Julio D. da Fonseca | Searcharxiv

arXiv2022

Density of instantaneous frequencies in the Kuramoto-Sakaguchi model

We obtain a formula for the statistical distribution of instantaneous frequencies in the Kuramoto-Sakaguchi model. This work is based on the Kuramoto-Sakaguchi's theory of globally coupled phase oscillators, which we review in full detail by discussing its assumptions and showing all steps behind the derivation of its main results. Our formula is a stationary probability density function with a complex mathematical structure, is consistent with numerical simulations and gives a description of the stationary collective states of the Kuramoto-Sakaguchi model.

nlin.AO↗

arXiv2020

Instantaneous frequencies in the Kuramoto model

Using the main results of the Kuramoto theory of globally coupled phase oscillators combined with methods from probability and generalized function theory in a geometric analysis, we extend Kuramoto's results and obtain a mathematical description of the instantaneous frequency (phase-velocity) distribution. Our result is validated against numerical simulations, and we illustrate it in cases where the natural frequencies have normal and Beta distributions. In both cases, we vary the coupling strength and compare systematically the distribution of time-averaged frequencies (a known result of Kuramoto theory) to that of instantaneous frequencies, focussing on their qualitative differences near the synchronized frequency and in their tails. For a class of natural frequency distributions with power-law tails, which includes the Cauchy-Lorentz distribution, we analyze rare events by means of an asymptotic formula obtained from a power series expansion of the instantaneous frequency distribution.

nlin.CD↗