Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley-Morse graphs. A Conley-Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.

中文翻译：

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组合动力系统中 Conley-Morse 图的持久性

多向量场为在组合框架中研究连续动力系统提供了途径。目前文献中有两种方法使用持久同源性来捕获组合动力系统中的变化。第一个捕获 Conley 指数的变化，而第二个捕获 Morse 分解的变化。然而，这样的方法有局限性。前一种方法只描述了 Conley 指数如何在选定的孤立不变集上变化，尽管动态可能比单个孤立不变集的行为复杂得多。同样，考虑莫尔斯分解会忽略关于单个莫尔斯集的许多信息。在本文中，我们提出了一种通过捕获所谓的 Conley-Morse 图中的变化来总结组合动力系统变化的方法。Conley-Morse 图包含有关所选 Morse 分解的结构和分解中每个 Morse 集的 Conley 指数的信息。因此，我们的方法以比以前的方法更精细的粒度总结了一系列动态系统的变化结构。