We introduce “state space persistence analysis” for deducing the symbolic dynamics of time series data obtained from high-dimensional chaotic attractors. To this end, we adapt a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits. By comparing the shapes along a chaotic trajectory to those of the periodic orbits, state space persistence analysis quantifies the shape similarity of chaotic trajectory segments and periodic orbits. We demonstrate the method by applying it to the three-dimensional Rössler system and a 30-dimensional discretization of the Kuramoto–Sivashinsky partial differential equation in $(1+1)$ dimensions.

中文翻译：

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从持久性推断混沌流的符号动力学

我们引入“状态空间持久性分析”以推导从高维混沌吸引子获得的时间序列数据的符号动力学。为此，我们采用一种称为持久性同源性的拓扑数据分析技术来表征混沌轨迹和周期轨道的状态空间投影。通过将沿混沌轨迹的形状与周期性轨道的形状进行比较，状态空间持久性分析可以量化混沌轨迹段和周期轨道的形状相似性。我们通过将其应用于三维Rössler系统和Kuramoto-Sivashinsky偏微分方程的30维离散化来证明该方法。$（\mathrm{1个}+\mathrm{1个}）$ 尺寸。