In this paper we develop a novel Topological Data Analysis (TDA) approach for
studying graph representations of time series of dynamical systems.
Specifically, we show how persistent homology, a tool from TDA, can be used to
yield a compressed, multi-scale representation of the graph that can
distinguish between dynamic states such as periodic and chaotic behavior. We
show the approach for two graph constructions obtained from the time series. In
the first approach the time series is embedded into a point cloud which is then
used to construct an undirected $k$-nearest neighbor graph. The second
construct relies on the recently developed ordinal partition framework. In
either case, a pairwise distance matrix is then calculated using the shortest
path between the graph's nodes, and this matrix is utilized to define a
filtration of a simplicial complex that enables tracking the changes in
homology classes over the course of the filtration. These changes are
summarized in a persistence diagram---a two-dimensional summary of changes in
the topological features. We then extract existing as well as new geometric and
entropy point summaries from the persistence diagram and compare to other
commonly used network characteristics. Our results show that persistence-based
point summaries yield a clearer distinction of the dynamic behavior and are
more robust to noise than existing graph-based scores, especially when combined
with ordinal graphs.

采用拓扑数据分析方法，我们从动力系统时间序列的图形化表示中提取了有关其周期性和混沌特征的信息，并使用序数分区框架构建了两种图形。该方法提供了更准确的动态特性描述，并具有更强的噪声鲁棒性。