@misc{10481/69591,
year = {2021},
url = {http://hdl.handle.net/10481/69591},
abstract = {We study the synchronization of a generalized Kuramoto system in which the coupling
weights are determined by the phase differences between oscillators. We employ the fast-learning
regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by
the approach of phases. First, we study the well-posedness problem for the singular weighted Kuramoto
systems in which the Lipschitz continuity fails to hold. We present the dynamics of the
system equipped with singular weights in all the subcritical, critical and supercritical regimes of
the singularity. A key fact is that solutions in the most singular cases must be considered in Filippov’s
sense. We characterize sticking of phases in the subcritical and critical case and we exhibit
a continuation criterion for classical solutions after any collision state in the supercritical regime.
Second, we prove that strong solutions to these systems of differential inclusions can be recovered
as singular limits of regular weights.We also study the emergence of synchronous dynamics for the
singular and regular weighted Kuramoto models.},
organization = {Basic Research Program through the National Research Foundation of Korea (NRF) - MSIT 

NRF-2020R1A4A3079066},
organization = {European Research Council (ERC)

639638},
organization = {MECD (Spain) 

FPU14/06304},
organization = {MINECO-Feder (Spain) 

RTI2018-098850-B-I00},
organization = {Junta de Andalucia

European Commission

PY18-RT-2422
A-FQM-311-UGR18},
publisher = {EMS Press},
keywords = {Kuramoto models},
keywords = {Adaptive coupling},
keywords = {Singular interactions},
keywords = {Hebbian learning},
keywords = {Filippov-type solutions},
keywords = {Clustering},
keywords = {Finite-time synchronization},
keywords = {Sticking},
keywords = {Cucker-Smale},
title = {Filippov trajectories and clustering in the Kuramoto model with singular couplings},
doi = {10.4171/JEMS/1081},
author = {Park, Jinyeong and Poyato Sánchez, Jesús David and Soler Vizcaino, Juan Segundo},
}