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标题：Persistence of the Conley Index in Combinatorial Dynamical Systems
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作者：Tamal K. Dey ; Marian Mrozek ; Ryan Slechta 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：164
页码：37:1-37:17
DOI：10.4230/LIPIcs.SoCG.2020.37
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman [R. Forman, 1998; R. Forman, 1998] and their recent generalization to multivector fields [Mrozek, 2017] have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to "track" features in a changing multivector field.
关键词：Dynamical systems; combinatorial vector field; multivector; Conley index; persistence