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Comparing the locking threshold for rings and chains of oscillators
Bertrand Ottino-Löffler1, Steven H Strogatz1
1Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
Physical Review. E
|January 14, 2017
Summary
Topology significantly impacts oscillator synchronization. Rings typically synchronize more readily than chains due to boundary conditions, though exceptions exist. This study analyzes synchronization in coupled phase oscillator networks.
Area of Science:
- Complex systems
- Nonlinear dynamics
- Network science
Background:
- Synchronization is a fundamental phenomenon in coupled oscillator systems.
- Network topology and boundary conditions critically influence synchronization dynamics.
- The Kuramoto model is a standard framework for studying synchronization in phase oscillators.
Purpose of the Study:
- To investigate the effect of topology (ring vs. chain) on the synchronization of phase oscillators.
- To compare the synchronization capabilities of ring and chain topologies under identical conditions.
- To derive theoretical bounds for the synchronization threshold ratio between ring and chain configurations.
Main Methods:
- Analysis of phase oscillator arrays with identical initial conditions and random natural frequencies.
- Comparison of periodic boundary conditions (ring) versus open boundary conditions (chain).
- Mathematical derivation of locking thresholds and their ratio for ring and chain topologies using a Kuramoto model variant.
Main Results:
- Stable phase-locked states exist when the natural frequency spread is below a topology-dependent locking threshold.
- Rings generally exhibit higher synchronization readiness (lower locking threshold) than matched chains.
- Rigorous bounds were established for the ratio of locking thresholds between ring and chain configurations.
Conclusions:
- Topology, specifically boundary conditions, plays a crucial role in determining synchronization efficiency.
- While rings usually synchronize more readily, the specific frequency distribution can lead to exceptions.
- The findings provide a deeper understanding of network structure's influence on collective dynamics in oscillatory systems.
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