Synchronization of coupled oscillators is a game

Huibing Yin, Prashant G. Mehta, Sean P. Meyn, Vinayak V. Shanbhag

Research output: Contribution to journalArticle

55 Citations (Scopus)

Abstract

The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, and economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population linear quadratic Gaussian (LQG) model of Huang 2007, comprised here of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function. The states are interpreted as phase angles for a collection of heterogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents opt out of the game, setting their controls to zero, resulting in the incoherence equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the partial differential equation (PDE) model about the incoherence equilibrium. A critical value of the control cost parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

Original languageEnglish (US)
Article number6018994
Pages (from-to)920-935
Number of pages16
JournalIEEE Transactions on Automatic Control
Volume57
Issue number4
DOIs
StatePublished - Apr 1 2012

Fingerprint

Synchronization
Partial differential equations
Bifurcation (mathematics)
Stochastic systems
System theory
Linearization
Cost functions
Dynamical systems
Differential equations
Phase transitions
Economics
Costs
Experiments

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

Cite this

Yin, Huibing ; Mehta, Prashant G. ; Meyn, Sean P. ; Shanbhag, Vinayak V. / Synchronization of coupled oscillators is a game. In: IEEE Transactions on Automatic Control. 2012 ; Vol. 57, No. 4. pp. 920-935.
@article{1b00b5a2a6354e6295a9d41a70591be9,
title = "Synchronization of coupled oscillators is a game",
abstract = "The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, and economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population linear quadratic Gaussian (LQG) model of Huang 2007, comprised here of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function. The states are interpreted as phase angles for a collection of heterogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents opt out of the game, setting their controls to zero, resulting in the incoherence equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the partial differential equation (PDE) model about the incoherence equilibrium. A critical value of the control cost parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.",
author = "Huibing Yin and Mehta, {Prashant G.} and Meyn, {Sean P.} and Shanbhag, {Vinayak V.}",
year = "2012",
month = "4",
day = "1",
doi = "10.1109/TAC.2011.2168082",
language = "English (US)",
volume = "57",
pages = "920--935",
journal = "IEEE Transactions on Automatic Control",
issn = "0018-9286",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "4",

}

Synchronization of coupled oscillators is a game. / Yin, Huibing; Mehta, Prashant G.; Meyn, Sean P.; Shanbhag, Vinayak V.

In: IEEE Transactions on Automatic Control, Vol. 57, No. 4, 6018994, 01.04.2012, p. 920-935.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Synchronization of coupled oscillators is a game

AU - Yin, Huibing

AU - Mehta, Prashant G.

AU - Meyn, Sean P.

AU - Shanbhag, Vinayak V.

PY - 2012/4/1

Y1 - 2012/4/1

N2 - The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, and economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population linear quadratic Gaussian (LQG) model of Huang 2007, comprised here of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function. The states are interpreted as phase angles for a collection of heterogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents opt out of the game, setting their controls to zero, resulting in the incoherence equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the partial differential equation (PDE) model about the incoherence equilibrium. A critical value of the control cost parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

AB - The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, and economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population linear quadratic Gaussian (LQG) model of Huang 2007, comprised here of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function. The states are interpreted as phase angles for a collection of heterogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents opt out of the game, setting their controls to zero, resulting in the incoherence equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the partial differential equation (PDE) model about the incoherence equilibrium. A critical value of the control cost parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=84859709769&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859709769&partnerID=8YFLogxK

U2 - 10.1109/TAC.2011.2168082

DO - 10.1109/TAC.2011.2168082

M3 - Article

AN - SCOPUS:84859709769

VL - 57

SP - 920

EP - 935

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 4

M1 - 6018994

ER -