title: Approximate solution for frequency synchronization in a finite-size Kuramoto model.
abstract: Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behavior, such as frequency synchronization (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, with arbitrary finite-variance distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behavior of networks.
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, AB24 3UE Aberdeen, United Kingdom.
authors.forename: Chengwei
authors.initials: C
authors.lastname: Wang
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, AB24 3UE Aberdeen, United Kingdom.|Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay.
authors.forename: Nicolás
authors.initials: N
authors.lastname: Rubido
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, AB24 3UE Aberdeen, United Kingdom.
authors.forename: Celso
authors.initials: C
authors.lastname: Grebogi
authors.affiliation: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, King's College, AB24 3UE Aberdeen, United Kingdom.
authors.forename: Murilo S
authors.initials: MS
authors.lastname: Baptista