Cycling Signatures: Identifying Cycling Motions in Time Series using Algebraic Topology

Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We develop algebraic topological notions for identifying and classifying elementary recurrent motions -- called cycling -- and the transitions between those. Statistics on these cycling motions can be computed from sampled trajectories (time series data), providing coarse global information on the structure of the recurrent behavior. We demonstrate this through three examples; in particular, we identify and analyze six cycling motions in a four dimensional system with a hyperchaotic attractor. We see this as a promising approach to reveal coarse-grained dynamical information on high-dimensional systems.