论文标题
组合动力学系统中康利指数的持久性
Persistence of the Conley Index in Combinatorial Dynamical Systems
论文作者
论文摘要
动态系统的组合框架为将经典动力学与面向数据的算法方法连接起来提供了途径。福尔曼（Forman）引入的组合向量字段及其最近对多人字段的概括为建立这种连接提供了一个起点。在这项工作中，我们通过将Conley指数放置在持续的同源性环境中来加强这种关系。 Conley指数是与所谓的孤立不变集相关的同源特征，因此，Conley指数的变化是对基础多生机场中扰动的响应。我们展示了如何使用曲折的持久性来总结Conley索引的变化，并开发了在噪声存在下捕获此类变化的技术。我们通过开发一种算法来跟踪变化的多派场中的特征来结束。
A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields 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.
