AI 中文总结
本文针对振荡系统提出周期稳态理论，利用分岔与奇点理论在参数化ODE模型中研究非退化霍普夫分岔产生的稳定周期解，连接了振荡产生与周期稳态并给出参数空间几何描述。
AI 中文摘要
稳态（Homeostasis）是一种生物学现象，指当输入参数变化时，系统状态的某一函数近似保持恒定。近期的数学研究已开发出利用奇点理论研究稳态的框架，在该框架中，系统状态函数近似恒定的要求被替换为该函数对输入参数的导数在孤立点处消失的条件，这一零导数条件被称为无穷小稳态（infinitesimal homeostasis）。现有理论大多关注稳态，而本文针对振荡系统发展了类似理论，其中稳态量是稳定周期解的周期，我们将该行为称为周期稳态（period homeostasis），对应的零导数条件称为无穷小周期稳态（infinitesimal period homeostasis）。蓝细菌及培养的人类细胞实验研究表明，随着温度降至临界值，昼夜节律可通过霍普夫分岔消失，振荡幅度趋于零。受该证据启发，本文聚焦非退化霍普夫分岔产生的稳定周期解，证明分岔理论与奇点理论可用于在参数化常微分方程（ODE）模型中寻找周期稳态，研究结果将振荡的产生与周期稳态相联系，并给出参数空间的几何描述，该描述可局部组织ODE解的定性行为。
英文摘要
Homeostasis is a biological phenomenon in which a function of the state of the system remains approximately constant as an input parameter varies. Recent mathematical work has developed a framework for studying homeostasis using methods from singularity theory. In this framework, the requirement that the function of the state remain approximately constant is replaced by the condition that the derivative of the function with respect to the input parameter vanishes at an isolated point. This zero-derivative condition is called infinitesimal homeostasis. Much of the existing theory concerns steady states. In this paper, we develop an analogous theory for oscillatory systems in which the homeostatic quantity is the period of a stable periodic solution. We refer to this behavior as period homeostasis and to the corresponding zero-derivative condition as infinitesimal period homeostasis. Experimental studies in cyanobacteria and cultured human cells suggest that circadian rhythms can disappear through Hopf bifurcation as temperature decreases to a critical value, with the oscillation amplitude tending to zero. Motivated by this evidence, we focus on stable periodic solutions arising from nondegenerate Hopf bifurcation. We show that bifurcation theory and singularity theory can be used to find period homeostasis in parameterized ODE models. Our results connect the onset of oscillations with homeostasis of the period and yield a geometric description of parameter space that locally organizes the qualitative behavior of solutions of the ODE.
