A study about complex motion caused by hierarchical structure and intermittency, in nonlinear dynamical system with large degree of freedom

大自由度非线性动力系统层次结构和间歇性引起的复杂运动研究

基本信息

批准号：
12834012
负责人：
HATA Hiroki
金额：
$ 1.73万
依托单位：
Kagoshima University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

Various hierarchical structures and motion hide in nonlinear dynamical systems, and so it is an important problem to build new methodology and general idea to understand it. We paid our attention to the hierarchical structure of invariant manifolds in nonlinear dynamical systems with the large degree of freedom as its approach, and understanding the motion theoretically and numerically is our aim of this research.( i ) To understand the complicated behavior of high dimensional chaos, we began studies invariant manifolds and On-off intermittency around them, already. We developed this study. Then we found that the complex behavior of the large degree of freedom chaos (chaotic itinerancy) is wondering motion between invariant manifolds and shown that we get clear characterization by statistical coarse-graining,( ii ) The properties of On-off intermittency that is the base of ( i ) had been argued with perturbation theory. We got new knowledge by treating them by non-preservative methods. … More ( iii ) Many non-chaotic attractors (fixed points, periodic orbits, and quasi-periodic orbits) can exist in a large degree of freedom (the number of attractor suddenly increases with increase of system size). We studied the properties from a viewpoint of response for noise in particular and was found that enough small noise tied attractor and caused intermittent phenomena. In addition, we found what we could describe the motion as anomalous diffusion in macro-quantities of system.( iv ) We studied also relation intermittency of chaos in conservative dynamical systems and low dimensional invariant structure.Next to a study about the behavior of dynamical systems mentioned above, we can develop approach for more complicated behavior. We started (a) a characterization of complicated behavior in generalized shift maps by anomalous diffusion, (b) an analysis of EEG pattern by time-series' entropy, (c) an analysis of intermittency of explosions of SAKURAJIMA volcano and (d) a study for the complicated behavior of dynamical systems with the hierarchical structure which seemed to have invariant manifold in an invariant manifold. Less

非线性动力系统中隐藏着各种层次结构和运动，因此建立新的方法和总体思路来理解它是一个重要的问题，我们关注大自由度非线性动力系统中不变流形的层次结构。作为其方法，从理论上和数值上理解运动是我们这项研究的目的。（i）为了理解高维混沌的复杂行为，我们已经开始研究不变流形及其周围的开关间歇性。然后我们发现大自由度混沌（混沌旅行）的复杂行为是不变流形之间的奇怪运动，并表明我们通过统计粗粒度得到了清晰的表征，（ii）开关的性质。作为 ( i ) 基础的间歇性已经通过微扰理论进行了争论，我们通过非保守方法处理它们获得了新的知识……更多 ( iii ) 许多非混沌吸引子（不动点，周期轨道和准周期轨道）可以以很大的自由度存在（吸引子的数量随着系统尺寸的增加而突然增加），我们特别从噪声响应的角度研究了这些特性，发现足够小。此外，我们发现了可以将运动描述为系统宏观量中的反常扩散。（iv）我们还研究了保守动力系统中混沌的间歇性与低维的关系。不变结构。接下来研究上述动力系统的行为，我们可以开发更复杂行为的方法，我们开始（a）通过反常扩散来表征广义转移图中的复杂行为，（b）脑电图分析。时间序列熵的模式，（c）对樱岛火山爆发的间歇性分析以及（d）对具有似乎具有不变性的分层结构的动力系统的复杂行为的研究不变流形中的流形。