  Recurrence is a fundamental characteristic of complicated deterministic
dynamical systems. Understanding its structure is challenging, especially if
the system of interest has many degrees of freedom so that visualizations of
trajectories are of limited use. To analyze recurrent phenomena, we propose a
computational method to identify oscillations and the transitions between them.
To this end, we introduce the concept of cycling signature, which is a
topological descriptor of a trajectory segment. The advantage of a topological
approach, in particular in an application setting, is that it is robust to
noise. Cycling signatures are computable from data and provide a comprehensive
global description of oscillations through their statistics over many
trajectory segments. We demonstrate this through three examples. In particular,
we identify and analyze six oscillations in a four-dimensional system with a
hyperchaotic attractor.
