The mathematics of asymptotic stability in the Kuramoto model

Helge Dietert, Bastien Fernandez

Published 2018-01-04Version 1

Now a standard in Nonlinear Sciences, the Kuramoto model epitomizes the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been predicted in early works on this model, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions (and also globally rotating ones, by symmetry) in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. Various examples, additional original results and some extensions to common developments of the model are also given, which complete thorough understanding of the dynamics over a broad range of parameters and situations.