Synchronization of diffusively-coupled limit cycle oscillators

S. Yusef Shafi; Murat Arcak; Mihailo Jovanović; Andrew K. Packard

doi:10.1016/j.automatica.2013.09.011

Synchronization of diffusively-coupled limit cycle oscillators

S. Yusef Shafi, Murat Arcak, Mihailo Jovanović, Andrew K. Packard

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction-diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.

Original language	English (US)
Pages (from-to)	3613-3622
Number of pages	10
Journal	Automatica
Volume	49
Issue number	12
DOIs	https://doi.org/10.1016/j.automatica.2013.09.011
State	Published - Dec 2013

Bibliographical note

Funding Information:
S.Y. Shafi and M. Arcak were supported in part by grants NSF ECCS-1101876 and AFOSR FA9550-11-1-0244 . M.R. Jovanovic was supported in part by NSF CAREER Award CMMI-0644793 . The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic.

Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

Keywords

Diffusively-coupled systems
Limit cycles
Structured singular value
Synchronization
Time-varying systems

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title = "Synchronization of diffusively-coupled limit cycle oscillators",

abstract = "We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction-diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.",

keywords = "Diffusively-coupled systems, Limit cycles, Structured singular value, Synchronization, Time-varying systems",

author = "Shafi, {S. Yusef} and Murat Arcak and Mihailo Jovanovi{\'c} and Packard, {Andrew K.}",

note = "Funding Information: S.Y. Shafi and M. Arcak were supported in part by grants NSF ECCS-1101876 and AFOSR FA9550-11-1-0244 . M.R. Jovanovic was supported in part by NSF CAREER Award CMMI-0644793 . The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic. Copyright: Copyright 2013 Elsevier B.V., All rights reserved.",

year = "2013",

month = dec,

doi = "10.1016/j.automatica.2013.09.011",

language = "English (US)",

volume = "49",

pages = "3613--3622",

journal = "Automatica",

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T1 - Synchronization of diffusively-coupled limit cycle oscillators

AU - Shafi, S. Yusef

AU - Arcak, Murat

AU - Jovanović, Mihailo

AU - Packard, Andrew K.

N1 - Funding Information: S.Y. Shafi and M. Arcak were supported in part by grants NSF ECCS-1101876 and AFOSR FA9550-11-1-0244 . M.R. Jovanovic was supported in part by NSF CAREER Award CMMI-0644793 . The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic. Copyright: Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/12

Y1 - 2013/12

N2 - We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction-diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.

AB - We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction-diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.

KW - Diffusively-coupled systems

KW - Limit cycles

KW - Structured singular value

KW - Synchronization

KW - Time-varying systems

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U2 - 10.1016/j.automatica.2013.09.011

DO - 10.1016/j.automatica.2013.09.011

M3 - Article

AN - SCOPUS:84888006027

VL - 49

SP - 3613

EP - 3622

JO - Automatica

JF - Automatica

SN - 0005-1098

IS - 12

ER -

Synchronization of diffusively-coupled limit cycle oscillators

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