标题： 动力系统的基本定理：同时且在同一地点 显示英文标题

标题： The Fundamental Theorem of Dynamical Systems: all at once and all in the same place

摘要： 所谓的动力系统基本定理——它(1)将吸引子和排斥子与链回转集相关联，(2)给出了一个完整李雅普诺夫函数的存在性——可以看作是分离“循环”和“瞬态”动力学的一种手段。 该定理的各种形式，包括连续时间/离散时间以及流/半流的概述被给出。 作为该概述的一部分，开发了一种统一的方法，用于同时处理拓扑动力学中的连续时间和离散时间框架。 此外，首次为连续时间流和半流提供了完整的李雅普诺夫函数。 显示英文摘要

摘要： The so-called Fundamental Theorem of Dynamical Systems -- which(1) relates attractors and repellers to the chain recurrent set and (2) gives the existence of a complete Lyapunov function -- can be seen as a means of separating out ``recurrent'' and ``transient'' dynamics. An overview of this theorem is given in its various guises, continuous-time/discrete-time and flows/semiflows. As part of this overview, a unified approach is developed for working simultaneously with both the continuous-time and discrete-time frameworks for topological dynamics. Additionally, a complete Lyapunov function is provided for the first time for continuous-time flows and semiflows.