We discuss the asymptotic frequency synchronization for the non-identical Kuramoto oscillators with time delayed interactions. We provide explicit lower bound on the coupling strength and upper bound on the time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimate by Choi et al. [Phys. D, 241(7):735–754, 2012] for the non-identical Kuramoto oscillators without time delays but also improves previous result by Schmidt et al. [Automatica, 48(12):3008–3017, 2012] in the case of homogeneous time delays where the initial phase diameter is assumed to be less than $\pi / 2$. The proof relies on a Lyapunov functional approach.

中文翻译：

---

具有延迟耦合的 Kuramoto 振荡器的指数同步

我们讨论具有时间延迟相互作用的非相同 Kuramoto 振荡器的渐近频率同步。我们在确保指数同步的初始配置方面提供了耦合强度的明确下限和时间延迟的上限。这不仅概括了 Choi等人的频率同步估计。[物理。D , 241(7):735–754, 2012] 用于没有时间延迟的非相同 Kuramoto 振荡器，但也改进了 Schmidt等人的先前结果。[ Automatica , 48(12):3008–3017, 2012] 在假设初始相位直径小于 $\pi / 2$ 的均匀时间延迟的情况下。证明依赖于李雅普诺夫泛函方法。