We study the emergence of asymptotic patterns in Winfree ensemble such as the partial/complete phase-locking and bump states under the effect of heterogeneous frustrations. Although the Winfree model is the first model for the synchronization of limit-cycle oscillators, there is little literature on the mathematical validity of asymptotic patterns compared to the vast literature of the well-studied Kuramoto model. Recently, it has received a renewed attention in nonlinear dynamics and statistical physics communities due to its diverse asymptotic patterns that it can generate. In particular, we provide a rigorous result on the existence of bump states in a homogeneous ensemble with the same natural frequency. Our provided results exhibit the robustness of emerging asymptotic patterns with respect to frustrations. We derive several sufficient frameworks for the unique existence of an equilibrium state, bump states and uniform stability with respect to initial data.

我们研究了 Winfree 集成中渐近模式的出现，例如在异构挫折的影响下的部分/完全锁相和碰撞状态。尽管 Winfree 模型是第一个用于同步极限循环振荡器的模型，但与经过充分研究的 Kuramoto 模型的大量文献相比，关于渐近模式的数学有效性的文献很少。最近，由于它可以生成多种渐近模式，它在非线性动力学和统计物理学界重新受到关注。特别是，我们提供了具有相同固有频率的均匀系综中存在碰撞状态的严格结果。我们提供的结果展示了新兴渐近模式在挫折方面的稳健性。