Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration
type: Item
instance of: scholarly article
title: Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration
zbMATH Open document ID: 1501.35389
author: Seung-Yeal Ha
author: Yinglong Zhang
author: Javier Morales
published in: Communications on Pure and Applied Analysis
publication date: 2022-10-24T00:00:00Z
review text: The authors of this interesting paper study the collective behaviors of the Kuramoto model with frustration. It was discussed by \textit{H. Sakaguchi} and \textit{Y. Kuramoto} [``A soluble active rotator model showing phase transitions via mutual entrainment'', Progr. Theor. Phys. 76, No. 3, 576--581 (1986; \url{doi:10.1143/PTP.76.576})] as a prototype example of synchronization of phase-coupled oscillators. Here the authors study the phase concentration for the kinetic Kuramoto-Sakaguchi (K-S) type equations with frustration via detailed estimates on the dynamics of ordered parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. It is introduced \(\theta_i=\theta_i(t)\in\mathbb{T}\) and \(\alpha \in (-\pi /2, \ \pi /2)\) which are the phase of the \(i\)-th Kuramoto oscillator and the uniform frustration between oscillators, respectively. \[ \begin{gathered} \dot{\theta}_i=\nu_i+(\kappa /N) \sum\limits_{j=1}^{N}\sin(\theta_j- \theta_i+\alpha ), \ \ t>0,\\ \theta (0)=\theta_i^{in}, \ \ i=1,\ldots ,N, \end{gathered}\tag{\(\ast\)} \] where \(\nu_i,\kappa >0\), \(N\in \mathbb{N}_{+}\) denote the natural frequency, the coupling strength, and the number of oscillators, respectively. Here the right-hand side of \((\ast )\) is Lipschitzean. The need of frustration is suggested from empirical facts in physics and biology. The presence of frustration causes some mathematical difficulties in the analysis of synchronization behaviors of oscillatory systems. Let \(f = f (t,\nu ,\theta )\) be an one-oscillator probability density function at phase \(\theta \) with natural frequency \(\nu \) at time \(t\). Then, the dynamics of \(f\) is governed by the following Cauchy problem to the K-S equation with frustration \begin{gather*} \partial_tf+\partial _{\theta } (V[f]f)=0, \ \ (\theta ,\nu ,t) \in \mathbb{T}\times\mathbb{R} \times\mathbb{R}_{+}, \\ V[f](t,\nu ,\theta )=\nu +\kappa \int_{\mathbb{T}\times\mathbb{R}} \sin{(\theta_{\ast }-\theta +\a…
Mathematics Subject Classification ID: 35Q70
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 70F99
Mathematics Subject Classification ID: 92B25
zbMATH DE Number: 7606063
zbMATH Keywords: emergent dynamics
zbMATH Keywords: Kuramoto model
zbMATH Keywords: frustration
zbMATH Keywords: order parameters
zbMATH Keywords: synchronization
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: cpaa.2021013
OpenAlex ID: W3127822866
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DOI: 10.3934/CPAA.2021013
description: scientific article; zbMATH DE number 7606063