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Finite-size effects in a stochastic Kuramoto model
1School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales 2006, Australia.
Chaos (Woodbury, N.Y.)
|November 3, 2017
Summary
Finite ensembles of Kuramoto oscillators exhibit Brownian diffusion in their mean phase, unlike infinite systems. This collective coordinate approach accurately captures this finite-size effect.
Area of Science:
- Physics
- Complex Systems
- Nonlinear Dynamics
Background:
- The Kuramoto model describes the synchronization of coupled oscillators.
- In the thermodynamic limit (N → ∞), synchronized oscillator clusters exhibit a constant mean phase.
- Finite-size effects in collective behavior are not fully understood.
Purpose of the Study:
- To develop a collective coordinate approach for studying finite ensembles of stochastic Kuramoto oscillators.
- To investigate the dynamics of shape and mean phase in these systems.
- To analyze the impact of finite size on oscillator synchronization.
Main Methods:
- Developed a collective coordinate approach using two degrees of freedom: oscillator shape dynamics and mean phase.
- Applied the approach to a finite ensemble of N stochastic Kuramoto oscillators.
- Analyzed the behavior of the mean phase in the finite-size regime.
Main Results:
- The mean phase of a finite-size cluster of synchronized oscillators exhibits Brownian diffusion.
- The variance of this Brownian diffusion is inversely proportional to the number of oscillators (1/N).
- The collective coordinate approach quantitatively captures this finite-size effect.
Conclusions:
- Finite-size ensembles of Kuramoto oscillators display distinct collective dynamics compared to the thermodynamic limit.
- The proposed collective coordinate approach provides a powerful tool for analyzing finite-size effects in synchronization phenomena.
- Understanding these finite-size effects is crucial for applications involving coupled oscillator systems.
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