Inferring symbolic dynamics of chaotic flows from persistence

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.