Models for Rhythmic Phenomena for Living Bodies and Their Applications

即使不考虑24小时的环境变化，睡眠-觉醒节律和体温的周期性变化也存在接近24小时的周期，这表明这些周期性变化是由某种内生机制产生的。在生物中观察到的这些周期性变化称为昼夜节律。为了帮助发现昼夜节律的机制，我们建立了数学模型并进行了分析，并用数值分析验证了分析结果。除了昼夜节律外，一些糖酵解中间产物的振荡也是在生物中观察到的其他周期性现象。生化振荡周期从几分钟到10分钟不等，昼夜节律周期长达100倍以上。尽管有很大的不同，但人们不能否认昼夜节律和生化反应之间的某种联系，因为昼夜节律存在于各种生物中，因为生化反应是生物获得能量所必需的。在这项研究中，我们假设生物的一部分被另一部分包围。在它们之间没有物质传输的情况下，内部物质浓度的变化由一个非线性振子来控制。然后，我们假设物质在两个部分之间有运输。该假设导致了一个系统，其中控制内部浓度的系统和控制外部浓度的系统耦合在一起。假设非振子具有一定的特性，我们证明了耦合系统的周期解是存在的。此外，我们还对系统进行了分析，找出了由较短周期的振荡产生长周期振荡的条件。对电路模型和生化反应模型进行了数值分析，并对分析结果进行了验证。

Sleep-wake rhythm and periodic body temperature change persist with period close to 24-hour even in isolation from 24-hour environmental change, which shows that these periodic changes are generated by some endogenous mechanism. These periodic changes observed in living things are called circadian rhythms. In order to help finding the mechanism of circadian rhythms, we set up mathematical models, analyze them, and verify analytical results with numerical analysis. Besides circadian rhythms, oscillations of some glycolytic intermediates are other periodic phenomena observed in living things. The period of biochemical oscillations ranges a few minutes to 10 minutes, and the period of circadian rhythms are more than 100 times as long. In spite of the significant difference, one can not deny some sort of relation between circadian rhythms and biochemical reactions, because circadian rhythms are observed in variety of living things, and because biochemical reactions are essential for living things to obtain energy. In this study, we suppose that a part of a living thing is surrounded by another part. In case there is no transport of substances between them, the change of concentrations of substances in the inner part is governed by a nonlinear oscillator. Then we assume that there is transport of the substances between the parts. The assumption leads to a system in which a system governing the concentrations in the inner part and a system governing the concentrations in the outer part are coupled. Assuming that a non oscillator has some characteristics, we show that periodic solutions of the coupled system can exist. Moreover, we analyze the system to find conditions under which oscillations of long period can be generated from oscillations of relatively short period. We analyze a model for a electrical circuit anda model for a biochemical reactions, numerically, and verify the results of analysis.