### Abstract

We discuss the asymptotic formation and nonlinear orbital stability of phase-locked states arising from the ensemble of non-identical Kuramoto oscillators. We provide an explicit lower bound for a coupling strength on the formation of phase-locked states, which only depends on the diameters of natural frequencies and initial phase configurations. We show that, when the phases of non-identical oscillators are distributed over the half circle and the coupling strength is sufficiently large, the dynamics of Kuramoto oscillators exhibits two stages (transition and relaxation stages). In a transition stage, initial configurations shrink to configurations whose diameters are strictly less than π2 in a finite-time, and then the configurations tend to phase-locked states asymptotically. This improves previous results on the formation of phase-locked states by ChopraSpong (2009) [26] and HaHaKim (2010) [27] where their attention were focused only on the latter relaxation stage. We also show that the Kuramoto model is ^{ℓ1}-contractive in the sense that the ^{ℓ1}-distance along two smooth Kuramoto flows is less than or equal to that of initial configurations. In particular, when two initial configurations have the same averaged phases, the ^{ℓ1}-distance between them decays to zero exponentially fast. For the configurations with different phase averages, we use the method of average adjustment and translation-invariant of the Kuramoto model to show that one solution converges to the translation of the other solution exponentially fast. This establishes the orbital stability of the phase-locked states. Our stability analysis does not employ any standard linearization technique around the given phase-locked states, but instead, we use a robust ^{ℓ1}-metric functional as a Lyapunov functional. In the formation process of phase-locked states, we estimate the number of collisions between oscillators, and lowerupper bounds of the transversal phase differences.

Original language | English |
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Pages (from-to) | 735-754 |

Number of pages | 20 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 241 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2012 Apr 1 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

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## Cite this

*Physica D: Nonlinear Phenomena*,

*241*(7), 735-754. https://doi.org/10.1016/j.physd.2011.11.011