时间序列动态分析的单纯多值映射与见证复形
Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series
摘要：
基于拓扑的分析动力系统的时间序列数据是强大的：它潜在地允许基于计算机的证明存在的各种类别的规则和混沌的高维动力学不变集。标准的方法是基于一个立方离散化的动态和使用的时间序列来构建一个外部近似的基础动力系统。由此产生的多值映射可以用来计算孤立不变集的立方体的Conley指数。在本文中，我们介绍了一种离散化，而不是使用单纯复杂的构造从一个里程碑的关系。我们的目标是获得与时间序列本身的不变密度更紧密联系的自然离散化。数据的时间顺序也直接导致了这个单纯复形上的一个映射，我们称之为见证映射。我们得到的条件下，这个证人映射给出了一个外部近似的动态，从而可以用来计算孤立不变集的康利指数。通过一个简单的例子说明了该方法使用的数据从经典的Hénon映射。
英文摘要：
Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical Hénon map.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
