We consider two identical oscillators with weak, time delayed coupling. We start with a general system of delay differential equations then reduce it to a phase model. With the assumption of large time delay, the resulting phase model has an explicit delay and phase shift in the argument of the phases and connection function, respectively. Using the phase model, we prove that for any type of oscillators and any coupling, the in-phase and anti-phase phase-locked solutions always exist and give conditions for their stability. We show that for small delay these solutions are unique, but with large enough delay multiple solutions of each type with different frequencies may occur. We give conditions for the existence and stability of other types of phase-locked solutions. We discuss the various bifurcations that can occur in the phase model as the time delay is varied. The results of the phase model analysis are applied to Morris–Lecar oscillators with diffusive coupling and compared with numerical studies of the full system of delay differential equations. We also consider the case of small time delay and compare the results with the existing ones in the literature.

我们考虑两个具有弱延时耦合的相同振荡器。我们从一个一般的时滞微分方程系统开始，然后将其简化为一个相位模型。假设有较大的时间延迟，则生成的相位模型在相位和连接函数的参数中分别具有显式的延迟和相移。使用相位模型，我们证明对于任何类型的振荡器和任何耦合，同相和反相锁相解始终存在，并为其稳定性提供了条件。我们表明，对于较小的延迟，这些解决方案是唯一的，但对于足够大的延迟，则可能会出现每种类型的具有不同频率的多个解决方案。我们给出了其他类型的锁相解的存在性和稳定性的条件。我们讨论了随着时间延迟的变化，相位模型中可能发生的各种分歧。相位模型分析的结果应用于具有扩散耦合的Morris-Lecar振荡器，并与整个时滞微分方程组的数值研究进行了比较。我们还考虑了时间延迟小的情况，并将结果与​​文献中的现有结果进行比较。