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Dynamics of clustering patterns in the Kuramoto model with unidirectional coupling

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Abstract

We study the synchronization transition in the Kuramoto model by considering a unidirectional coupling with a chain structure. The microscopic clustering features are characterized in the system. We identify several clustering patterns for the long-time evolution of the effective frequencies and reveal the phase transition between them. Theoretically, the recursive approach is developed in order to obtain analytical insights; the essential bifurcation schemes of the clustering patterns are clarified and the phase diagram is illustrated in order to depict the various phase transitions of the system. Furthermore, these recursive theories can be extended to a larger system. Our theoretical analysis is in agreement with the numerical simulations and can aid in understanding the clustering patterns in the Kuramoto model with a general structure.

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Acknowledgements

This work was supported partly by the National Natural Science Foundation of China (Grant Nos. 11605055 and 11475022), the Fundamental Research Funds for the Central Universities of China (Grant No. 2017MS054), and the Scientific Research Funds of Huaqiao University (Grant Nos. 600005- Z17Y0064 and 15BS401) and China Scholarship Council (CSC).

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Authors and Affiliations

  1. School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

    Xia Huang & Jin Dong

  2. Cardiovascular Research Laboratories, Departments of Medicine and Biomathematics, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, CA, 90095, USA

    Xia Huang

  3. Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen, 361021, China

    Wen-Jing Jia, Zhi-Gang Zheng & Can Xu

Authors
  1. Xia Huang
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  2. Jin Dong
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  3. Wen-Jing Jia
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  4. Zhi-Gang Zheng
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  5. Can Xu
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Correspondence to Xia Huang or Can Xu.

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Huang, X., Dong, J., Jia, WJ. et al. Dynamics of clustering patterns in the Kuramoto model with unidirectional coupling. Front. Phys. 13, 130506 (2018). https://doi.org/10.1007/s11467-018-0783-1

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