Emergent dynamics of the kuramoto ensemble under the effect of inertia

Young Pil Choi; Seung Yeal Ha; Javier Morales

doi:10.3934/dcds.2018213

Emergent dynamics of the kuramoto ensemble under the effect of inertia

Young Pil Choi, Seung Yeal Ha, Javier Morales

Department of Mathematics

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12 Citations (Scopus)

Abstract

We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.

Original language	English
Pages (from-to)	4875-4913
Number of pages	39
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	38
Issue number	10
DOIs	https://doi.org/10.3934/dcds.2018213
Publication status	Published - 2018 Oct

Bibliographical note

Funding Information:
The work of Y.-P. Choi was supported by NRF grant(No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation, the work of J. Morales was supported by the ERC grant Regularity and Stability in Partial Differential Equations(RSPDE), and the work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project(Number SSTF-BA1401-03).

Funding Information:
Acknowledgments. The work of Y.-P. Choi was supported by NRF grant(No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation, the work of J. Morales was supported by the ERC grant Regularity and Stability in Partial Differential Equations(RSPDE), and the work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project(Number SSTF-BA1401-03).

Publisher Copyright:
© 2018 American Institute of Mathematical Sciences. All rights reserved.

All Science Journal Classification (ASJC) codes

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2018213

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title = "Emergent dynamics of the kuramoto ensemble under the effect of inertia",

abstract = "We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.",

author = "Choi, {Young Pil} and Ha, {Seung Yeal} and Javier Morales",

note = "Funding Information: The work of Y.-P. Choi was supported by NRF grant(No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation, the work of J. Morales was supported by the ERC grant Regularity and Stability in Partial Differential Equations(RSPDE), and the work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project(Number SSTF-BA1401-03). Funding Information: Acknowledgments. The work of Y.-P. Choi was supported by NRF grant(No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation, the work of J. Morales was supported by the ERC grant Regularity and Stability in Partial Differential Equations(RSPDE), and the work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project(Number SSTF-BA1401-03). Publisher Copyright: {\textcopyright} 2018 American Institute of Mathematical Sciences. All rights reserved.",

year = "2018",

month = oct,

doi = "10.3934/dcds.2018213",

language = "English",

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pages = "4875--4913",

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AU - Choi, Young Pil

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PY - 2018/10

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N2 - We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.

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Emergent dynamics of the kuramoto ensemble under the effect of inertia

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