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<p>Neutral delay differential equations (NDDEs) play a crucial role in the mathematical modeling of dynamical systems across various fields, including physics, engineering, biology, and economics. These equations capture time-delay effects, where the rate of change depends on both the current state and previous states of the system. Despite their theoretical and practical significance, the discretization of NDDEs remains relatively underexplored, leading to challenges in their numerical analysis and computational implementation. In this work, we propose a transformation that produces a consistent discretized system corresponding to the original NDDE. This transformation ensures that when the delay parameter and the coefficient of the neutral term are set to zero, the discretized system simplifies to the standard discretization of the corresponding non-delay differential equation. We apply the nearly exact discretization scheme (NEDS) to convert a neural network model in neutral form into a 2<i>m</i>-dimensional discrete-time system. A detailed theoretical investigation of the system’s local stability and Neimark-Sacker bifurcation is carried out to better understand its discrete dynamics. Additionally, we present a simple and effective approach for applying the hybrid control method to stabilize the discretized system, while avoiding the technical complexities often involved in conventional stability analysis. To support our theoretical results, we consider a four-dimensional discrete system and provide numerical simulations that demonstrate the effectiveness of the proposed methods.</p>

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A Nearly Exact Discretization of a Neutral-Form Neural Network Model

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月25日

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  • A. M. A. Abo-Bakr

摘要

Neutral delay differential equations (NDDEs) play a crucial role in the mathematical modeling of dynamical systems across various fields, including physics, engineering, biology, and economics. These equations capture time-delay effects, where the rate of change depends on both the current state and previous states of the system. Despite their theoretical and practical significance, the discretization of NDDEs remains relatively underexplored, leading to challenges in their numerical analysis and computational implementation. In this work, we propose a transformation that produces a consistent discretized system corresponding to the original NDDE. This transformation ensures that when the delay parameter and the coefficient of the neutral term are set to zero, the discretized system simplifies to the standard discretization of the corresponding non-delay differential equation. We apply the nearly exact discretization scheme (NEDS) to convert a neural network model in neutral form into a 2m-dimensional discrete-time system. A detailed theoretical investigation of the system’s local stability and Neimark-Sacker bifurcation is carried out to better understand its discrete dynamics. Additionally, we present a simple and effective approach for applying the hybrid control method to stabilize the discretized system, while avoiding the technical complexities often involved in conventional stability analysis. To support our theoretical results, we consider a four-dimensional discrete system and provide numerical simulations that demonstrate the effectiveness of the proposed methods.