The Kuramoto model (KM) of coupled phase oscillators on graphs provides the most influential framework for studying collective dynamics and synchronization. It exhibits a rich repertoire of dynamical regimes. Since the work of Strogatz and Mirollo, the mean field equation derived in the limit as the number of oscillators in the KM goes to infinity, has been the key to understanding a number of interesting effects, including the onset of synchronization and chimera states. In this work, we study the mathematical basis of the mean field equation as an approximation of the discrete KM. Specifically, we extend the Neunzert's method of rigorous justification of the mean field equation to cover interacting dynamical systems on graphs. We then apply it to the KM on convergent graph sequences with non-Lipschitz limit. This family of graphs includes many graphs that are of interest in applications, e.g., nearest-neighbor and small-world graphs.
Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type
National Tsing Hua University（国立清华大学）；National Center for Theoretical Sciences National Taiwan University（国立台湾大学理论科学中心）；Institut für Mathematik Freie Universität Berlin（柏林自由大学数学研究所）；Department of Mathematics National Taiwan University（国立台湾大学数学系）
