Global analysis of a continuum model for monotone pulse-coupled oscillators

Research output: Contribution to journalArticle

Abstract

We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupledmodels, the stability results obtained in the continuum limit are global. For the nonlinear transport equation governing the evolution of the oscillators, we propose (under technical assumptions) a global Lyapunov function which is induced by a total variation distance between quantile densities. The monotone time evolution of the Lyapunov function completely characterizes the dichotomic behavior of the oscillators: either the oscillators converge in finite time to a synchronous state or they asymptotically converge to an asynchronous state uniformly spread on the circle. The results of the present paper apply to popular phase oscillators models (e.g., the well-known leaky integrate-and-fire model) and show a strong parallel between the analysis of finite and infinite populations. In addition, they provide a novel approach for the (global) analysis of pulse-coupled oscillators.

Original languageEnglish
Pages (from-to)1154-1166
Number of pages13
JournalIEEE Transactions on Automatic Control
Volume58
Issue number5
DOIs
Publication statusPublished - 2013
Externally publishedYes

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Keywords

  • Global stability
  • Impulsive coupling
  • Lyapunov function
  • Partial differential equations
  • Phase oscillators
  • Synchronization
  • Total variation distance
  • Transport equation

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