Filippov trajectories and clustering in the Kuramoto model with singular couplings Park, Jinyeong Poyato Sánchez, Jesús David Soler Vizcaino, Juan Segundo Kuramoto models Adaptive coupling Singular interactions Hebbian learning Filippov-type solutions Clustering Finite-time synchronization Sticking Cucker-Smale The work of J. Park was supported by the Basic Research Program through the National Research Foundation of Korea (NRF) funded by the MSIT(NRF-2020R1A4A3079066). The work of D. Poyato (D.P.) and J. Soler (J.S.) has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 639638), the MECD (Spain) research grant FPU14/06304 (D.P), the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Projects PY18-RT-2422 & A-FQM-311-UGR18 (D.P., J.S.). 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. 2021-07-07T10:35:05Z 2021-07-07T10:35:05Z 2021 info:eu-repo/semantics/article Park, J., Poyato, D., & Soler, J. (2021). Filippov trajectories and clustering in the Kuramoto model with singular couplings. Journal of the European Mathematical Society. DOI: [10.4171/JEMS/1081] http://hdl.handle.net/10481/69591 10.4171/JEMS/1081 eng info:eu-repo/grantAgreement/EC/H2020/639638 http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess Atribución 3.0 España EMS Press