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By characterizing the phase dynamics in coupled oscillators, we gain insightsinto the fundamental phenomena of complex systems. The collective dynamics inoscillatory systems are often described by order parameters, which areinsufficient for identifying more specific behaviors. To improve thissituation, we propose a topological approach that constructs the quantitativefeatures describing the phase evolution of oscillators. Here, the phase dataare mapped into a high-dimensional space at each time, and the topologicalfeatures describing the shape of the data are subsequently extracted from themapped points. These features are extended to time-variant topological featuresby adding the evolution time as an extra dimension in the topological featurespace. The time-variant features provide crucial insights into the evolution ofphase dynamics. Combining these features with the kernel method, wecharacterize the multi-clustered synchronized dynamics during the earlyevolution stages. Finally, we demonstrate that our method can qualitativelyexplain chimera states. The experimental results confirmed the superiority ofour method over those based on order parameters, especially when the availabledata are limited to the early-stage dynamics.

Evaluating the phase dynamics of coupled oscillators via time-variant topological features