Finite-time Stepanov almost periodic synchronization for fractional-order stochastic high-order Hopfield neural networks

Yisen Zhou; Yongkun Li; Yisen Zhou; Yongkun Li

doi:10.3934/math.2026432

AIMS Mathematics

2026, Volume 11, Issue 4: 10478-10517. doi: 10.3934/math.2026432

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Research article

Finite-time Stepanov almost periodic synchronization for fractional-order stochastic high-order Hopfield neural networks

Yisen Zhou ,
Yongkun Li ^,

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

Received: 01 February 2026 Revised: 07 April 2026 Accepted: 09 April 2026 Published: 17 April 2026
MSC : 34K14, 34K50, 92B20

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.
Citation: Yisen Zhou, Yongkun Li. Finite-time Stepanov almost periodic synchronization for fractional-order stochastic high-order Hopfield neural networks[J]. AIMS Mathematics, 2026, 11(4): 10478-10517. doi: 10.3934/math.2026432

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Abstract

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.

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