Some problems arising from the dynamics of the Kuramoto oscillators

Abstract

In this thesis, we study several problems on the ensemble of Kuramoto oscillators. We present the nonlinear stability of the phase-locked states using a robust $\ell_1$-metric as a Lyapunov functional. We show that the phase-locked states are congruent each other in the sense that one phase-locked state is the simply translation of the other and phase-shift is the difference of averaged initial phases. We also show the contraction property for measure valued solutions of the kinetic Kuramoto model. We next consider the effect of interaction frustration on the complete synchronization of Kuramoto oscillators. In general, interaction frustration hinders the formation of complete frequency synchronization. For more quantitative estimates, we considered three Kuramoto-type models. Our first model is for an ensemble of Kuramoto oscillators with uniform interaction frustration. Our second model is, as a special case of the first model, a mixture of two identical Kuramoto oscillator groups with distinct natural frequencies. Our third model is like the Kuramoto model for identical oscillators on the bipartite graph. Finally, we investigate the intricate interplay between the inertia and frustration in an ensemble of Kuramoto oscillators. We cannot apply the explicit macro-micro decomposition to reduce the dynamics of initial phases to that of fluctuations. However, we can still derive second-order differential inequalities for the phase or frequency diameters so that the second-order Gronwall inequality method still works well. Moreover, both the analytical and numerical studies demonstrate this fact.