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Remarks on the stability properties of the Kuramoto–Sakaguchi–Fokker–Planck equation with frustration

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Abstract

We present stability properties of the Kuramoto–Sakaguchi–Fokker–Planck (in short, KS–FP) equation with frustration arising from the synchronization modeling of a large ensemble of weakly coupled oscillators under the effect of uniform frustration (phase-lag). Even in the absence of Fokker–Planck term (i.e., the diffusion term), the existence of frustration destroys a gradient flow structure of the original kinetic Kuramoto equation. Thus, useful machineries from the gradient flow theory cannot be used in the large-time behavior of the equation as it is. In this paper, we address two stability estimates for the KS–FP equation. First, we present a sufficient framework leading to the asymptotic stability of the incoherent state. Our stability framework has been formulated in terms of the probability density function of natural frequencies, coupling strength, diffusion coefficient, size of frustration and initial data. Second, we show that the KS–FP equation is structurally stable with respect to the size of frustration. More precisely, we show that the solution to the KS–FP equation tends to the corresponding solution to the KS–FP with zero frustration, as the size of frustration tends to zero under a suitable framework.

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Acknowledgements

The work of S.-Y. Ha is supported by the NRF Grant (2017R1A2B2001864).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea

    Seung-Yeal Ha

  2. Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Republic of Korea

    Seung-Yeal Ha

  3. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea

    Doheon Kim, Jaeseung Lee & Yinglong Zhang

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  1. Seung-Yeal Ha
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Correspondence to Doheon Kim.

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Ha, SY., Kim, D., Lee, J. et al. Remarks on the stability properties of the Kuramoto–Sakaguchi–Fokker–Planck equation with frustration. Z. Angew. Math. Phys. 69, 94 (2018). https://doi.org/10.1007/s00033-018-0984-z

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  • DOI: https://doi.org/10.1007/s00033-018-0984-z

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