标题： 半动力系统中的Morse分解及其在状态依赖延迟微分方程系统中的应用 显示英文标题

标题： Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations

摘要： 理解全局吸引子的结构在动力系统领域至关重要，其中Morse分解通过将吸引子划分为有限多个不变的Morse集和类似梯度的连接轨道，提供了一种强大的工具。 在Mallet-Paret开创性地使用离散Lyapunov函数来构建延迟微分方程中的Morse分解的基础上，类似的方法已被扩展到各种延迟系统，也包括状态依赖延迟。 在本文中，我们提出一个统一的框架，假设在任意度量空间上的半动力系统存在一个离散Lyapunov函数及其某些性质，并在此一般情况下构造全局吸引子的Morse分解。 我们证明了我们的结果推广了以前的结果；此外，我们将定理应用于具有阈值型状态依赖延迟的微分方程循环系统。 显示英文摘要

摘要： Understanding the structure of the global attractor is crucial in the field of dynamical systems, where Morse decompositions provide a powerful tool by partitioning the attractor into finitely many invariant Morse sets and gradient-like connecting orbits. Building on Mallet-Paret's pioneering use of discrete Lyapunov functions for constructing Morse decompositions in delay differential equations, similar approaches have been extended to various delay systems, also including state-dependent delays. In this paper, we develop a unified framework assuming the existence and some properties of a discrete Lyapunov function for a semi-dynamical system on an arbitrary metric space, and construct a Morse decomposition of the global attractor in this general setting. We demonstrate that our findings generalize previous results; moreover, we apply our theorem to a cyclic system of differential equations with threshold-type state-dependent delay.