Model reduction for networks of coupled oscillators

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Statistical Physics

Background:

• Coupled phase oscillators are fundamental in modeling synchronization phenomena across various scientific disciplines.
• The Kuramoto model is a canonical model for studying synchronization, but its high dimensionality poses computational challenges.

Purpose of the Study:

• To develop a novel collective coordinate approach for simplifying the analysis of coupled phase oscillators.
• To apply this method to the Kuramoto model for efficient and accurate synchronization studies.
• To investigate the behavior of synchronized clusters and finite-size scaling effects.

Main Methods:

• Reduction of an N-dimensional Kuramoto model to a lower-dimensional (n-dimensional) ordinary differential equation system using collective coordinates.
• Introduction of two collective coordinates to describe interactions between partially synchronized clusters.
• Comparison of analytical results with numerical simulations for various frequency distributions.

Main Results:

• The collective coordinate approach accurately reproduces the onset of local and global synchronization.
• Both soft and hard transitions in synchronization are effectively described.
• The interaction of two partially synchronized clusters with bimodal frequency distributions is successfully modeled.
• Accurate description of finite-size scaling for critical coupling strength is achieved.

Conclusions:

• The collective coordinate approach offers a significant reduction in complexity for analyzing coupled phase oscillator systems like the Kuramoto model.
• This method provides accurate insights into synchronization dynamics, including cluster interactions and transition types.
• The approach is validated through comparison with numerical simulations, demonstrating its broad applicability.