Analysis of Coupled Oscillator Networks and Application to Chemical Reaction Oscillations and Biological Rhythms

化学反应的宏观行为、觉醒-睡眠周期的昼夜节律以及神经网络的振荡模式通常由非线性动力系统来描述。在本研究中，我们试图分析来自不同科学领域的数学模型，并试图统一这些模型的定性行为，如同步、分叉现象、多相振荡和混沌振荡。分析了由盐水振子和BZ反应产生的耦合振子的数学模型。我们发现，不同的耦合会导致不同类型的多相振荡，如同相、反相或三相振荡。运用分叉理论进行更详细的分析，留待以后的研究问题。利用分叉理论分析了两种不同的人体昼夜节律模型。我们发现了许多不同类型的同步。我们还分析了一些对应于准周期态的非同步态，并与Kronauer等人的结果进行了比较。这些结果也与Daan等人提出的弛豫振子进行了比较。从分支的角度研究了线性耦合和连接耦合的神经网络的振动性。我们发现一些振荡图案正在沿线和环连接传播。对于这一新现象，将通过改变振荡器和连接的类型来进行进一步的研究。这本小册子是出版的，其中收集了主要成果。

Macroscopic behavior of chemical reactions, circadian rhythms of wake-sleep cycles, and oscillation patterns of neural networks is generally described by nonlinear dynamical systems. In this research we tried to analyze the mathematical models derived from these different fields of science and try to unify the qualitative behavior of these models, such as synchronization, bifurcation phenomena, multi-phase oscillations, and chaotic oscillations.1. Mathematical models of coupled oscillators derived from salt-water oscillators and BZ reactions are analyzed. We found that the different couplings cause different types of multi-phase oscillations, such as in-phase, anti-phase, or tri-phase oscillations. More detailed analysis by using the bifurcation theory is left to the future research problems.2. Two different kinds of models to human circadian rhythems are analyzed by using bifurcation theory. We found many different types od synchronizations. We analyzed also some non-synchronized states corresponding to quasi-periodic states which are compared with the results obtained by Kronauer et al. These results are also compared with the relaxation oscillator proposed by Daan et al.3. Oscillations of neural networks coupled as linear and connections are investigated from bifurcational point of view. We found some oscillation patterns are propagating along the lines and ring connection. For this new phenomenon further studies will be expected by changing the type of oscillators and connections.4. The booklet is published in which main results are collected together.