| This thesis seeks to study synchronization of coupled phase oscillators in different systems. First, we study coupled chaotic systems. We prove a sufficient condition of synchronization for coupled one-dimensional maps and estimate the size of the window of parameters where synchronization takes place. We also derive a condition of synchronization for coupled chaotic ODEs. The analysis of synchronization in these systems is illustrated with numerical experiments.;Then, the Kuramoto model of coupled phase oscillators on complete and Paley graphs is analyzed. Specifically, we identify twisted states, steady-state solutions of the Kuramoto model on complete and Paley graphs. Furthermore, we determine the stability of q-twisted state solutions (both q = 0 and q ≠ 0) for both attractive and repulsive Kuramoto models on Paley graph.;In the last chapter, we extend our study to the Kuramoto model on weighted graphs. If the coupling strength is not uniform, we find solutions composed of both synchronized and desynchronized oscillators. We numerically show that these solutions belong to the big family, the set of states for which the Kuramoto order parameter is equal to 0. The q-twisted states (q ≠ 0) also belong to this family. We show that it is a stable family of equilibria. |
