Global polynomial synchronization and stability of discrete-time inertial neural networks with proportional delays and its application to image encryption

Fengzhao Wang; Xingjian chen; Cheng Yang; Xuan Chen; Fengzhao Wang; Xingjian chen; Cheng Yang; Xuan Chen

doi:10.3934/math.2026980

AIMS Mathematics

2026, Volume 11, Issue 8: 24282-24303. doi: 10.3934/math.2026980

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Research article

Global polynomial synchronization and stability of discrete-time inertial neural networks with proportional delays and its application to image encryption

1.
School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550025, China
2.
Key Laboratory of Automotive Electronics Technology of Guizhou Provincial Department of Education, Guiyang 550025, China

Received: 03 June 2026 Revised: 20 July 2026 Accepted: 05 August 2026 Published: 10 August 2026
MSC : 39A30, 93C43, 94A60

This paper investigates global polynomial synchronization (GPS) of discrete-time inertial neural networks with proportional delays (PD-DTINNs). A discrete-time model is formulated on the sampling grid $ \hslash k $, where the proportional-delay state is directly represented by $ x_j(\lfloor \mathfrak {p}_{j}k \rfloor \hslash) $, thereby avoiding the continuous-time logarithmic transformation commonly used for proportional delays. A simple feedback controller is designed for the corresponding drive–response PD-DTINNs. By constructing appropriate auxiliary functions and combining difference relations with norm estimates, a direct analytical approach is developed without constructing Lyapunov functionals or employing the matrix-measure technique. Based on this approach, delay-dependent and delay-independent criteria are established to guarantee GPS, from which the corresponding global asymptotic synchronization (GAS) results are obtained. The developed framework is further extended to the stability analysis of a single discrete-time inertial neural network with proportional delays, thus yielding criteria for global polynomial stability and global asymptotic stability. Finally, one numerical example is provided to verify the proposed results, and the obtained synchronization behavior is applied to image encryption and decryption, thus illustrating its potential application in image security.
- discrete-time inertial neural networks,
- proportional delays,
- global polynomial synchronization,
- image encryption
Citation: Fengzhao Wang, Xingjian chen, Cheng Yang, Xuan Chen. Global polynomial synchronization and stability of discrete-time inertial neural networks with proportional delays and its application to image encryption[J]. AIMS Mathematics, 2026, 11(8): 24282-24303. doi: 10.3934/math.2026980

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Abstract

This paper investigates global polynomial synchronization (GPS) of discrete-time inertial neural networks with proportional delays (PD-DTINNs). A discrete-time model is formulated on the sampling grid $ \hslash k $, where the proportional-delay state is directly represented by $ x_j(\lfloor \mathfrak {p}_{j}k \rfloor \hslash) $, thereby avoiding the continuous-time logarithmic transformation commonly used for proportional delays. A simple feedback controller is designed for the corresponding drive–response PD-DTINNs. By constructing appropriate auxiliary functions and combining difference relations with norm estimates, a direct analytical approach is developed without constructing Lyapunov functionals or employing the matrix-measure technique. Based on this approach, delay-dependent and delay-independent criteria are established to guarantee GPS, from which the corresponding global asymptotic synchronization (GAS) results are obtained. The developed framework is further extended to the stability analysis of a single discrete-time inertial neural network with proportional delays, thus yielding criteria for global polynomial stability and global asymptotic stability. Finally, one numerical example is provided to verify the proposed results, and the obtained synchronization behavior is applied to image encryption and decryption, thus illustrating its potential application in image security.

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