# en fr Synchronization hypothesis in the Winfree model: Synchronization in Winfree model with N oscillators Hypothèse de synchronisation dans le modèle de Winfree

1 USTHB - Université des Sciences et de la Technologie Houari Boumediene Alger 2 IMB - Institut de Mathématiques de Bordeaux

Abstract : 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.

Keywords : synchronization desynchronization Winfree Model Coupled oscillators periodic orbit

Author: W Oukil - A Kessi - Ph Thieullen -

Source: https://hal.archives-ouvertes.fr/