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Phase diagram for the Kuramoto model with van Hemmen interactions
Isabel M Kloumann1, Ian M Lizarraga1, Steven H Strogatz1
1Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
Summary
This study analyzes a Kuramoto model with random interactions, revealing four distinct collective behaviors in coupled oscillators. The findings map out states from complete incoherence to mixed synchronization patterns.
Area of Science:
- Complex Systems
- Statistical Physics
- Nonlinear Dynamics
Background:
- The Kuramoto model is a fundamental framework for studying synchronization in coupled oscillator systems.
- Quenched random interactions, inspired by spin glass models, introduce complex coupling patterns.
- Understanding emergent collective behaviors in such systems is crucial for various scientific domains.
Purpose of the Study:
- To analytically investigate the phase diagram of a Kuramoto model with quenched random interactions.
- To characterize the different collective states emerging from the interplay of attractive and random coupling.
- To explore the system's behavior under specific conditions: zero noise and Lorentzian frequency distribution.
Main Methods:
- Analytical derivation of the phase diagram for the specified model.
- Application of techniques used in spin glass theory (van Hemmen model).
- Mathematical analysis for a system with zero noise and a Lorentzian distribution of natural frequencies.
Main Results:
- Identification of four distinct macroscopic states: complete incoherence, partial synchronization, partial antiphase synchronization, and a mixed synchronization state.
- The phase diagram is determined by the relative strengths of attractive and random coupling terms.
- The system's behavior transitions between these states based on coupling parameters.
Conclusions:
- The Kuramoto model with quenched random interactions exhibits rich collective dynamics.
- The interplay between attractive and random couplings dictates the emergent synchronization patterns.
- This work provides a theoretical framework for understanding complex synchronization phenomena in diverse systems.
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