Existence of attractor and quasi-steady state for time-discrete infinite Winfree model

We study the discrete-time dynamics of infinite Winfree model which corresponds to the infinite counterpart of the Winfree model. The time-discrete model can be realized via the first-order Euler discretization of the continuous infinite model. For this proposed time-discrete model, we discuss the emergence of complete oscillator death phenomenon which commonly occurs with a bounded trajectory of solutions. We study how large the coupling strength should be for the emergence of death, and suggest two different scenarios that the phase configuration tends to the equilibrium exponentially fast or gets stuck with a steady phase diameter due to the infinite number of oscillators. After we specify a sufficient condition that the solution converges to an equilibrium, under similar circumstances, we provide a continuous transition from the time-discrete model to the time-continuous model in an infinite-time horizon.