This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincaré-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique steady state is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Two different approaches are discussed to rule out period orbits, one based on direct linearization and another one based on the theory of second additive compound matrices. Among the examples that illustrate the theoretical results is the classical Goldbeter model of the circadian rhythm.

中文翻译：

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带负反馈的单调三对角系统的全局稳定性条件

本文研究具有负反馈的单调三对角系统。这些系统具有庞加莱-本迪克森性质，这意味着，如果轨道是有界的，如果有一个唯一的稳态并且这个唯一的稳态是渐近稳定的，如果可以排除周期性轨道，那么稳态是全局渐近的稳定的。讨论了两种不同的方法来排除周期轨道，一种基于直接线性化，另一种基于第二加法复合矩阵的理论。在说明理论结果的例子中，有经典的昼夜节律的 Goldbeter 模型。