Relaxation dynamics of the Inertial Winfree model

We prove two synchronization theorems for the second-order (inertial) Winfree model of coupled oscillators. The first result is a pathwise oscillator-death theorem with explicit smallness thresholds on the natural frequencies, initial velocities, and inertia, scaling as $R_0^{3/2}$ in the initial order parameter $R_0$. The second result is a qualitative zero-inertia synchronization statement: under generic initial data, if the intrinsic and initial velocity spreads are small compared to $\kappa$ and the inertia $m$ is small, then the limiting order parameter can be made arbitrarily close to 2. The proof of the first result is organized around three mechanisms, namely inertial gradient flow and the {\L}ojasiewicz theorem, an initial layer argument, and an order-parameter bootstrapping argument. The proof of the second result involves approximation to the first-order case via a quantitative Tikhonov theorem.