Synchronization hypothesis in the Winfree model

We consider \(N\) oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength \(\kappa\) and the spectrum width \(\gamma\) of the frequencies of each oscillator. In the uncoupled regime, \(\kappa=0\), each oscillator possesses its own n...

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Bibliographic Details
Published in:arXiv.org 2016-07
Main Authors: Oukil, W, Kessi, A, Thieullen, Ph
Format: Article
Language:English
Subjects:
Coupling
Hypotheses
Mathematical models
Oscillators
Resonant frequencies
Synchronism
Online Access:Get full text
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Summary:We consider \(N\) oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength \(\kappa\) and the spectrum width \(\gamma\) of the frequencies of each oscillator. In the uncoupled regime, \(\kappa=0\), each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. We say that \(N\) oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in \((\gamma,\kappa)\) of synchronization contains coupling values that are both weak and strong. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We give a numerical counter-example which shows that this hypothesis is necessary for the existence of synchronization.
ISSN:2331-8422