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Fractal Fract
Volume 9
Issue 9
10.3390/fractalfract9090559
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Dynamical Analysis of Fractional-Order Quaternion-Valued Neural Networks with Leakage and Communication Delays

by
Qiankun Wang
2,
Tianzeng Li
2,
Yu Wang
1,* and
Xiaowen Tan
2
1
School of Mathematics and Statistics, Hainan Normal University, Haikou 571127, China
2
School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(9), 559; https://doi.org/10.3390/fractalfract9090559
Submission received: 25 July 2025 / Revised: 19 August 2025 / Accepted: 21 August 2025 / Published: 25 August 2025
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)
Versions Notes

Abstract

This paper investigates the stability and Hopf bifurcation problems of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage delay and communication delay. Utilizing the Hamilton rule of quaternions, the fractional-order quaternion-valued time-delay neural network model is transformed into an equivalent fractional-order real-valued time-delay neural network system. Then, employing the stability theory and bifurcation theory of fractional-order dynamical systems, novel sufficient criteria are derived to ensure system stability and to induce Hopf bifurcation, respectively, using the leakage delay and the communication delay as bifurcation parameters. Furthermore, the influences of both delay types on the bifurcation behavior of FOQVNNs are analyzed in depth. To verify the correctness of the theoretical results, bifurcation diagrams and simulation results generated using MATLAB are presented. The theoretical results established in this paper provide a significant theoretical basis for the analysis and design of FOQVNNs.
Keywords: fractional-order quaternion-valued neural networks; leakage delay; communication delay; hopf bifurcation fractional-order quaternion-valued neural networks; leakage delay; communication delay; hopf bifurcation

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MDPI and ACS Style

Wang, Q.; Li, T.; Wang, Y.; Tan, X. Dynamical Analysis of Fractional-Order Quaternion-Valued Neural Networks with Leakage and Communication Delays. Fractal Fract. 2025, 9, 559. https://doi.org/10.3390/fractalfract9090559

AMA Style

Wang Q, Li T, Wang Y, Tan X. Dynamical Analysis of Fractional-Order Quaternion-Valued Neural Networks with Leakage and Communication Delays. Fractal and Fractional. 2025; 9(9):559. https://doi.org/10.3390/fractalfract9090559

Chicago/Turabian Style

Wang, Qiankun, Tianzeng Li, Yu Wang, and Xiaowen Tan. 2025. "Dynamical Analysis of Fractional-Order Quaternion-Valued Neural Networks with Leakage and Communication Delays" Fractal and Fractional 9, no. 9: 559. https://doi.org/10.3390/fractalfract9090559

APA Style

Wang, Q., Li, T., Wang, Y., & Tan, X. (2025). Dynamical Analysis of Fractional-Order Quaternion-Valued Neural Networks with Leakage and Communication Delays. Fractal and Fractional, 9(9), 559. https://doi.org/10.3390/fractalfract9090559

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