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Determination of the critical coupling for oscillators in a ring
Hassan F El-Nashar1, Hilda A Cerdeira
1Department of Physics, Faculty of Science, Ain Shams University, Cairo, Egypt.
Chaos (Woodbury, N.Y.)
|October 2, 2009
Summary
We identified key oscillators that dictate complete synchronization in coupled systems. A new formula accurately predicts the critical coupling strength needed for all oscillators to synchronize, aiding in cluster size determination.
Area of Science:
- Complex Systems
- Nonlinear Dynamics
- Statistical Physics
Background:
- Coupled oscillator systems are fundamental in various scientific domains.
- Understanding synchronization phenomena is crucial for modeling collective behavior.
- Periodic boundary conditions introduce unique challenges in synchronization dynamics.
Purpose of the Study:
- To identify the critical factors determining stable phase-locked solutions in coupled oscillators.
- To develop a predictive formula for the critical coupling strength leading to complete synchronization.
- To analyze the role of boundary conditions and initial frequencies in synchronization.
Main Methods:
- Modeling coupled oscillators with bidirectional nearest-neighbor interactions.
- Analyzing systems with periodic boundary conditions.
- Locating critical oscillators based on initial frequencies.
- Deriving a formula for critical coupling strength, incorporating boundary condition corrections.
- Comparing theoretical predictions with numerical simulations.
Main Results:
- Identified specific oscillators at cluster borders as key to stable phase-locking.
- Developed an accurate formula for critical coupling strength predicting complete synchronization.
- The formula shows high accuracy when compared to numerical simulation results.
- The study provides insights into the sizes of major clusters near full synchronization.
Conclusions:
- The behavior of oscillators at cluster boundaries is pivotal for achieving complete synchronization.
- The derived formula offers a robust method for predicting synchronization thresholds.
- This work enhances the understanding of synchronization in systems with periodic boundary conditions.
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