M. Darwish; Fahad S. Alshammari; Atef M. AboElkher; Tarek M. Abed-Elhameed; Mohamed M. Darwish; Fahad S. Alshammari; Atef M. AboElkher

doi:10.3934/math.2026654

AIMS Mathematics

2026, Volume 11, Issue 6: 15887-15911. doi: 10.3934/math.2026654

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Different orders jerk models: Dynamics, synchronization and their application in image encryption

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Computer Science, University College of Alwajh, University of Tabuk, Al Wajh 48721, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

Received: 30 March 2026 Revised: 21 May 2026 Accepted: 27 May 2026 Published: 05 June 2026
MSC : 34D06, 37B55, 37N35, 65P20

This work introduced three different versions of a complex model called the chaotic hidden attractor jerk model. These versions were categorized as commensurate, non-commensurate, and distributed-order models. They can be applied in various practical fields like physics and image encryption. We explored the characteristics of these models such as fixed points, symmetry, and dissipation. These models exhibit chaotic behaviors, as evidenced by the Lyapunov exponent (LLE) and bifurcation diagram. We introduced a novel combination synchronization (CS) between these models using a tracking control method. We established a theorem for achieving synchronization among these different models. We presented numerical computations to validate the analytical results. Our research primarily focused on the encryption and decryption processes of grayscale images by the proposed synchronization method. We evaluated the efficacy of image encryption and decryption through various metrics like information entropy and histograms to ensure the accuracy and security of the process.
- commensurate,
- distributed-order,
- jerk model,
- stability,
- chaotic,
- image encryption
Citation: Tarek M. Abed-Elhameed, Mohamed M. Darwish, Fahad S. Alshammari, Atef M. AboElkher. Different orders jerk models: Dynamics, synchronization and their application in image encryption[J]. AIMS Mathematics, 2026, 11(6): 15887-15911. doi: 10.3934/math.2026654

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Abstract

This work introduced three different versions of a complex model called the chaotic hidden attractor jerk model. These versions were categorized as commensurate, non-commensurate, and distributed-order models. They can be applied in various practical fields like physics and image encryption. We explored the characteristics of these models such as fixed points, symmetry, and dissipation. These models exhibit chaotic behaviors, as evidenced by the Lyapunov exponent (LLE) and bifurcation diagram. We introduced a novel combination synchronization (CS) between these models using a tracking control method. We established a theorem for achieving synchronization among these different models. We presented numerical computations to validate the analytical results. Our research primarily focused on the encryption and decryption processes of grayscale images by the proposed synchronization method. We evaluated the efficacy of image encryption and decryption through various metrics like information entropy and histograms to ensure the accuracy and security of the process.

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