Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling

Area of Science:

• Complex Systems
• Dynamical Networks
• Statistical Physics

Background:

• Coupling heterogeneity is a natural feature in systems with many interacting agents.
• The Kuramoto model is a standard framework for studying synchronization in networks.

Purpose of the Study:

• To investigate the dynamics of the 3D Kuramoto model with heterogeneous couplings.
• To theoretically predict the degree of global coherence in such systems.

Main Methods:

• Linear stability analysis of an incoherent state.
• Development of a self-consistency approach for predicting coherence.
• Numerical simulations to validate theoretical predictions.

Main Results:

• The critical coupling strength for the instability of incoherence was rigorously derived to be zero.
• A theoretical prediction for the degree of global coherence was developed for positive coupling strengths.
• Theoretical analyses showed good agreement with numerical simulations.

Conclusions:

• Heterogeneous couplings significantly influence the collective behavior of dynamical networks.
• The developed self-consistency approach provides a valuable tool for understanding coherence in complex systems.
• This work deepens the understanding of emergent collective behaviors in heterogeneously coupled high-dimensional networks.