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.

中文翻译：

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具有奇异耦合的 Kuramoto 模型中的 Filippov 轨迹和聚类

我们研究了广义 Kuramoto 系统的同步，其中耦合权重由振荡器之间的相位差决定。我们在类似赫布的可塑性规则中采用快速学习机制，以便通过相位方法增强振荡器之间的相互作用。首先，我们研究了 Lipschitz 连续性不成立的奇异加权 Kuramoto 系统的适定性问题。我们展示了在奇点的所有亚临界、临界和超临界状态下配备奇异权重的系统的动力学。一个关键事实是，必须按照菲利波夫的意思来考虑最奇异情况下的解决方案。我们表征了亚临界和临界情况下的相粘连，并在超临界状态下的任何碰撞状态后展示了经典解的延续标准。其次，我们证明了这些微分包含系统的强解可以恢复为规则权重的奇异极限。我们还研究了奇异和规则加权 Kuramoto 模型的同步动力学的出现。