# Long time dynamics for interacting oscillators on dense graphs

Abstract : The long time dynamics of the stochastic Kuramoto model defined on a graph is analyzed in the subcritical regime. The emphasis is posed on the relationship between the mean field behavior and the connectivity of the underlying graph: we give an explicit deterministic condition on the sequence of graphs such that, for any initial condition, even dependent on the network, the system approaches the unique stable stationary solution and it remains close to it, up to almost exponential times. The condition on the sequence of graphs is expressed through a concentration in $\ell_{\infty}\to \ell_1$ norm and it is shown to be satisfied by a large class of graphs, random and deterministic, provided that the number of neighbors per site diverges, as the size of the system tends to infinity.
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Cited literature [28 references]

https://hal.archives-ouvertes.fr/hal-02263485
Contributor : Fabio Coppini <>
Submitted on : Sunday, August 4, 2019 - 10:28:54 PM
Last modification on : Wednesday, August 7, 2019 - 1:11:02 AM

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• HAL Id : hal-02263485, version 1
• ARXIV : 1908.01520

### Citation

Fabio Coppini. Long time dynamics for interacting oscillators on dense graphs. 2019. ⟨hal-02263485⟩

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