Bounds for the asymptotic order parameter of the stochastic Kuramoto model

TL;DR

This paper derives sharp bounds and approximations for the asymptotic order parameter of the stochastic Kuramoto model using inequalities for Bessel functions and mathematical approximation techniques.

Contribution

It introduces new bounds and approximation methods for the order parameter, enhancing understanding of the stochastic Kuramoto model's behavior.

Findings

Sharp lower and upper bounds for the asymptotic order parameter

Approximate formulas using Lagrange inversion and rational approximation

Improved analytical tools for studying stochastic synchronization

Abstract

Tur\'an type inequalities for modified Bessel functions of the first kind are used to deduce some sharp lower and upper bounds for the asymptotic order parameter of the stochastic Kuramoto model. Moreover, approximation from the Lagrange inversion theorem and a rational approximation are given for the asymptotic order parameter.