Instantaneous frequencies in the Kuramoto model

TL;DR

This paper extends Kuramoto's theory to describe the distribution of instantaneous frequencies in coupled oscillators, validated through simulations and analyzed for various natural frequency distributions, including power-law tails.

Contribution

It provides a mathematical description of instantaneous frequency distribution in the Kuramoto model, incorporating probability and geometric analysis, and explores rare events for specific distributions.

Findings

Validated the theoretical distribution against numerical simulations.

Compared time-averaged and instantaneous frequency distributions.

Analyzed rare events for distributions with power-law tails.

Abstract

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.