From pulsars to pedestrians and bacteria to brain cells, objects that exhibit cyclical behavior, called oscillators, are found in a variety of different settings. When oscillators adjust their behavior in response to nearby oscillators, they often achieve a state of synchrony, in which they all have the same phase and frequency. Here, we explore the Kuramoto model, a simple and general model which describes oscillators as dynamical systems on a graph and has been used to study synchronization in systems ranging from firefly swarms to the power grid. We discuss analytical and numerical methods used to investigate the governing system of differential equations and the conditions that lead to synchronization, and demonstrate that perfect synchronization occurs only under strict conditions and for specific graph structures. We also present results from an experiment with coupled metronomes in which spontaneous emergence of synchronization, consistent with the mathematical theory, can be observed in a real-world setting
Linville, Dylan and Fontes, Daniel Trugillo Martins, "Spontaneous synchrony on graphs and the emergence of order from disorder" (2015). Mathematical Sciences Technical Reports (MSTR). 152.