Abstract: Periodic orbits in nonlinear systems often serve as organizing blocks of the dynamics on the attractors of these systems. However, these orbits are not directly observed in practice and must be inferred from time series data subject to perturbation and noise. Adapting ideas from algebraic topology and persistent homology, we present a novel method due to Bauer et al. of automatically decomposing time series data into elementary near-periodic orbits and transitions between them based on the inclusion of their first-rank homology into that of a joint comparison space. Faithful embeddings of the attractor of dynamical systems are often high-dimensional, rendering standard methods of homology computation infeasible. We present a discrete Morse-theoretic algorithm adapted for sparse cubical complexes embedded in high dimension with promising results compared to previous algorithms.