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

Global stability; Impulsive coupling; Lyapunov function; Partial differential equations; Phase oscillators; Synchronization; Total variation distance; Transport equation; Instantaneous couplings; Leaky integrate-and-fire model; Nonlinear transport equations; Pulse-coupled oscillators; Total variation; Continuum mechanics; Nonlinear equations; Oscillators (mechanical); Phase modulation; Lyapunov functions

Abstract :

[en] 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. © 2012 IEEE.

Disciplines :

Engineering, computing & technology: Multidisciplinary, general & others
Mathematics

Author, co-author :

Mauroy, Alexandre ; Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation

Sepulchre, Rodolphe ; Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation

Language :

English

Title :

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

Publication date :

2013

Journal title :

IEEE Transactions on Automatic Control

ISSN :

0018-9286

Publisher :

Institute of Electrical and Electronics Engineers, United States - New Jersey

Volume :

Issue :

Pages :

1154-1166

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 03 April 2014

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