Fast and Slow Relaxations to Bi-cluster Configurations for the Ensemble of Kuramoto Oscillators

We present asymptotic relaxation estimates to bi-cluster configurations for the ensemble of Kuramoto oscillators with two different natural frequencies which have been observed in numerical simulations. We provide a set of initial configurations with a positive Lebesgue measure in T leading to bi-(point) cluster configurations consisting of linear combinations of two Dirac measures in super-threshold and threshold-coupling regimes. In a super-threshold regime where the coupling strength is larger than the difference of two natural frequencies, we use the 1-contraction property of the Kuramoto model to derive exponential convergence toward bi-cluster configurations. The exact location of bi-cluster configurations is explicitly computable using the coupling strength, the difference of natural frequencies, and the total phase. In contrast, for the thresholdcoupling regime where the coupling strength is exactly equal to the difference of natural frequencies, the mixed ensemble of Kuramoto oscillators undergoes two dynamic phases. First, the initial configuration evolves to the segregated phase (two segregated subconfigurations consisting of the same natural frequency) in a finite time. After this segregation phase, each subconfiguration relaxes to the asymptotic phase algebraically slowly. Our analytical results provide a rigorous framework for the observed numerical simulations.