Persistence of Morse decompositions over grid resolution for maps and time series

图多向量持久性：动态系统的统一框架

Graph Multivector Persistence: A Unified Framework for Dynamic Systems

本文提出一种基于组合多向量动力学的图多向量持久性不变量，用于分析动态系统的多尺度结构特性。

Persistent Homology of Morse Decompositions in Combinatorial Dynamics

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to "vector cloud" data. We present experimental results on two numerical examp…

基于组合多向量场的马尔可夫链中Morse分解的持续同调

Persistent homology of Morse decomposition in Markov chains based on combinatorial multivector fields

本文提出一种基于组合多向量场的持续框架，用于马尔可夫链中的Morse分解，通过转移概率定义过滤，构建多向量场并应用Morse分解，得到不变集的层次结构及持续图，以分析有限状态随机过程的复发性和稳定性。

通过类持续性归约计算连接矩阵

Computing Connection Matrices via Persistence-like Reductions

本文提出一种单次遍历的类持续性算法，用于从单纯复形上的组合向量场高效计算连接矩阵，比现有两次遍历递归算法更简单高效。

拓扑-数值分析离散时间双基因Andrecut-Kauffman模型的全局动力学

Topological-numerical analysis of global dynamics in the discrete-time two-gene Andrecut-Kauffman model

该研究通过拓扑-数值分析方法，揭示了离散时间双基因Andrecut-Kauffman模型中多稳态和混沌吸引子等复杂动力学现象。

Topological computation analysis of meteorological time-series data

A topological computation method, called the MGSTD method, is applied to time-series data obtained from meteorological measurement. The method gives decomposition of the dynamics into invariant sets and gradient-like transitions between them, by dividing the phase space into grids and representing the time-series as a combinatorial multi-valued map over the grids. Since the time-series is highly stochastic, the multi-valued map is statistically determined by taking preferable transitions between the grids into account. The time-series data are principal components of pressure pattern in tropos…

Persistence and material coherence of a mesoscale ocean eddy

Ocean eddies play an important role in the transport and mixing processes of the ocean due to their ability to transport material, heat, salt, and other tracers across large distances. They exhibit at least two timescales; an Eulerian lifetime associated with persistent identifiable signatures in gridded fields such as vorticity or sea-surface height, and multiple Lagrangian or material coherence timescales that are typically much shorter. We propose a method to study the multi-timescale material transport, leakage, and entrainment by eddies with their surroundings by constructing sequences of…