Inferring symbolic dynamics of chaotic flows from persistence

TL;DR

This paper presents a novel topological data analysis method called state space persistence analysis, which infers symbolic dynamics from high-dimensional chaotic systems by comparing trajectory shapes to periodic orbits.

Contribution

The paper adapts persistent homology to characterize and compare the shapes of chaotic trajectories and periodic orbits in state space, enabling symbolic dynamics inference.

Findings

Successfully applied to Rössler system and Kuramoto--Sivashinsky PDE.

Quantifies shape similarity between chaotic trajectories and periodic orbits.

Provides a new topological approach for analyzing high-dimensional chaos.

Abstract

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 the periodic orbits. We demonstrate the method by applying it to the three-dimensional R\"{o}ssler system and a thirty-dimensional discretization of the Kuramoto--Sivashinsky partial differential equation in $(1+1)$ dimensions.