KURAMOTO ORDER PARAMETERS AND PHASE CONCENTRATION FOR THE KURAMOTO-SAKAGUCHI EQUATION WITH FRUSTRATION

We study phase concentration for the Kuramoto-Sakaguchi(K-S) equation with frustration via detailed estimates on the dynamics of order parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. When the coupling strength is sufficiently large and the size of frustration parameter is sufficiently small, we show that the amplitude order parameter has a positive lower bound uniformly in time, and we also show that the total mass concentrates on the translated phase order parameter by a frustration parameter asymptotically, whereas the mass in the region around the antipodal point decays to zero exponentially fast.