  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 is deprived. 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 of 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 provide the emergence of synchronous dynamics for the singular
and regular weighted Kuramoto models.
