# Paper on Entrainment of Winfree Oscillators

In a recent work with Igor Hoveijn and Yongjiao Zhang, motivated by the question of entrainment with the daily dark–light cycle, we considered a model consisting of globally coupled Winfree oscillators under the influence of a periodic external forcing term. The paper has just appeared in Chaos. From the paper:

Coupled oscillators have been the paradigmatic model for the study of synchronization since the works of Winfree and Kuramoto. In this paper, motivated by the question of entrainment with the daily dark–light cycle, we consider globally coupled Winfree oscillators under the influence of a periodic external forcing term. To describe the effect of the external forcing, we introduce and study in detail the entrainment degree which is the proportion of oscillators that synchronize to the forcing. The numerical study of the entrainment degree reveals clear trends on the influence of the strength of the external forcing that we discuss in detail. These results are accompanied by a theoretical study for the case of identical oscillators. In the case of non- identical oscillators whose natural frequencies follow a Lorentz distribution, to compute the entrainment degree, we apply the Ott–Antonsen Ansatz to obtain a low-dimensional dynamical description of the order parameter. However, the dynamics of the order parameter does not provide direct information about the entrainment degree. To overcome this problem, we simulate the dynamics of individual oscillators in the time-dependent mean-field predicted by the Ott–Antonsen Ansatz, and we use this to estimate the proportion of oscillators that synchronize with the external forcing.