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Nonlinear Instability of the Incoherent State for the Kuramoto–Sakaguchi–Fokker–Plank Equation

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Abstract

We study the nonlinear instability of the incoherent solution to the Kuramoto–Sakaguchi–Fokker–Plank (KSFP) equation in a large coupling strength regime. For our instability analysis, we construct an approximate, exponentially growing perturbation mode using an elementary energy method. This method does not require spectral information from the linearized KSFP equation or an explicit growing solution for the corresponding linear equation. When the distribution function of oscillator’s natural frequencies is either a Dirac measure or a bounded function with a compact support (in a small interval around the origin), the incoherent solution is nonlinearly unstable depending on the relative sizes of the coupling strength and diffusion coefficient.

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Acknowledgments

The work of S.-Y. Ha is supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03, and the work of Q.-H. Xiao is partially supported by NRF-2009-0083521(SRC) and NSFC 11171340. Part of Q.-H. Xiao’s work was done when he visited Seoul National University as a postdoctoral fellow.

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Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Seung-Yeal Ha

  2. Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Qinghua Xiao

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  1. Seung-Yeal Ha
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  2. Qinghua Xiao
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Correspondence to Qinghua Xiao.

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Ha, SY., Xiao, Q. Nonlinear Instability of the Incoherent State for the Kuramoto–Sakaguchi–Fokker–Plank Equation. J Stat Phys 160, 477–496 (2015). https://doi.org/10.1007/s10955-015-1270-5

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  • DOI: https://doi.org/10.1007/s10955-015-1270-5

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