Synchronization scenarios in the Winfree model of coupled oscillators

Rafael Gallego, Ernest Montbrió, and Diego Pazó
Phys. Rev. E 96, 042208 – Published 17 October 2017

Abstract

Fifty years ago Arthur Winfree proposed a deeply influential mean-field model for the collective synchronization of large populations of phase oscillators. Here we provide a detailed analysis of the model for some special, analytically tractable cases. Adopting the thermodynamic limit, we derive an ordinary differential equation that exactly describes the temporal evolution of the macroscopic variables in the Ott-Antonsen invariant manifold. The low-dimensional model is then thoroughly investigated for a variety of pulse types and sinusoidal phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.

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  • Received 24 May 2017

DOI:https://doi.org/10.1103/PhysRevE.96.042208

©2017 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear Dynamics

Authors & Affiliations

Rafael Gallego1, Ernest Montbrió2, and Diego Pazó3

  • 1Departamento de Matemáticas, Universidad de Oviedo, Campus de Viesques, 33203 Gijón, Spain
  • 2Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
  • 3Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain

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Issue

Vol. 96, Iss. 4 — October 2017

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