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Finite-size scaling in the Kuramoto model
Tommaso Coletta1, Robin Delabays1,2, Philippe Jacquod1
1School of Engineering, University of Applied Sciences of Western Switzerland, CH-1951 Sion, Switzerland.
Physical Review. E
|May 17, 2017
Summary
For finite N oscillators in the Kuramoto model, the order parameter and Lyapunov exponent scale with coupling strength near the locking threshold. The coupling range for this scaling shrinks as N increases.
Area of Science:
- Complex systems
- Nonlinear dynamics
- Statistical physics
Background:
- The Kuramoto model describes synchronization in coupled oscillator systems.
- Understanding the transition to synchrony and finite-size effects is crucial.
Purpose of the Study:
- Investigate scaling properties of the order parameter and largest Lyapunov exponent in the fully locked state.
- Analyze finite N effects and their convergence to the infinite N limit.
Main Methods:
- Analytical investigation of scaling laws.
- Numerical confirmation using evenly spaced oscillator frequencies.
- Analysis of crossover behavior and spectral properties.
Main Results:
- For finite N, order parameter and Lyapunov exponent scale as (K-K_{L})^{1/2} near the locking threshold.
- The coupling range for this scaling shrinks as δK∼N^{-1.5}.
- A crossover to the infinite-N behavior (K-K_{L})^{2/3} is observed away from the threshold.
Conclusions:
- The crossover is linked to the spectral properties of the fully locked state.
- Results clarify the convergence to the infinite N limit in the Kuramoto model.
- Provides insights into finite-size scaling in synchronization phenomena.
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