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A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators

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Abstract

Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength \(K_c\) for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between \(\langle K_c\rangle \) and \(N\), which is \(\langle K_c\rangle \sim \sqrt{N}\), not only for small \(N\), but also for large \(N\). We also investigate the heavy-tailed distribution of \(K_c\) for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of \(K_c\) for large \(N\) from our algorithm to get the empirical density of \(K_c\). Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with \(N\) and would provide the theoretical support for engineering applications.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Lin Zhang, Ye Wu, Xia Shi, Zuguo He & Jinghua Xiao

  2. State Key Lab of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Ye Wu & Jinghua Xiao

Authors
  1. Lin Zhang
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  2. Ye Wu
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  3. Xia Shi
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  4. Zuguo He
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  5. Jinghua Xiao
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Corresponding author

Correspondence to Jinghua Xiao.

Additional information

This work was jointly supported by the National Natural Science Foundation of China (Grant Nos. 61104152, 11375033, 11262006, and 11272065) and the Fundamental Research Funds for the Central Universities.

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Zhang, L., Wu, Y., Shi, X. et al. A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators. Nonlinear Dyn 77, 99–105 (2014). https://doi.org/10.1007/s11071-014-1276-6

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