  Complex systems are commonly modeled using nonlinear dynamical systems. These
models are often high-dimensional and chaotic. An important goal in studying
physical systems through the lens of mathematical models is to determine when
the system undergoes changes in qualitative behavior. A detailed description of
the dynamics can be difficult or impossible to obtain for high-dimensional and
chaotic systems. Therefore, a more sensible goal is to recognize and mark
transitions of a system between qualitatively different regimes of behavior. In
practice, one is interested in developing techniques for detection of such
transitions from sparse observations, possibly contaminated by noise. In this
paper we develop a framework to accurately tag different regimes of complex
systems based on topological features. In particular, our framework works with
a high degree of success in picking out a cyclically orbiting regime from a
stationary equilibrium regime in high-dimensional stochastic dynamical systems.
