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The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas
arXiv (Cornell University)
19 Dec 2016
Abstract
In his classical work on synchronization, Kuramoto derived the formula for
the critical value of the coupling strength corresponding to the transition to
synchrony in large ensembles of all-to-all coupled phase oscillators with
randomly distributed intrinsic frequencies. We extend the Kuramoto's result to
a large class of coupled systems on convergent families of deterministic and
random graphs. Specifically, we identify the critical values of the coupling
strength (transition points), between which the incoherent state is linearly
stable and is unstable otherwise. We show that the transition points depend on
the largest positive or/and smallest negative eigenvalue(s) of the kernel
operator defined by the graph limit. This reveals the precise mechanism, by
which the network topology controls transition to synchrony in the Kuramoto
model on graphs. To illustrate the analysis with concrete examples, we derive
the transition point formula for the coupled systems on Erd\H{o}s-R\'{e}nyi,
small-world, and $k$-nearest-neighbor families of graphs. As a result of
independent interest, we provide a rigorous justification for the mean field
limit for the Kuramoto model on graphs. The latter is used in the derivation of
the transition point formulas.
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Details
- Title
- The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas
- Creators
- Hayato ChibaGeorgi S Medvedev
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862734104721