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.
Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay
Institute of Science Tokyo (former Tokyo Tech)（东京科学大学（原东京工业大学））；University of Namur（那慕尔大学）；Tokyo University of Science（东京理科大学）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心 (iTHEMS)）；Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
Stability of Hopf bifurcations in Time-delayed fully-connected PLL networks
时滞微分方程（DDEs）的动力学是一个很好的研究问题，主要是因为DDEs出现在许多科学领域的模型中，包括生物学、生理学、种群动力学和工程学。在\textit{Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators}中研究了锁相环振荡器网络参数空间解的性质变化。Communications in Nonlinear Science and数值模拟，Elsevier BV，第22卷，第1-3期，2015年5月，第793-820页，其中，对于两种情况，即保序同步和破序同步，都很好地建立了Hopf分叉的存在性。在这项工作中，我们继续分析探讨的稳定性的小振幅周期解附近出现的Hopf分支的不动点子空间，基于减少到一个二维中心流形的无限维空间。为了证实我们的分析结果，给出了数值模拟。虽然我们探讨了二阶振子的网络动力学，但结果可扩展到更高阶的节点。
