Synchronization hypothesis in the Winfree model

Hypothèse de synchronisation dans le modèle de Winfree

W Oukil (1, 2) , A Kessi (3) , Ph Thieullen (1)

1 IMB - Institut de Mathématiques de Bordeaux
2 Université Yahia Fares de Médéa
3 USTHB - Université des Sciences et de la Technologie Houari Boumediene = University of Sciences and Technology Houari Boumediene [Alger]

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.

Mots clés: desynchronization; synchronization; Winfree Model; Coupled oscillators; periodic orbit

Domaines: Systèmes dynamiques [math.DS]

W Oukil, A Kessi, Ph Thieullen. Synchronization hypothesis in the Winfree model: Synchronization in Winfree model with N oscillators. 2016. ⟨hal-01175759v4⟩