Collective behaviors of a Winfree ensemble on an infinite cylinder

Abstract

<p style='text-indent:20px;'>The Winfree model is the first phase model for synchronization and it exhibits diverse asymptotic patterns that cannot be observed in the Kuramoto model. In this paper, we propose a Winfree type model describing the aggregation of particles on the surface of an infinite cylinder. For a special case, our proposed model is in fact equivalent to the complex Winfree model. For the proposed model, we present a sufficient framework leading to the complete oscillator death and uniform <inline-formula><tex-math id="M1">\begin{document}$ \ell_p $\end{document}</tex-math></inline-formula>-stability in a large coupling regime. We also derive the corresponding kinetic model via uniform-in-time mean-field limit. In addition, we also provide several numerical simulations for the particle and compare them with analytical results.