组合动力系统中Conley指标的持久性
Persistence of the Conley Index in Combinatorial Dynamical Systems
摘要：
动力系统的组合框架为经典动力学与面向数据的算法方法的连接提供了一条途径。组合向量场介绍了福尔曼和他们最近推广到多向量场提供了一个起点，建立这样一个连接。在这项工作中，我们加强了这种关系，把康利指数在持久的同源设置。康利指数是与所谓的孤立不变集相关的同调特征，因此康利指数的变化是对底层多向量场扰动的响应。我们展示了如何可以使用之字形持久性来总结康利指数的变化，我们开发的技术来捕捉这种变化中存在的噪声。最后，我们开发了一种算法来跟踪功能在不断变化的多向量场。
英文摘要：
A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman 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.
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