English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study the asymptotic clustering (phase-locking) dynamics for the Kuramotomodel. For the analysis of emergent asymptotic patterns in the Kuramoto flow,we introduce the pathwise critical coupling strength which yields a sharptransition from partial phase-locking to complete phase-locking, and providenontrivial upper bounds for the pathwise critical coupling strength. Numericalsimulations suggest that multi- and mono-clusters can emerge asymptotically inthe Kuramoto flow depending on the relative magnitude of the coupling strengthcompared to the sizes of natural frequencies. However, theoretical and rigorousanalysis for such phase-locking dynamics of the Kuramoto flow still lacks acomplete understanding, although there were some recent progress on thecomplete synchronization of the Kuramoto model in a sufficiently large couplingstrength regime. In this paper, we present sufficient frameworks for partialphase-locking of a majority ensemble and the complete phase-locking in terms ofthe initial phase configuration, coupling strength and natural frequencies. Asa by-product of our analysis, we obtain nontrivial upper bounds for thepathwise critical coupling strength in terms of the diameter of naturalfrequencies, initial Kuramoto order parameter and the system size $N$. We alsoshow that phase-locked states whose order parameters are less than$N^{-\frac{1}{2}}$ are linearly unstable.

Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model