SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 3, Page 2135-2161, January 2020.
The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical system consisting of a central circadian oscillator that drives a peripheral circadian oscillator. We develop techniques to reduce the higher-dimensional phase space of the coupled system to derive a generalized two-dimensional entrainment map. Determining the nature of various fixed points, together with an understanding of their stable and unstable manifolds, leads to conditions for existence and stability of periodic orbits of the circadian system. We use the map to investigate how various properties of solutions depend on parameters and initial conditions including the time to and direction of entrainment. We show that the concepts of phase advance and phase delay need to be carefully assessed when considering hierarchical systems.

中文翻译：

---

二维映射揭示的强迫分层昼夜节律系统的携带动力学

SIAM应用动力系统杂志，第19卷，第3期，第2135-2161页，2020年1月。
昼夜节律系统携带24小时明暗循环的能力是其最重要的特性之一。最近引入了一种称为夹带图的新工具来研究单个振荡器的这一过程。在这里，我们概括了该图，以研究由中心昼夜节律振荡器驱动外围昼夜节律振荡器组成的分层系统中的明暗强迫效应。我们开发了减少耦合系统的高维相空间的技术，以得出广义的二维夹带图。确定各个不动点的性质以及对它们的稳定和不稳定流形的理解，将为生物钟系统的周期轨道的存在和稳定性提供条件。我们使用该图调查溶液的各种属性如何取决于参数和初始条件，包括到达时间和夹带的方向。我们表明，在考虑分层系统时，需要仔细评估相位超前和相位延迟的概念。