Predicting Finite Size Effects in the Kuramoto Model

Synchronization is a key phenomenon of coupled nonlinear oscillators. The phase transition evident in many oscillator models has been thoroughly studied in the Kuramoto model, in large part because the order parameter can be obtained analytically for a number of infinite-sized distributions. Nearly all schemes for predicting the order parameter only apply to specific distributions and infinite populations. Schemes for finite populations are rarer and have focused on more tightly prescribed distributions. Finite size effects are essentially uncharacterized. In this talk I will discuss finite size effects in the Kuramoto model. I will begin by illustrating that subsets of oscillators form coherent clumps. For a given population and coupling strength these clumps are reproducible, but they vary widely from one population to the next. Approximations based on the existence and structure of these subsets leads to population specific predictions for subset composition. From these, one can obtain predictions for the average order parameter and its deviation as a function of coupling strength. The predictions exhibit qualitative agreement for a range of distributions, with statistically significant agreement for certain population sizes and coupling strengths.