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

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Network Science

Background:

• Collective dynamics in oscillatory systems are crucial for understanding complex phenomena.
• Traditional order parameters often fail to capture specific behaviors in phase dynamics.
• Characterizing phase evolution in coupled oscillators requires advanced analytical tools.

Purpose of the Study:

• To develop a novel topological approach for quantitative analysis of phase dynamics in coupled oscillators.
• To overcome limitations of order parameters in describing complex oscillatory behaviors.
• To provide deeper insights into the evolution of phase dynamics and emergent phenomena.

Main Methods:

• Mapping oscillator phase data into a high-dimensional space at each time step.
• Extracting topological features describing the shape of the mapped data.
• Extending features to time-variant topological features by incorporating evolution time.
• Utilizing kernel methods for characterizing synchronized dynamics.

Main Results:

• The proposed topological method effectively quantifies phase evolution.
• Time-variant topological features reveal crucial insights into dynamics.
• Multiclustered synchronized dynamics were characterized during early evolution stages.
• The method qualitatively explains chimera states in oscillatory systems.

Conclusions:

• The topological approach offers superior characterization of phase dynamics compared to order parameters.
• This method is particularly advantageous when analyzing limited or early-stage dynamics data.
• The findings advance the understanding of complex systems and emergent behaviors in coupled oscillators.