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Abstract <p>We suggest a system of heterogeneous limit-cycle oscillators with a nonlinear global coupling that is equivalent to the Kuramoto phase model in its long-term dynamics. The continuum limit of our system yields directly the well-known equation for the Ott-Antonsen ansatz in a consistent way, confirming implicitly the validity of the ansatz assumption. For a Lorentzian frequency distribution of the individual oscillators, in particular, the nonlinear evolution of the order parameter is determined simply by a low-dimensional dynamical system, without assuming the Ott-Antonsen ansatz, and we present a complete spectrum analysis for the linear stability. For arbitrary frequency distributions, we derive the characteristic equations for the discrete spectrum as well as the continuous spectrum for the partially synchronized state. Possible extensions to variants of the Kuramoto model and linearly coupled oscillators are discussed.</p>

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Stability of Networks of Heterogeneous Stuart–Landau Oscillators with Nonlinear Global Coupling

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • R.-S. Kim,
  • J.-S. Kang,
  • Ch.-U. Choe

摘要

Abstract

We suggest a system of heterogeneous limit-cycle oscillators with a nonlinear global coupling that is equivalent to the Kuramoto phase model in its long-term dynamics. The continuum limit of our system yields directly the well-known equation for the Ott-Antonsen ansatz in a consistent way, confirming implicitly the validity of the ansatz assumption. For a Lorentzian frequency distribution of the individual oscillators, in particular, the nonlinear evolution of the order parameter is determined simply by a low-dimensional dynamical system, without assuming the Ott-Antonsen ansatz, and we present a complete spectrum analysis for the linear stability. For arbitrary frequency distributions, we derive the characteristic equations for the discrete spectrum as well as the continuous spectrum for the partially synchronized state. Possible extensions to variants of the Kuramoto model and linearly coupled oscillators are discussed.