状态依赖与时滞阈值微分方程的实用性问题 (Practicalities of State-Dependent and Threshold Delay Differential Equations)

from arxiv, Based on presentations at "Delays and Structures in Dynamical Systems: Modeling, Analysis and Numerical Methods" at the International Centre for Mechanical Sciences (CISM) in November 2023 in Udine

Delays are ubiquitous in applied problems, but often do not arise as the simple constant discrete delays that analysts and numerical analysts like to treat. In this chapter we show how state-dependent delays arise naturally when modeling and the consequences that follow. We treat discrete state-dependent delays, and delays implicitly defined by threshold conditions. We will consider modeling, formulation as dynamical systems, linearization, and numerical techniques. For discrete state-dependent delays we show how breaking points can be tracked efficiently to preserve the order of numerical methods for simulating solutions. For threshold conditions we will discuss how a velocity ratio term arises in models, and present a heuristic linearization method that avoids Banach spaces and sun-star calculus, making the method accessible to a wider audience. We will also discuss numerical implementations of threshold and distributed delay problems which allows them to be treated numerically with standard software.

翻译：时滞在应用问题中普遍存在，但通常并非分析学者和数值分析学者所偏好的简单恒定离散时滞形式。本章将阐述状态依赖时滞在建模过程中如何自然产生及其相应影响。我们将处理离散状态依赖时滞，以及由阈值条件隐式定义的时滞。内容涵盖建模方法、动力系统表述形式、线性化处理及数值计算技术。针对离散状态依赖时滞，我们将展示如何高效追踪断点以保持数值模拟方法的精度阶数。关于阈值条件，我们将讨论速度比项在模型中的产生机制，并提出一种避免使用巴拿赫空间与日星演算的启发式线性化方法，使该技术更易于被广泛研究者掌握。此外，我们还将探讨阈值问题与分布时滞问题的数值实现方案，使其能够通过标准软件进行数值处理。