Filippov trajectories and clustering in the Kuramoto model with singular couplings

We study the synchronization of a generalized Kuramoto system in which coupling weights are determined by phase differences between oscillators. employ fast-learning regime Hebbian-like plasticity rule so that interaction oscillators is enhanced approach phases. First, we well-posedness problem for singular weighted systems Lipschitz continuity deprived. present dynamics equipped with all subcritical, critical and supercritical regimes singularity. A key fact solutions most cases must be considered Filippov's sense. characterize sticking phases subcritical case exhibit continuation criterion classical after any collision state regime. Second, prove strong to these differential inclusions can recovered as limits regular weights. also provide emergence synchronous models.