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Synthesis of Symmetrical Nonlinear-Circuits with Finite Group Representaion

有限群表示的对称非线性电路的综合

基本信息

批准号：
11650383
负责人：
KAWAKAMI Hiroshi
金额：
$ 1.79万
依托单位：
The University of Tokushima
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

We considered possible combination of periodic solutions and their bifurcations of given nonlinear dynamical system for topologically classification. With symmetry point of view, corresponding finite group representations and some mathematical results are obtained. We would like to report and publish about these results after finishing preparation. Concretely we applied these mathematical results to neural networks and electric circuits. Firstly we reinvestigated symmetrically coupled BVP oscillators. The coupled oscillators can be chaotic under a reasonable assumption ; each oscillator has different rhythm. We clarified bifurcation diagrams in detail. All phenomena have been confirmed in laboratory experiments.Next, we investigated BVP oscillators synaptically coupled by an alpha-function to simulate delayed coupling system. Since generated periodic solutions can be differentiable, the Poincare mapping is well-defined. Thus bifurcation phenomena and chaos are numerically calculated. Even though the BVP osillator is introduced as a firing model of Hodgkin-Huxley dynamics, qualified analogy between this synaptically coupled system and HH equation have never treated before. We found out there exist similar bifurcation structures between them. Also possible periodic Solutions and non-periodic solutions are classified and enumerated by using finite group representations, then some synchronization phenomena and the state transition of bursting responses are clarified.We are also developing intensively some mathematical tools for interrupted dynamical systems since we found out that the switching point of the periodic orbit can be transferred into one of periodic point of the Poincare mapping.

考虑了给定的非线性动力系统的周期解及其分支的可能组合，并进行了拓扑分类。利用对称性的观点，得到了相应的有限群表示和一些数学结果。准备完毕后，我们将报告并发布这些结果。具体地说，我们将这些数学结果应用于神经网络和电路。首先，我们重新研究对称耦合BVP振子。在合理的假设下，耦合振子可以是混沌的，每个振子有不同的节奏。我们详细阐明了分叉图。这些现象都在实验室实验中得到了证实。接下来，我们研究了用α函数突触耦合的BVP振子来模拟延迟耦合系统。由于生成的周期解可以是可微的，所以Poincare映射是明确定义的。从而对分岔现象和混沌现象进行了数值计算。尽管BVP振荡器是作为Hodgkin-Huxley动力学的一个放电模型引入的，但这个突触耦合系统与HH方程之间的合格类比以前从未处理过。我们发现它们之间存在着相似的分叉结构。利用有限群表示对可能的周期解和非周期解进行了分类和计数，阐明了一些同步现象和突发响应的状态转移，并且由于发现周期轨道的切换点可以转化为Poincare映射的周期点，我们也在积极发展一些研究间断动力系统的数学工具。