In this talk, we investigate emergent behaviors in infinite ensembles of weakly coupled oscillators, focusing on two prototypes: the Winfree model and the Lohe matrix model. Synchronization, where interacting oscillators adjust their rhythms through weak coupling, is a fundamental phenomenon observed across natural and engineered systems.

In the first part, we examine the infinite-size Winfree model in three formulations: a continuous-time system over a countable set of oscillators, its discrete-time approximation via the first-order Euler scheme, and a continuum model governed by an integro-differential equation. For each setting, we establish sufficient conditions for the emergence of asymptotic patterns such as boundedness, quasi-steady states, global equilibrium, stability, and phase-locking. We also prove uniform-in-time convergence of the discrete dynamics to the continuous one and the continuum limit.

In the second part, we turn to the infinite Lohe matrix model, which generalizes the Kuramoto model to the space of unitary matrices. We analyze the emergence of quasi-steady states, complete synchronization, and orbital stability under suitable conditions. Our results reveal the distinct emergent behaviors specific to the infinite-dimensional setting, which have no finite-dimensional counterparts. We also use nonlinear functionals to describe the system's relaxation toward synchronized states.

Overall, this talk aims to provide a comprehensive understanding of emergent collective dynamics in an infinite set of weakly coupled oscillators without relying on the mean-field approximation.