  By characterizing the phase dynamics in coupled oscillators, we gain insights
into the fundamental phenomena of complex systems. The collective dynamics in
oscillatory systems are often described by order parameters, which are
insufficient for identifying more specific behaviors. To improve this
situation, we propose a topological approach that constructs the quantitative
features describing the phase evolution of oscillators. Here, the phase data
are mapped into a high-dimensional space at each time, and the topological
features describing the shape of the data are subsequently extracted from the
mapped points. These features are extended to time-variant topological features
by adding the evolution time as an extra dimension in the topological feature
space. The time-variant features provide crucial insights into the evolution of
phase dynamics. Combining these features with the kernel method, we
characterize the multi-clustered synchronized dynamics during the early
evolution stages. Finally, we demonstrate that our method can qualitatively
explain chimera states. The experimental results confirmed the superiority of
our method over those based on order parameters, especially when the available
data are limited to the early-stage dynamics.
