Yongkun Li

1. The concept of almost automorphic functions was proposed by Bochner [1], which is a natural extension of Bohr’s concept of almost periodic functions [2] and plays an important role in a profound...

This paper aims to address significant gaps in the theory of almost periodic functions and their applications in differential equations. Although Weyl almost periodic functions generalize Bohr almo...

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.

This paper investigates the dynamics of high-order Hopfield neural networks incorporating fractional-order derivatives, stochastic disturbances, and time-varying delays. To address the more realistic scenario of discontinuous or weakly regular time-varying parameters, the analysis is conducted within the framework of Stepanov almost-periodicity. First, sufficient criteria for the existence and uniqueness of a Stepanov almost periodic solution in distribution for the considered network are established using Banach's fixed point theorem and inequality techniques. Subsequently, by treating the studied network as a drive system, a corresponding response system is constructed. Effective control strategies are designed to achieve finite-time synchronization between these two systems. Finally, a numerical example is provided to illustrate the validity of the theoretical results. This study offers new theoretical insights for analyzing almost periodic oscillations in complex fractional-order stochastic systems with delays and has potential applications in fields requiring precise temporal coordination, such as secure communication and cooperative control.