Synchronization of diffusively-coupled limit cycle oscillators

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

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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 languageEnglish (US)
Pages (from-to)3613-3622
Number of pages10
Issue number12
StatePublished - 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 2013 Elsevier B.V., All rights reserved.


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


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