Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model

Young Pil Choi, Seung Yeal Ha, Sungeun Jung, Yongduck Kim

Research output: Contribution to journalArticle

79 Citations (Scopus)

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 languageEnglish
Pages (from-to)735-754
Number of pages20
JournalPhysica D: Nonlinear Phenomena
Volume241
Issue number7
DOIs
Publication statusPublished - 2012 Apr 1

Fingerprint

Kuramoto Model
Orbital Stability
orbitals
configurations
oscillators
Configuration
Linearization
Natural frequencies
linearization
resonant frequencies
Linearization Techniques
adjusting
Phase Difference
Nonlinear Stability
Lyapunov Functional
Less than or equal to
Natural Frequency
collisions
Stability Analysis
decay

All Science Journal Classification (ASJC) codes

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

Cite this

Choi, Young Pil ; Ha, Seung Yeal ; Jung, Sungeun ; Kim, Yongduck. / Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. In: Physica D: Nonlinear Phenomena. 2012 ; Vol. 241, No. 7. pp. 735-754.
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Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. / Choi, Young Pil; Ha, Seung Yeal; Jung, Sungeun; Kim, Yongduck.

In: Physica D: Nonlinear Phenomena, Vol. 241, No. 7, 01.04.2012, p. 735-754.

Research output: Contribution to journalArticle

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