单调脉冲耦合振子连续介质模型的全局分析
Global analysis of a continuum model for monotone pulse-coupled oscillators
摘要：
我们考虑一个连续的相位振子的圆通过脉冲瞬时耦合相互作用。与以往的研究相比，相关的脉冲耦合模型，在连续极限的稳定性结果是全球性的。对于控制振子演化的非线性输运方程，我们提出了一个由分位数密度之间的全变差距离诱导的全局李雅普诺夫函数.李雅普诺夫函数的单调时间演化完全表征了振子的二分行为：要么振子在有限时间内收敛到同步状态，要么它们渐近收敛到均匀分布在圆周上的异步状态。本文的结果适用于流行的相位振荡器模型（例如，著名的泄漏积分和火灾模型），并绘制了一个强大的平行有限和无穷总体的分析。此外，他们提供了一种新的方法（全球）分析脉冲耦合振荡器。
英文摘要：
We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupled models, the stability results obtained in the continuum limit are global. For the nonlinear transport equation governing the evolution of the oscillators, we propose (under technical assumptions) a global Lyapunov function which is induced by a total variation distance between quantile densities. The monotone time evolution of the Lyapunov function completely characterizes the dichotomic behavior of the oscillators: either the oscillators converge in finite time to a synchronous state or they asymptotically converge to an asynchronous state uniformly spread on the circle. The results of the present paper apply to popular phase oscillators models (e.g. the well-known leaky integrate-and-fire model) and draw a strong parallel between the analysis of finite and infinite populations. In addition, they provide a novel approach for the (global) analysis of pulse-coupled oscillators.
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