Continuum limit of the adaptive Kuramoto model

We investigate the dynamics of the adaptive Kuramoto model with slow adaptation in the continuum limit, $N\to \mathrm{\infty }$. This model is distinguished by dense multistability, where multiple states coexist for the same system parameters. The underlying cause of this multistability is that some oscillators can lock at different phases or switch between locking and drifting depending on their initial conditions. We identify new states, such as two-cluster states. To simplify the analysis we introduce an approximate reduction of the model via row-averaging of the coupling matrix. We derive a self-consistency equation for the reduced model and present a stability diagram illustrating the effects of positive and negative adaptation. Our theoretical findings are validated through numerical simulations of a large finite system. Comparisons to previous work highlight the significant influence of adaptation on synchronization behavior.