Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling

In this paper, we investigate the local synchronization problem for delayed phase-coupled oscillators with plastic coupling (also referred as a generalized Kuramoto model). By linearizing the phase-coupled oscillators system at a special solution, an approximation phase-coupled system with delay effects is deduced. Moreover, we find the multiplicity of semisimple zero eigenvalue for coupling strength coefficients matrix and the rank of Vandermonde matrix associated initial values are two sensitive factors. The former is closely related to the maximum number of clusters phase of linear coupled system, and the latter determines clusters phase synchronization. As results, both frequency synchronization criteria and phase synchronization criteria are established in this literature. Also, the final synchronization frequency and phase are formulated in terms of coupling strength, initial values and time delay.