Phase holonomy underlies puzzling temporal patterns in Kuramoto models with two sub-populations

Area of Science:

• Complex Systems Dynamics
• Theoretical Physics

Background:

• Kuramoto models are widely used to study synchronization in coupled oscillator systems.
• Previous studies reported unusual dynamical behaviors like chimeras and traveling waves in specific Kuramoto model configurations.
• The underlying mechanisms for these behaviors remained incompletely understood.

Purpose of the Study:

• To geometrically investigate the origins of chimeras and traveling waves in Kuramoto models with two sub-populations.
• To explore the potential role of geometric phase in these complex system dynamics.
• To establish a novel connection between geometric phase and emergent phenomena in oscillator ensembles.

Main Methods:

• Geometric analysis of the Kuramoto model dynamics.
• Investigation of phase space geometry.
• Mathematical formulation connecting oscillator behavior to geometric phase concepts.

Main Results:

• Chimeras and traveling waves in two-sub-population Kuramoto models are directly linked to the emergence of geometric phase.
• This study provides the first documented instance of geometric phase appearing in Kuramoto oscillator ensembles.
• The findings reveal a new geometric perspective on complex system dynamics.

Conclusions:

• The birth of geometric phase is a key factor driving chimeras and traveling waves in these Kuramoto models.
• This work extends the known applications of geometric phase to the field of complex systems and network dynamics.
• The geometric framework offers a novel approach for understanding emergent behaviors in coupled oscillator systems.