On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators

Abstract

<p style='text-indent:20px;'>We investigate the collective behavior of synchrony for the Kuramoto and Winfree models. We first prove the <i>global convergence</i> of frequency synchronization for the non-identical Kuramoto system of three oscillators. It is shown that the uniform boundedness of the diameter of the phase functions implies complete frequency synchronization. In light of this, we show, under a suitable condition on the coupling strength and deviation of the intrinsic frequencies, that the diameter function of the phases is uniformly bounded. In a similar spirit, we also prove the <i>global convergence</i> of phase-locked synchronization for the Winfree model of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> oscillators for <inline-formula><tex-math id="M2">\begin{document}$ N\ge2 $\end{document}</tex-math></inline-formula>.