Partially Phase-Locked Solutions to the Kuramoto Model

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The Kuramoto model is a canonical model for understanding phase-locking phenomenon. It is well-understood that, in the usual mean-field scaling, full phase-locking is unlikely and that it is partially phase-locked states that are important in applications. Despite this, while there has been much attention given to existence and stability of fully phase-locked states in the finite N Kuramoto model, the partially phase-locked states have received much less attention. In this paper we present two related results. Firstly, we derive an analytical criteria that, for sufficiently strong coupling, guarantees the existence of a partially phase-locked state by proving the existence of an attracting ball around a fixed point of a subset of the oscillators. We also derive a larger invariant ball such that any point in it will asymptotically converge to the attracting ball. Secondly, we consider the large N behavior of the finite N Kuramoto system with randomly distributed frequencies. In the case where the frequencies are independent and identically distributed we use a result of De Smet and Aeyels on partial entrainment to derive a condition giving (with high probability) the existence of a partially entrained cluster. We also derive upper and lower bounds on the size of the largest entrained cluster, together with a lower bound on the order parameter. Interestingly in a series of numerical experiments we find that the observed size of the largest entrained cluster is predicted extremely well by the upper bound.

Original languageEnglish (US)
Article number46
JournalJournal of Statistical Physics
Issue number3
StatePublished - Jun 2021


  • Mean-field scaling
  • Phase-locking
  • Synchronization
  • Thermodynamic limit

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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