Interplay of Delayed Inter-Coupling and Synchronous Coupling in a Kuramoto–Winfree Ensemble

We study the collective dynamics of a mixed Kuramoto-Winfree ensemble with delayed inter-couplings and instantaneous intra-couplings. For homogeneous ensembles with zero natural frequencies, an energy method yields asymptotic frequency death under a small-delay condition. For heterogeneous ensembles, we establish frameworks for complete or partial phase boundedness. The rotation numbers of the corresponding oscillators therefore exist and vanish. Complete phase boundedness, together with an additional small-delay condition, also yields asymptotic frequency death. Assuming eventual small-amplitude bounds, suitable coercivity conditions, and a sufficiently small delay, we prove exponential decay of the velocity energy and exponential convergence to an equilibrium. The general boundedness conclusions alone do not guarantee these eventual bounds. Under a further small-amplitude restriction, the equilibrium is unique within the prescribed region.