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Sufficiently dense Kuramoto networks are globally synchronizing
Martin Kassabov1, Steven H Strogatz1, Alex Townsend1
1Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.
Chaos (Woodbury, N.Y.)
|August 3, 2021
Summary
This study refines the critical connectivity threshold for Kuramoto oscillator networks, proving it
Area of Science:
- Complex Systems
- Nonlinear Dynamics
- Network Science
Background:
- The Kuramoto model describes synchronization in coupled oscillator networks.
- Network connectivity influences the emergence of synchronized states.
- A critical connectivity threshold (μc) determines guaranteed convergence to the all-in-phase state.
Purpose of the Study:
- To determine the precise value of the critical connectivity threshold (μc) for Kuramoto oscillator networks.
- To establish a rigorous upper bound for μc.
- To investigate the limitations of linear stability analysis in determining μc.
Main Methods:
- Analysis of a network of n identical Kuramoto oscillators with bidirectional coupling.
- Application of linear stability analysis to determine the critical connectivity.
- Mathematical proof to establish the upper bound for μc.
Main Results:
- The critical connectivity threshold (μc) is proven to be less than or equal to 0.75.
- This upper bound is shown to be the best achievable using purely linear stability analysis.
- Previous bounds of μc ≤ 0.7889 and μc > 0.6838 are considered.
Conclusions:
- The study establishes a tight upper bound for the critical connectivity in Kuramoto networks.
- Linear stability analysis has inherent limitations in precisely determining the synchronization threshold.
- The findings contribute to a deeper understanding of synchronization phenomena in complex systems.
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