Global exponential stability of pseudo almost automorphic solutions of octonion-valued stochastic high-order Hopfield neural networks with delays

Yuwei Cao; Yongkun Li; Yuwei Cao; Yongkun Li

doi:10.3934/era.2026180

Electronic Research Archive

2026, Volume 34, Issue 6: 4005-4036. doi: 10.3934/era.2026180

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Research article

Global exponential stability of pseudo almost automorphic solutions of octonion-valued stochastic high-order Hopfield neural networks with delays

Yuwei Cao ,
Yongkun Li ^,

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received: 29 March 2026 Revised: 24 April 2026 Accepted: 06 May 2026 Published: 14 May 2026

Octonion-valued neural networks (OVNNs) provide a powerful framework for modeling and processing high-dimensional data due to their eight-dimensional normed division algebraic structure. However, their inherent non-commutativity and non-associativity, coupled with ubiquitous time delays and stochastic disturbances in real-world systems, make the analysis of their dynamical behavior a significant challenge. This paper focuses on the complex oscillatory dynamics, specifically pseudo almost automorphy, within a class of stochastic higher-order Hopfield neural networks (NNs) based on octonions. To adequately characterize the stochastic processes involved, a novel concept of pseudo almost automorphic stochastic processes in finite-dimensional distributions is first proposed. Subsequently, by fixed point theorems and employing inequality techniques, sufficient criteria are established for the existence and global exponential stability of pseudo almost automorphic solutions in finite-dimensional distributions for the considered octonion-valued stochastic higher-order Hopfield NNs with time-varying delays. The obtained results are not only new for the octonionic system, but also remain novel even when the system degenerates to its real-valued counterpart. Furthermore, the analytical framework developed herein offers a general methodology applicable to studying pseudo almost automorphic dynamics in other types of complex-valued NNs.
Citation: Yuwei Cao, Yongkun Li. Global exponential stability of pseudo almost automorphic solutions of octonion-valued stochastic high-order Hopfield neural networks with delays[J]. Electronic Research Archive, 2026, 34(6): 4005-4036. doi: 10.3934/era.2026180

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Abstract

Octonion-valued neural networks (OVNNs) provide a powerful framework for modeling and processing high-dimensional data due to their eight-dimensional normed division algebraic structure. However, their inherent non-commutativity and non-associativity, coupled with ubiquitous time delays and stochastic disturbances in real-world systems, make the analysis of their dynamical behavior a significant challenge. This paper focuses on the complex oscillatory dynamics, specifically pseudo almost automorphy, within a class of stochastic higher-order Hopfield neural networks (NNs) based on octonions. To adequately characterize the stochastic processes involved, a novel concept of pseudo almost automorphic stochastic processes in finite-dimensional distributions is first proposed. Subsequently, by fixed point theorems and employing inequality techniques, sufficient criteria are established for the existence and global exponential stability of pseudo almost automorphic solutions in finite-dimensional distributions for the considered octonion-valued stochastic higher-order Hopfield NNs with time-varying delays. The obtained results are not only new for the octonionic system, but also remain novel even when the system degenerates to its real-valued counterpart. Furthermore, the analytical framework developed herein offers a general methodology applicable to studying pseudo almost automorphic dynamics in other types of complex-valued NNs.

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