EMERGENT ASYMPTOTIC PATTERNS FOR THE DISCRETE AND CONTINUOUS WINFREE MODELS WITH INERTIA*

We study emergent dynamics of the continuous Winfree model with inertia and its discrete analogue. The Winfree model is the first mathematical model for weakly-coupled oscillators modeling a collective synchronization of pulse-coupled oscillators. Unlike the Kuramoto-type models, the Winfree model does not conserve the total phase, so that its emergent dynamics becomes more interesting. In this paper, we provide sufficient conditions for the complete oscillator death to the Winfree model in the presence of inertia and the discrete-time analogue with or without inertia. Moreover, we also present a uniform-in-time convergence from the discrete model to the continuous model for zero inertia case, as the time-step tends to zero.